# A thorough introduction to non-relativistic matrix mechanics in multi-qudit
systems with a study on quantum entanglement and quantum quantifiers

Lucas Camponogara Viera  Institute of Electro-Optical Engineering, National
Taiwan Normal University, Taipei 116, Taiwan<EMAIL_ADDRESS>https://camponogaraviera.github.io/homepage/ Shu-Hsien Liao  Institute of
Electro-Optical Engineering, National Taiwan Normal University, Taipei 116,
Taiwan

###### Abstract

Quantum computing is among the most far-reaching technologies of the 21st
century, tackling challenges at the cutting edge of physics. This new paradigm
in computer science harnesses quantum entanglement, one striking non-intuitive
feature of quantum mechanics and a cornerstone of quantum information, to
provide computation with a quantum speed-up over the best-known classical
algorithms and to enable encrypted data communication against eavesdropping.
The bulk of this article is focused on providing a deep and abiding
understanding of non-relativistic matrix mechanics by demonstrating the
fundamental mathematical identities of the contemporary postulatory approach
of quantum mechanics within the state vector and density operator formalism in
multipartite systems. In addition to that, we derive and analyze the
respective 1-qubit, 1-qutrit, 2-qubit, and 2-qudit coherent and incoherent
density operators using Bloch’s parametrization for generalized
$d$-dimensional $N$-qudit states embedded in the $SU(d)$ Lie group with
associate generalized Gell Mann’s matrices spanning the $\mathfrak{su}(d)$ Lie
algebra. We also address the fundamental concepts of quantum nondemolition
measurements, quantum decoherence and, particularly, quantum entanglement
providing for the latter a systematic view on its historical development and
mathematical description in multipartite systems. We conclude our review by
introducing some of the ubiquitous quantum quantifiers required to measure
degrees of quantum entanglement and quantum coherence, deriving the $p$-norm
quantum coherence measure for a 1-qubit state.

##  1 Introduction

The historical development of the so-called old quantum mechanics broadly
stems from 1900 with the seminal work of Max Planck [1] on the correction of
the Rayleigh-Jeans law regarding the ultraviolet catastrophe. Max Planck
proposed that the electromagnetic spectrum of black bodies occur according to
the emission and absorption of discreet quantities of energy, called quanta,
whose separation is proportional to the frequency $\nu$ of the radiation and
to a quantum of action $h$ commonly known as Planck constant. Building on
Planck’s realization, Albert Einstein explained the photoelectric effect,
proposing that light waves are quantized by the same amount $h\nu$ carried by
a discreet pack of energy termed a photon. The field advanced with a series of
atomic models by Ernest Rutherford [2], Niels Bohr [3][4], and Arnold
Sommerfeld [5]. The first attempt to craft a self-consistent theory of quantum
mechanics emerged with the Bohr-Sommerfeld model whose goal was to describe
the physics of atomic data from spectroscopy. The data was associated with
electronic transitions defined as discontinuous and unpredictable quantum
jumps in the atom since its proposal by Niels Bohr in 1913 [3] and backed by
the first experimental observations in 1986 [6]. Bohr’s 1913 model was able to
explain the Balmer series in the hydrogen spectral lines. The 1916 Sommerfeld
model [5], a special-relativistic extension of the aforementioned Bohr’s
planetary model of the atom, correctly described the Stark effect and the fine
structure in the emission spectrum of the hydrogen atom. In spite of its
theoretical success, the model was inconsistent and unable to accurately
explain the “anomalous” Zeeman effect, the case where there is a non-zero net
spin, for spin had not yet been discovered.

The new quantum mechanics began to soar with the contributions of Werner
Heisenberg, Max Born, Pascual Jordan, and Erwin Schrödinger [7]. Although the
term “quantum mechanics” first appeared in Born’s 1924 publication [8], the
leapfrog from the classical to the quantum world only happened with
Heisenberg’s 1925 breakthrough publication [9] entitled “quantum-theoretical
reinterpretation of kinematic and mechanical relations”. Heisenberg’s pivotal
idea was to promote the classical physical observables, such as the position
coordinate, to analog quantum-theoretical quantities named quantum
observables. The realization that Heisenberg’s rule for multiplying the
aforementioned quantities was equivalent to a matrix multiplication led to the
first formulation of a consistent theory of quantum mechanics by Max Born and
Pascual Jordan [10]. The Born-Jordan paper [10] laid a set of postulates, such
as the commutation law, and proofs of conservation theorems, while considering
quantum observables as Hermitian operators. The sequel “Zur Quantenmechanik
II” (On quantum mechanics II) by Born, Heisenberg, and Jordan [11] matured
Heisenberg’s idea into the first complete formalism of the quantum mechanical
theory now regarded as matrix mechanics. During the same year of 1926, Erwin
Schrödinger [12] presented his wave theory formalism describing the electron
in the atom as an oscillating cloud of probability evolving continuously in
space and time according to a second-order differential wave equation. Max
Born then proposed that the square of the absolute value of the complex-valued
wave-function should represent the probability density associated with a
measurement outcome, thus providing a consistent statistical interpretation of
quantum mechanics. Being non-relativistic by nature, a feature easily
confirmed by the different order of its partial derivatives in space and time,
Schrödinger’s wave equation provided an analytical solution for the non-
relativistic hydrogen atom, and a general form to describe atoms with more
than one electron by simply updating its Hamiltonian. However, the
shortcomings of spin shared by both models suggested a revamp. Paul Dirac’s
1928 relativistic equation [13] naturally yields the spin quantum number,
rather than imposing one in an ad hoc fashion, and predicted the existence of
antimatter. A year earlier, Heisenberg’s uncertainty principle [14] stated
that conjugate quantum observables have an intrinsic non-simultaneous
characteristic: a feature often regarded as the hallmark of the quantum
mechanical theory for which quantum mechanics strongly differs from classical
mechanics.

Then, in 1935, Albert Einstein, Boris Podolsky and Nathan Rosen published an
article [15] describing a thought experiment known today as the EPR paradox.
The EPR paper claimed that quantum mechanics was an incomplete theory of
nature that should be framed as a deterministic rather than a probabilistic
model. To assess the existence of such a deterministic framework, John Bell in
1964 [16] proposed a local hidden variable model (LHVM) of quantum mechanics
and showed that the expectation value of certain observables, if they were to
obey a LHVM, should satisfy some sort of correlation inequality. One such Bell
inequality is the ubiquitous CHSH inequality [17] derived by Clauser, Horne,
Shimony, and Holt. The fact that some entangled systems violate Bell’s
inequalities confirmed the incompatibility of quantum mechanics with a LHVM, a
result backed by the first loophole-free experiment realized only eighty years
later in 2015 [18]. Furthermore, the violation of Bell’s inequalities has
shown that either locality or realism known together as local-realism [19][20]
is an incorrect assumption about quantum mechanics, and one must be adopted in
place of the other.

It was only through the intertwined history of quantum mechanics that quantum
information science (QIS) was allowed to thrive. QIS is an interdisciplinary
field that extends the classical theory of information to explain quantum
phenomena. The primary concern regarding the transmission of classical
information was how to circumvent the loss of information over noisy
communication channels. In 1948, Claude Shannon [21] published two remarkable
theorems. The first, Shannon’s noiseless channel coding theorem, quantifies
the physical resources required to store information. The second, Shannon’s
noisy channel coding theorem, provided an upper bound on how much classical
information can be protected from noise using an error-correcting code. A
quantum analog to Shannon’s first theorem appeared in 1995 in the work of
Benjamin Schumacher [22], who also coined the name “qubit”. The first
successful protocol in QIS, named superdense coding, was devised in 1992 by
Charles Bennett and Stephen Wiesner [23] with experimental realization in 1996
[24]. The vanilla version of the protocol uses a single qubit (particle or
artificial atom) of an entangled qubit pair to realize the transmission of two
bits of classical information through a quantum channel. Prior to that was the
recognition that quantum mechanics could be used for encrypted communication
without the possibility of eavesdropping. This procedure, namely quantum
cryptography or quantum key distribution (QKD), strongly relies on the
property known as quantum entanglement. The first consistent quantum
cryptographic protocol, the so-called BB84 QKD protocol, was devised by
Charles Bennett and Gilles Brassard [25] in 1984.

Quantum mechanics truly took a leap towards quantum computing (QC) between
1980 and 1982, when Paul Benioff [26][27] spun the idea of a reversible
Turing-based machine [28] operating according to the principles of quantum
mechanics. At about the same time, in 1982, Richard Feynman [29], realizing
the difficulties in simulating quantum systems on classical computers,
proposed a universal quantum simulator. David Deutsch [30] carried out on the
previous contributions and in 1985 claimed that quantum computers could
outstrip classical computers by efficiently solving certain oracle problems
believed to have no efficient solution on a classical Turing machine. The
first quantum algorithms to show a quantum speed-up over their classical
counterparts were devised by Peter Shor [31] in 1994 and Lov Grover [32] in
1996. Shor’s algorithm is based on the quantum Fourier transform for finding
the prime factors of an integer and solving the discrete logarithm problem
with an exponential speedup. While Grover’s algorithm performs a quantum
searching database with a quadratic speedup in time.

The QIS community laid the primary ideas [29][30] and requirements [33] for
the universal QC platform. To date, the Josephson-based superconducting
quantum processor architecture is considered to be the best candidate for a
large-scale ($N$-qudits), fault-tolerant [34][35][36], and error-corrected
[37] QC technology operating within the framework of circuit quantum
electrodynamics (QED) [38]. In the rapidly growing field of circuit QED, Minev
et. al [39] in 2019 experimentally demonstrated a continuous, coherent, and
deterministic evolution in the complete transition from the ground state to an
excited state of a superconducting artificial three-level atom (a.k.a qutrit):
a remarkable feat that runs counter the original beliefs of discontinuous and
unpredictable atomic transitions proposed by Bohr [3]. In this sense, the
authors demonstrated the possibility to predict at which times a transition is
about to occur, and reverse it before it happens: a potentially invaluable
tool in quantum error correction schemes that can be used to prevent bit-flip
errors during gate operations. The experiment also provided a strong support
to the modern quantum trajectory theory [40], which describes the trajectories
of individual particles in open quantum systems, i.e, in systems interacting
with their environment.

Along with Heisenberg’s uncertainty principle, Bohr’s complementarity
principle [41] provides another important foundation of quantum mechanics, one
whose quantitatively formulation [42][43] open venues for the quantification
of the predictability, a feature regarded as a quantum resource just like
quantum coherence and quantum entanglement are quantum properties considered
as resources for certain tasks in the field of QIS and QC. The purpose of this
article is to provide a thorough introduction to the mathematical formalism
used in QIS by demonstrating its usage through an example-oriented approach in
various quantum information tasks.

The remainder of this article is structured as follows. In section 2 we
provide a quick review on linear algebra. In section 3 we address the
Heisenberg uncertainty principle, the postulates of quantum mechanics, and the
characteristic of quantum nondemolition measurements. In section 4 we
introduce the mathematical formalism of coherent quantum superposition within
the state vector formalism and derive the Bloch’s sphere representation for
1-qubit states. Particular attention is given to the historical development of
quantum entanglement as we address the EPR hypothesis and demonstrate the
violation of a particular Bell inequality known as the CHSH inequality. In
section 5 we frame the postulates of quantum mechanics within the density
operator formalism and derive the criterion to distinguish pure and mixed
density operators. In section 6 we derive the mathematical relations for
measurements in the Hilbert space of bipartite systems with the ubiquitous
partial trace function for density operators. In section 7 we describe the
phenomenon of quantum decoherence. In section 8 we derive particular coherent
and incoherent density operators from the generalized $N$-qudit density matrix
defined by Bloch’s parametrization. In section 9, the Schmidt decomposition
and the Peres-Horodecki criterion are outlined as two important tests used to
discriminate between separable and entangled bipartite density operators. In
section 10, we give the definition of a quantum entanglement quantifier and
apply some of the ubiquitous entanglement measures to quantify the degree of
entanglement in particular multipartite systems. In section 11 we introduce
the requirements for a quantum coherence quantifier and derive the $p$-norm
quantum coherence measure for a 1-qubit state. Finally, in section 12 we
provide a worked example of a quantum entanglement application known as
quantum teleportation. We demonstrate perfect quantum teleportation with
maximally entangled pure states and briefly discuss the necessity for
teleportation with nonmaximally multipartite entangled states.

##  2 Linear algebra: a primer

The mathematical framework employed in the remainder of this article is the
usual matrix mechanics formalism of quantum mechanics initially developed by
Werner Heisenberg, Max Born, and Pascual Jordan in a sequel of articles
[9][10][11] between 1925 and 1926. Within this formalism, the physical
observables (e.g., spin, electric charge, magnetic flux, position, momentum,
etc.) are represented by self-adjoint operators as the central pivots of the
theory. Therefore, to master the intricacies of the quantum information theory
and the way quantum phenomena are hemmed in, a solid background knowledge is
required. This section provides a skill-reaffirming background on elementary
linear algebra [44] in the ubiquitous bra-ket notation of Paul Dirac.

###  2.1 Hilbert space

The equivalence between wave mechanics and matrix mechanics is primarily
attributed to John Von Neumann [45], who in 1932 proved them to be different
realizations of an abstract Hilbert space. Let
$\\{|\lambda_{j}\rangle\\}_{j=1}^{d}$ denote any set of $d$ linearly
independent orthonormal column vectors $|\lambda_{j}\rangle$ in a
$d$-dimensional complex vector space $V^{d}$ defined in some finite
dimensional inner product space $\mathcal{H}$, known as Hilbert space, over
the field (set) $\mathbb{C}$ of the complex numbers $\lambda_{j}\in\mathbb{C}$
(voiced $\lambda_{j}$ is an element of $\mathbb{C}$). The number of linearly
independent vectors in the set defines the dimension $d$ of the space.
Moreover, linear independence states that no vector in the set can be given as
a linear combination of the remaining ones and, therefore, the set form a
basis for the space $V^{d}$. As a consequence, any arbitrary state vector
(a.k.a ket-vector or column vector) $|\psi\rangle\in V^{d}$ can be represented
as a linear combination of the vectors of the basis set according to:

$\displaystyle|\psi\rangle\doteq\sum_{j=1}^{d}\lambda_{j}|\lambda_{j}\rangle=\begin{pmatrix}\lambda_{1}\\\
\lambda_{2}\\\ \vdots\\\ \lambda_{d}\end{pmatrix}.$ (1)

And the transposed conjugate state vector also known as bra-vector or row
vector is defined as:

$\displaystyle\langle\psi|$ $\displaystyle=$
$\displaystyle(|\psi\rangle)^{\dagger}=(|\psi\rangle^{T})^{*}$ (2)
$\displaystyle=$
$\displaystyle\left(\sum_{j=1}^{d}\lambda_{j}|\lambda_{j}\rangle\right)^{\dagger}=\sum_{j=1}^{d}\lambda_{j}^{*}\langle\lambda_{j}|$
(3) $\displaystyle=$
$\displaystyle\begin{pmatrix}\lambda_{1}^{*}&\lambda_{2}^{*}&\cdots&\lambda_{d}^{*}\end{pmatrix}.$
(4)

The transpose conjugate (a.k.a Hermitian conjugate, self-adjoint or dagger)
operation has the following property:

$\displaystyle(A+B)^{\dagger}=A^{\dagger}+B^{\dagger}.$ (5)

Here, $|\psi\rangle^{T}$ denotes the transpose of $|\psi\rangle$ swapping rows
for columns. The complex scalar quantity $\lambda_{j}$ is defined in the
standard way $\lambda_{j}\doteq a+bi$ with real scalars $a$ and $b$, and
imaginary number $i\doteq\sqrt{-1}$. Whereas $(*)$ denotes the complex
conjugate, such that $\lambda_{j}^{*}=a-bi$. Orthonormality condition of the
basis vectors entails:

$\displaystyle\langle\lambda_{j}|\lambda_{k}\rangle\doteq\delta_{jk}=\begin{cases}0,&\mbox{if
}j\neq k,\\\ 1,&\mbox{if }j=k,\end{cases}$ (6)

where $\delta_{jk}$ is termed “Kronecker delta”. In this sense, the basis set
$\\{|\lambda_{j}\rangle\\}_{j=1}^{d}$ is termed an orthonormal basis set. In
tensorial notation, the $d$x$d$-dimensional unit matrix (a.k.a identity
matrix) reads:

$\displaystyle\delta_{\mu\nu}=\begin{bmatrix}1&\dots&0\\\
\vdots&\ddots&\vdots\\\ 0&\dots&1\end{bmatrix}.$ (7)

###  2.2 Inner product space

An inner product complex space is a $d$-dimensional complex vector space
$V^{d}$ over the field $\mathbb{C}$ of the complex numbers endowed with an
inner product function defined by a map $(\cdot,\cdot):V^{d}$ x
$V^{d}\rightarrow\mathbb{C}$. The map is voiced “a function $(\cdot,\cdot)$,
namely inner product, takes as input two vectors from a vector space and
produces a scalar quantity, in general a complex number, as output”. The inner
product function then satisfies the following conditions for all vectors
$x,y,z\in V^{d}$ and all scalars $\alpha\in\mathbb{C}$:

  * •

$(x,\alpha y)=\alpha(x,y)$, (Linearity in the second argument).

  * •

$(x,y)=(y,x)^{*}$, (Conjugate symmetry).

  * •

$(x,x)\geq 0$. (Positive semi-definiteness).

Within Dirac’s bra-ket notation, the inner product
$(\cdot,\cdot)\doteq\langle\cdot|\cdot\rangle$ between vectors
$|q\rangle=\sum_{i}q_{k}|k\rangle$ and
$|\phi\rangle=\sum_{j}\phi_{j}|j\rangle$ is defined as:

$\displaystyle(|q\rangle,|\phi\rangle)$ $\displaystyle\doteq$
$\displaystyle|q\rangle^{\dagger}|\phi\rangle=\langle q|\phi\rangle$ (8)
$\displaystyle=$
$\displaystyle\left(\sum_{k=1}^{d}q_{k}|k\rangle\right)^{\dagger}\left(\sum_{j=1}^{d}\phi_{j}|j\rangle^{\dagger}\right)$
(9) $\displaystyle=$
$\displaystyle\sum_{k=1}^{d}\sum_{j=1}^{d}q_{k}^{*}\phi_{j}\langle
k|j\rangle)$ (10) $\displaystyle=$
$\displaystyle\sum_{k=1}^{d}\sum_{j=1}^{d}q_{k}^{*}\phi_{j}\delta_{kj}$ (11)
$\displaystyle=$ $\displaystyle\sum_{j=1}^{d}q_{j}^{*}\phi_{j}$ (12)
$\displaystyle=$ $\displaystyle
q_{1}^{*}\phi_{1}+q_{2}^{*}\phi_{2}+\cdots+q_{d}^{*}\phi_{d}$ (13)
$\displaystyle=$
$\displaystyle\begin{pmatrix}q_{1}^{*}&q_{2}^{*}&\cdots&q_{d}^{*}\end{pmatrix}\begin{pmatrix}\phi_{1}\\\
\phi_{2}\\\ \vdots\\\ \phi_{d}\end{pmatrix}.$ (14)

And the corresponding outer product in the space $V^{dxd}$ is defined as:

$\displaystyle|\phi\rangle|q\rangle^{\dagger}$ $\displaystyle:=$
$\displaystyle|\phi\rangle\langle q|$ (15) $\displaystyle=$
$\displaystyle\sum_{j=1}^{d}\sum_{k=1}^{d}\phi_{j}q_{k}^{*}|j\rangle\langle
k|$ (16) $\displaystyle=$
$\displaystyle\begin{bmatrix}\phi_{1}q_{1}^{*}&\dots&\phi_{1}q_{d}^{*}\\\
\vdots&\ddots&\vdots\\\
\phi_{d}q_{1}^{*}&\dots&\phi_{d}q_{d}^{*}\end{bmatrix}.$ (17)

###  2.3 Completeness relation

Suppose {|$j\rangle\\}_{j=1}^{d}$ is any orthonormal basis set for a
$d$-dimensional vector space $V^{d}$ with an arbitrary vector
$|v\rangle=\sum_{j=1}^{d}v_{j}|j\rangle$, then

$\displaystyle\left(\sum_{j=1}^{d}|j\rangle\langle j|\right)|v\rangle$
$\displaystyle=$ $\displaystyle\sum_{j=1}^{d}|j\rangle\langle j|v\rangle$ (18)
$\displaystyle=$ $\displaystyle\sum_{j=1}^{d}v_{j}|j\rangle=|v\rangle.$ (19)

And, therefore,

$\displaystyle\sum_{j=1}^{d}|j\rangle\langle j|={\hbox{1\kern-1.49994pt\vrule
width=0.8pt,depth=-0.5pt}}_{d},$ (20)

and known as the completeness relation.

###  2.4 Linear operator

Suppose $\\{|v_{j}\rangle\\}_{j=1}^{d_{a}}$ and
$\\{|w_{k}\rangle\\}_{k=1}^{d_{b}}$ are any orthonormal basis set for vector
spaces $V^{d_{a}}$ and $W^{d_{b}}$, respectively. A linear operator is any
function $\hat{O}$, with a map $\hat{O}$ : $V^{d_{a}}$ $\rightarrow$
$W^{d_{b}}$, that is linear in its inputs:

$\displaystyle\hat{O}\left(\sum_{j=1}^{d_{a}}v_{j}|v_{j}\rangle\right)=\sum_{j=1}^{d_{a}}v_{j}\hat{O}(|v_{j}\rangle).$
(21)

The map is voiced “a function $\hat{O}$, namely linear operator, acts on a
vector $V^{d_{a}}$ and produces a vector $W^{d_{b}}$”. By definition,

$\displaystyle\hat{O}\doteq\sum_{j=1}^{d_{a}}v_{j}|w_{j}\rangle\langle v_{j}|$
(22)

is a linear operator which, using the completeness relation, yields the
following outer product representation:

$\displaystyle\hat{O}=I_{v}\hat{O}I_{w}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{d_{a}}|v_{j}\rangle\langle
v_{j}|\hat{O}\sum_{k=1}^{d_{b}}|w_{k}\rangle\langle w_{k}|$ (23)
$\displaystyle=$ $\displaystyle\sum_{jk=1}^{d_{a},d_{b}}\langle
v_{j}|\hat{O}|w_{k}\rangle|v_{j}\rangle\langle w_{k}|.$ (24)

###  2.5 Hermitian Operator

Suppose $\hat{H}$ is a linear operator on a Hilbert space $\mathcal{H}$. There
exists a unique linear operator $\hat{H}^{\dagger}\in\mathcal{H}$ such that
for all vectors $|v\rangle,|w\rangle\in\mathcal{H}$:

$\displaystyle(|v\rangle,\hat{H}|w\rangle)=(\hat{H}^{\dagger}|v\rangle,|w\rangle)=\langle
v|\hat{H}|w\rangle.$ (25)

Where the operator satisfying $\hat{H}=\hat{H}^{\dagger}$ is dubbed a
Hermitian operator.

###  2.6 Positive operator

Positive operators are a special subclass of Hermitian operators. A Positive
operator $\hat{J}$ is defined as a linear operator such that, for any vector
$|\psi\rangle$, the inner product $(|\psi\rangle,\hat{J}|\psi\rangle)$ is a
real and non-negative number. If the inner product
$(|\psi\rangle,\hat{J}|\psi\rangle)$ is strictly greater than zero for all
$|\psi\rangle\neq 0$ then $\hat{J}$ is termed a positive-definite operator.
Within Dirac’s bra-ket notation, a positive definite-operator $\hat{J}$ reads

$\displaystyle\langle\psi|\hat{J}|\psi\rangle>0,$ (26)

for all $|\psi\rangle\neq 0$, while a positive-semidefinite operator $\hat{J}$
reads

$\displaystyle\langle\psi|\hat{J}|\psi\rangle\geq 0,$ (27)

for all $|\psi\rangle$.

###  2.7 Unitary operator

A bounded linear operator $\hat{U}$ in the Hilbert space $\mathcal{H}$ with a
map $\hat{U}:\mathcal{H}$ $\rightarrow$ $\mathcal{H}$ is dubbed a Unitary
operator if it satisfies:

$\displaystyle\hat{U}^{\dagger}\hat{U}\text{ (isometry)}$ $\displaystyle=$
$\displaystyle\hat{U}\hat{U}^{\dagger}\text{
(coisometry)}={\hbox{1\kern-1.49994pt\vrule width=0.8pt,depth=-0.5pt}}$
$\displaystyle\implies$ $\displaystyle\hat{U}^{\dagger}=\hat{U}^{-1}\text{
(inverse)}.$

Therefore, a Unitary operator is both an isometry and a coisometry, or,
equivalently, a surjective isometry.

Let $\hat{U}:V^{d}$ $\rightarrow$ $W^{d}$ be a map of a bounded unitary
operator $\hat{U}$, and $\\{|v_{j}\rangle\\}_{j=1}^{d}$ any orthonormal basis
set for the inner product vector space $V^{d}$ with an arbitrary vector
$|v\rangle=\sum_{j=1}^{d}v_{j}|v_{j}\rangle$. Then
$|w_{j}\rangle=\hat{U}|v_{j}\rangle$ forms an orthonormal basis since
$\hat{U}$ preserves the inner product:

$\displaystyle(\hat{U}|v\rangle,\hat{U}|w\rangle)$ $\displaystyle=$
$\displaystyle\langle v|\hat{U}^{\dagger}\hat{U}|w\rangle$ (28)
$\displaystyle=$ $\displaystyle\langle v|I|w\rangle=\langle v|w\rangle.$ (29)

Therefore, one can write the following elegant outer product representation
for any Unitary operator $\hat{U}$:

$\displaystyle\hat{U}=\sum_{j=1}^{d}|w_{j}\rangle\langle v_{j}|,$ (30)

such that

$\displaystyle\hat{U}|v_{j}\rangle=\sum_{j=1}^{d}|w_{j}\rangle\langle
v_{j}|v_{j}\rangle=\sum_{j=1}^{d}|w_{j}\rangle.$ (31)

Moreover, if $\hat{\mathcal{H}}$ $\doteq$ $-ilog(\hat{U})$ is Hermitian for
any Unitary $\hat{U}$, than:

$\displaystyle\hat{U}=e^{i\hat{\mathcal{H}}},$ (32)

for some Hermitian $\hat{\mathcal{H}}$.

###  2.8 Normal operator

Let $\hat{\mathcal{O}}$ be a linear operator on a finite-dimensional inner
product space. Then $\hat{\mathcal{O}}$ is said to be Normal if it satisfies

$\displaystyle\hat{\mathcal{O}}^{\dagger}\hat{\mathcal{O}}=\hat{\mathcal{O}}\hat{\mathcal{O}}^{\dagger}.$
(33)

######  Theorem 1 (Spectral decomposition)

Any Normal operator $\hat{\mathcal{O}}$ on a vector space $V$ is diagonal with
respect to some orthonormal basis for $V$. Conversely, any diagonalizable
operator is normal.

From Theorem (1), any Normal operator $\hat{\mathcal{O}}$ can be diagonalized
as

$\displaystyle\hat{\mathcal{O}}=\sum_{j=1}^{d}o_{j}|o_{j}\rangle\langle
o_{j}|,$ (34)

where $\\{|o_{j}\rangle\\}$ is any orthonormal set of eigenvectors for
$\hat{\mathcal{O}}$ with eigenvalues $o_{j}$. It is straightforward seeing
that Hermitian and Unitary operators are also Normal operators and, therefore,
have a espectral decomposition according to Eq. (34).

###  2.9 Projector operator

Suppose {|$j\rangle\\}_{j=1}^{k}$ is any orthonormal basis set of a
$k$-dimensional vector subspace $W^{k}$ of a $d$-dimensional vector space
$V^{d}$ with the completeness relation set by Eq. (20). The projection of
$V^{d}$ onto the subspace $W^{k}$ is given by the projector operator

$\displaystyle\hat{P}\doteq\sum_{j=1}^{k}|j\rangle\langle j|,$ (35)

It can be shown that such an operator is Hermitian satisfying
$P^{\dagger}P=P^{2}=P$ and, therefore, Normal, hence it can be given a
spectral decomposition according to Eq. (34).

$\displaystyle\hat{P}^{\dagger}$
$\displaystyle=\left(\sum_{j=1}^{k}|j\rangle\langle
j|\right)^{\dagger}=\left(\sum_{j=1}^{k}|1\rangle\langle
1|+\cdot\cdot\cdot+|k\rangle\langle k|\right)^{\dagger}$
$\displaystyle=\left(|1\rangle\langle
1|\right)^{\dagger}+\cdot\cdot\cdot+\left(|k\rangle\langle
k|\right)^{\dagger}=\sum_{j=1}^{k}\left(|j\rangle\langle j|\right)^{\dagger}$
$\displaystyle=\sum_{j=1}^{k}\langle
j|^{\dagger}|j\rangle^{\dagger}=\sum_{j=1}^{k}|j\rangle\langle j|=\hat{P}.$

###  2.10 Eingenvalue-eigenvector equation

A general operator $\hat{\mathcal{O}}$ with any orthonormal basis set
{|$o_{j}\rangle\\}_{j=1}^{d}$ obeys the following eingenvalue-eigenvector
equation:

$\displaystyle\hat{\mathcal{O}}|o_{j}\rangle=o_{j}|o_{j}\rangle\implies(\hat{\mathcal{O}}-o_{j}{\hbox{1\kern-1.49994pt\vrule
width=0.8pt,depth=-0.5pt}}_{d})|o_{j}\rangle=0,$ (36)

meaning that the matrix $(o_{j}{\hbox{1\kern-1.49994pt\vrule
width=0.8pt,depth=-0.5pt}}_{d}-\hat{\mathcal{O}})$ is not invertible, hence
its determinant must be zero. Therefore, the eigenvalues $o_{j}$ of
$\hat{\mathcal{O}}$ are roots of the characteristic polynomial equation

$\displaystyle det(\hat{\mathcal{O}}-o_{j}{\hbox{1\kern-1.49994pt\vrule
width=0.8pt,depth=-0.5pt}}_{d})=0.$ (37)

The fundamental theorem of algebra secures that every polynomial has at least
one complex root, so that every operator $\hat{\mathcal{O}}$ has at least one
eigenvalue $o_{j}$ with a corresponding eigenvector $|o_{j}\rangle$. However,
when the operator has only one eigenvalue a diagonal representation cannot be
given.

It can be shown that all eigenvalues of a Unitary matrix have modulus 1, that
is, they can be written in the form $e^{i\theta}$ for some real number
$\theta$. While the eigenvalues of a Projector operator are either 0 or 1.
Moreover, a Normal matrix is Hermitian if and only if it has real eigenvalues
($o_{j}\in\mathbb{R}$). Whereas the eigenvectors of a Hermitian operator with
different eigenvalues are necessarily orthogonal. Here, we show that the
eigenvalues of a positive operator are all non-negative, as follows:

$\displaystyle\langle\phi|\hat{\mathcal{O}}|\phi\rangle$
$\displaystyle=\sum_{j,k}c_{j}^{*}c_{k}\langle\phi_{j}|\hat{\mathcal{O}}|\phi_{k}\rangle$
(38)
$\displaystyle=\sum_{j,k}c_{j}^{*}c_{k}\langle\phi_{j}|o_{k}|\phi_{k}\rangle$
(39)
$\displaystyle=\sum_{j,k}c_{j}^{*}c_{k}o_{k}\langle\phi_{j}|\phi_{k}\rangle$
(40) $\displaystyle=\sum_{j,k}c_{j}^{*}c_{k}o_{k}\delta_{jk}$ (41)
$\displaystyle=\sum_{j=1}^{2}|c_{j}|^{2}o_{j}\geq 0\implies o_{j}\geq 0.$ (42)

###  2.11 Tensor product and Kronecker product

Suppose $\\{|a_{j}\rangle\\}_{j=1}^{d_{a}}$ and
$\\{|b_{k}\rangle\\}_{k=1}^{d_{b}}$ are any orthonormal basis set for vector
spaces $V^{d_{a}}$ and $W^{d_{b}}$, respectively. Then
$\\{|a_{j}\rangle\otimes|b_{k}\rangle\doteq|a_{j}b_{k}\rangle\\}_{j,k=1}^{d_{a},d_{b}}$
is a basis set for a $d_{a}$x$d_{b}$-dimensional vector space
$V^{d_{a}}\otimes W^{d_{b}}$, whose elements are given by the tensor product
with the property:

$\displaystyle a_{j}|a_{j}\rangle\otimes b_{k}\langle b_{k}|b_{l}\rangle$
$\displaystyle=a_{j}b_{k}(|a_{j}\rangle\otimes\delta_{kl})$ (43)
$\displaystyle=a_{j}b_{l}|a_{j}\rangle.$ (44)

In matrix representation, the abstract tensor product becomes the Kronecker
product. If $A\in M^{m,n}$ and $B\in M^{o,p}$ are $mxn$ and $oxp$ matrices,
respectively, then their Kronecker product yelds the following block matrix:

$\displaystyle(A\otimes B)=\begin{bmatrix}A_{11}B&&\cdots&&A_{1n}B\\\
\vdots&&\ddots&&\vdots\\\ A_{m1}B&&\cdots&&A_{mn}B\end{bmatrix},$

with dimension $(mo)$ x $(np)$. In general, for a complex number
$z\in\mathbb{C}$, matrices $A\in M^{m,n}$, $B\in M^{o,p}$, $C\in M^{q,r}$,
$D\in M^{s,t}$ and unit matrix ${\hbox{1\kern-1.49994pt\vrule
width=0.8pt,depth=-0.5pt}}=\delta_{\mu\nu}\in M^{u,u}$, the following
identities are satisfied:

  1. 1.

$(A\otimes{\hbox{1\kern-1.49994pt\vrule
width=0.8pt,depth=-0.5pt}}+{\hbox{1\kern-1.49994pt\vrule
width=0.8pt,depth=-0.5pt}}\otimes B)\in
M^{mu,nu}=A_{ij}\delta_{\mu\nu}+\delta_{ij}B_{\mu\nu}$.

  2. 2.

$(A\otimes{\hbox{1\kern-1.49994pt\vrule
width=0.8pt,depth=-0.5pt}}+{\hbox{1\kern-1.49994pt\vrule
width=0.8pt,depth=-0.5pt}}\otimes
B)(|u\rangle\otimes|v\rangle)=(A|u\rangle\otimes|v\rangle)+(|u\rangle\otimes
B|v\rangle)$.

  3. 3.

$(A\otimes B)(C\otimes D)=AC\otimes BD$.

  4. 4.

$(A\otimes
B)\sum_{j}(|a_{j}\rangle\otimes|b_{j}\rangle)=\sum_{j}A|a_{j}\rangle\otimes
B|b_{j}\rangle$.

  5. 5.

$(A\otimes B)^{\dagger}=A^{\dagger}\otimes B^{\dagger}$.

  6. 6.

$(A\otimes B)\otimes C=A\otimes(B\otimes C)$.

  7. 7.

$z(A\otimes B)=(zA)\otimes B=A\otimes(zB)$.

  8. 8.

$(A+B)\otimes C=A\otimes C+B\otimes C$.

  9. 9.

$A\otimes(B+C)=A\otimes B+A\otimes C$.

The aforementioned identities can be applied to all operators covered so far,
as they are all described by matrices. Moreover, the Kronecker product of two
Hermitian operators always results in another Hermitian operator. The same
applies to Unitary, Positive, and Projector operators. This contrasts with
matrix multiplication between operators of the same kind, as it does not
always result in an operator of the same kind.

###  2.12 Trace

Let $\hat{O}$ be a linear operator represented in a basis
$|\psi_{j}\rangle\in\mathcal{H}$ of a $d$-dimensional Hilbert space
$\mathcal{H}$. The trace of $\hat{O}$ is a function with a map
$tr:\mathcal{L}(\mathcal{H})\rightarrow\mathbb{C}$ defined as:

$\displaystyle
tr(\hat{O})\doteq\sum_{j=1}^{d}\langle\psi_{j}|\hat{O}|\psi_{j}\rangle,$ (45)

which is the sum of all diagonal elements of the $dxd$ square matrix
representation of $\hat{O}$. Another common definition appearing in the
literature is:

$\displaystyle tr(\hat{O})\doteq\sum_{j=1}^{d}\hat{O}_{jj}.$ (46)

As a word of caution, one should be wary that the trace of a non-square matrix
is undefined. For any arbitrary matrices $A$ and $B$, Unitary matrix $U$,
$d$-dimensional identity matrix ${\hbox{1\kern-1.49994pt\vrule
width=0.8pt,depth=-0.5pt}}_{d}$, and complex number $z$, the following
identities are satisfied:

  1. 1.

$tr({\hbox{1\kern-1.49994pt\vrule width=0.8pt,depth=-0.5pt}}_{d})=d$.

  2. 2.

$tr(z)=z$.

  3. 3.

$tr(A^{\dagger})=(tr(A))^{\dagger}$.

  4. 4.

$tr(A)=tr(A^{T})$.

  5. 5.

$tr(zA)=ztr(A)$.

  6. 6.

$tr(\sum_{j}A_{j})=\sum_{j}tr(A_{j})$.

  7. 7.

$tr(AB)=tr(BA)$.

  8. 8.

$tr(A^{T}B)=tr(BA^{T})=tr(B^{T}A)$.

  9. 9.

$tr(UAU^{\dagger})=tr(UU^{\dagger}A)=tr(A)$.

  10. 10.

$tr(A\otimes B)=tr(B\otimes A)=tr(A)tr(B)$.

  11. 11.

$tr(e^{(A\otimes{\hbox{1\kern-1.04996pt\vrule
width=0.8pt,depth=-0.5pt}}+{\hbox{1\kern-1.04996pt\vrule
width=0.8pt,depth=-0.5pt}}\otimes B)})=tr(e^{A})tr(e^{B})$.

From the above relations, considering a unit state vector $|\psi\rangle$ and
some orthonormal basis set $\\{|o_{j}\rangle\\}_{j=1}^{d}$, one can derive:

  1. 1.

$tr(A|\psi\rangle\langle\psi|)=tr(\langle\psi|A|\psi\rangle)=\langle\psi|A|\psi\rangle$.

  2. 2.

$\sum_{j=1}^{d}tr\left(|\psi_{j}\rangle\langle\psi_{j}|\right)=tr\left(\sum_{j=1}^{d}|\psi_{j}\rangle\langle\psi_{j}|\right)$.

  3. 3.

$tr(|o_{j}\rangle\langle o_{k}|)=tr(\langle o_{k}|o_{j}\rangle)=\langle
o_{k}|o_{j}\rangle=\delta_{kj}$.

  4. 4.

$tr(|o_{j}\rangle\langle o_{j}|)=1.$

##  3 Quantum mechanics road to quantum information

In this section we introduce the quantum mechanical theory that governs the
dynamics of quantized systems, in general. In subsection 3.1 is introduced the
commutation relation and the Heisenberg uncertainty principle. In subsection
3.2 we explore the postulates of quantum mechanics and derive the uncertainty
(a.k.a variance) of a measurement given the frequentist probability
interpretation. And subsection 3.3 address the concept of quantum
nondemolition measurements.

###  3.1 Conjugate observables

To help usher in the field of quantum mechanics, Werner Heisenberg [9], then
Bohr’s assistant in Copenhagen, promoted the classical observable quantities
to analog quantum observables represented by linear operators, such as the
position $x\to\hat{x}$ and momentum $p\to\hat{p}$. Max Born [10] then realized
that Heisenberg’s multiplication rule for quantum observables were equivalent
to that of matrix calculus, and that the observables satisfied a non-
commutative algebra. In the modern formalism of matrix mechanics, the non-zero
commutator of two conjugate observables (a.k.a Fourier transform duals)
represented by quantum operators $\hat{A}$ and $\hat{B}$ is defined as:

$\displaystyle[\hat{A},\hat{B}]\doteq\left(\hat{A}\hat{B}-\hat{B}\hat{A}\right)\neq
0.$ (47)

In contrast to classical theory, in quantum mechanics the commutator can be
order dependent, i.e, operators representing conjugate observables in quantum
mechanics are not defined in the algebra of commutative groups (a.k.a Abelian
group) where the commutator would to be zero (see Theorem (2)). Moreover, for
Hermitian operators, the commutator $[A,B]$ yields an antihermitian operator,
while $i[A,B]$ results in a Hermitian operator.

######  Theorem 2 (Simultaneous diagonalization)

Suppose $A$ and $B$ are two Hermitian operators. Then $[A,B]=AB-BA$ is equal
to the zero matrix (they commute) if and only if there exists an orthonormal
basis such that both $A$ and $B$ are diagonal with respect to that basis, i.e,
they share the same basis. We say that $A$ and $B$ are simultaneously
diagonalizable in this case.

The canonical commutation relation is a hallmark of quantum mechanics and has
been widely used to classify the different types of superconducting circuit-
based qubits [46] used in superconducting quantum computing [33][47] within
the framework of circuit quantum electrodynamics [38][48]. In particular, for
conjugate observables such as electric charge $Q\to\hat{Q}$ and magnetic flux
$\Phi\to\hat{\Phi}$, the commutation relation yields the same result as the
prime example of position and momentum:

$\displaystyle[\hat{\Phi},\hat{Q}]=\left(\hat{\Phi}\hat{Q}-\hat{Q}\hat{\Phi}\right)=i\hbar{\hbox{1\kern-1.49994pt\vrule
width=0.8pt,depth=-0.5pt}}_{d}.$ (48)

Here, $\hbar=\frac{h}{2\pi}$ denotes the reduced Planck constant (a.k.a
quantum of action), whereas ${\hbox{1\kern-1.49994pt\vrule
width=0.8pt,depth=-0.5pt}}_{d}$ denotes the $d$-dimensional identity matrix.
This result, known as the canonical commutation relation, follows immediately
from Heisenberg’s uncertainty principle.

Heisenberg’s uncertainty principle is a consequence of one striking aspect
that sets havoc on classical intuition: the intrinsic non-simultaneous nature
of conjugate observables. This feature, first described by Werner Heisenberg
[14] in the form of epistemological uncertainty relations, states that the
expectation values of two conjugate observables cannot be simultaneously
measured within the same measurement accuracy/precision, but only up to some
characteristic inaccuracy. This is often regarded as the most distinctive
feature between classical and quantum mechanics. Heisenberg’s prime example
considered the measurement of the position and momentum of an electron by a
microscope. Historically, the uncertainty relation has been extended to other
conjugate observables, and the modern version of the principle is regarded as
the Robertson uncertainty relation [49][50] stated as the following general
inequality for observables $\hat{A}$ and $\hat{B}$:

$\displaystyle\sigma_{\hat{A}}\sigma_{\hat{B}}\geq\frac{|\langle\psi|[A,B]|\psi\rangle|}{2},$
(49)

where the $\sigma$’s are the corresponding standard deviations of each
observable according to Eq. (100).

###  3.2 The fundamental postulates of quantum mechanics

Influenced by Georg Cantor’s set theory, German mathematician David Hilbert’s
perspective on the scientific world is rooted in the axiomatic approach as the
basis for any scientific theory to be developed independently of the need for
intuition and free from arbitrariness. This realization entails the
requirement of a formal and rigorous logical system of mathematical proof
based on set theory, in which theorems are derived from a set of axioms via
rules of inference conveyed with a symbolic logical language. A formal
symbolic logical system of this fashion emerged between 1910-1913 with
Bertrand Russell and Alfred Whitehead three-volume “Principia Mathematica”
[51]. In physics, however, Axioms are called Postulates and in the same ways
are starting points for any physical theory.

Historically, the postulates of quantum mechanics in matrix notation were
first devised by Born and Jordan in 1925 [10]. The original Born–Jordan
postulates, as they are commonly known, were devised in close analogy with
classical mechanics within a framework where the physical observables are
Hermitian operators satisfying a non-commutative algebra (see subsection 3.1).
Nielsen and Chuang [44] provide a systematic view of the contemporary
postulatory approach for closed systems within the matrix mechanics formalism,
and we now extend.

####  3.2.1 Postulate 1 (State space)

Any isolated (close) physical system is embedded in a complex $d$-dimensional
Hilbert space $\mathcal{H}^{d}$ known as the state space of the system endowed
with an inner product $(\cdot,\cdot)\doteq\langle\cdot|\cdot\rangle$. The
system is completely described by a normalized state vector
$|\psi\rangle\in\mathcal{H}^{d}$ which is a unit vector in the system’s state
space. In the orthonormal basis set $\\{|o_{j}\rangle\\}_{j=1}^{d}$, the state
vector reads

$\displaystyle|\psi\rangle\doteq\sum_{j=1}^{d}c_{j}|o_{j}\rangle,$ (50)

where $c_{j}=\langle o_{j}|\psi\rangle$ denotes the probability amplitude
associated with the preparation of an eigenstate $|o_{j}\rangle$. Normality
condition of $|\psi\rangle$ entails

$\displaystyle\langle\psi|\psi\rangle$ $\displaystyle=$
$\displaystyle\left(\sum_{j=1}^{d}c_{j}^{*}\langle
o_{j}|\right)\left(\sum_{k=1}^{d}c_{k}|o_{k}\rangle\right)$ (51)
$\displaystyle=$
$\displaystyle\sum_{j=1}^{d}\sum_{k=1}^{d}c_{j}^{*}c_{k}\langle
o_{j}|o_{k}\rangle$ (52) $\displaystyle=$
$\displaystyle\sum_{j=1}^{d}\sum_{k=1}^{d}c_{j}^{*}c_{k}\delta_{jk}$ (53)
$\displaystyle=$ $\displaystyle\sum_{j=1}^{d}c_{j}^{*}c_{j}$ (54)
$\displaystyle=$ $\displaystyle\sum_{j=1}^{d}|c_{j}|^{2}=1.$ (55)

The state vector is often referred to as a pure state to distinguish it from a
density operator (see section 5).

####  3.2.2 Postulate 2 (Evolution)

A closed quantum system in the initial $|\psi_{0}\rangle$ evolves, within the
Schrödinger picture formalism, according to a unitary transformation, such
that the state of the system after a time $t$ shall be
$|\psi_{t}\rangle=\hat{U}_{t}|\psi_{0}\rangle$ given by the action of the
propagator $\hat{U}_{t}$, which is a Unitary operator. Solution for
$\hat{U}_{t}$ comes from the general time-dependent Schrödinger equation of
the time evolution operator:

$i\hbar\frac{\partial\hat{U}_{t}}{\partial t}=\hat{H}_{t}\hat{U}_{t},$ (56)

with a time-dependent Hamiltonian operator $\hat{H}_{t}$ describing the total
energy of the system, i.e, the sum of all kinetic and potential energies. For
the particular case of closed systems, the Hamiltonian operator is Hermitian.
However, open systems (systems interacting with its surroundings), in general,
do not allow Hermitian Hamiltonians.

Solution for equation Eq. (56) depends mostly on the characteristic of the
Hamiltonian. We shall consider three particular cases:

  1. 1.

Time-independent Hamiltonian. It is possible to construct any Unitary operator
by means of a Hermitian observable according to Eq. (32). In this sense, for a
time-independent Hermitian Hamiltonian $\hat{H}$, the propagator now writes:

$\hat{U_{t}}=e^{-i\hat{H}t/{\hbar}},$ (57)

which satisfies Eq. (56) as can be shown by expanding the exponential function
in Taylor series and differentiating term by term with respect to time.

  2. 2.

Time-dependent Hamiltonian with commuting operators. For a time dependent
Hermitian Hamiltonian, where the operators describing the Hamiltonian in
different moments of time commute $[\hat{H}(t_{1}),\hat{H}(t_{2})]=0$, the
solution for the propagator becomes:

$\displaystyle\hat{U}_{t}=e^{-\frac{i}{\hbar}\int_{t_{0}}^{t}dt^{\prime}\hat{H}(t^{\prime})}.$
(58)

  3. 3.

Time-dependent Hamiltonian with non-commuting operators. When the operators
describing the Hamiltonian in different moments of time do not commute, the
solution for the propagator becomes:

$\displaystyle\hat{U}_{t}$
$\displaystyle=1+\sum_{n=0}^{\infty}{\frac{(-i)^{n}}{\hbar}}\left(\prod_{k=1}^{n}\int_{t_{0}}^{t}dt_{k}\right)$
(59)
$\displaystyle=\int_{t_{0}}^{t_{n}-1}dt_{n}\hat{H}(t_{1})\hat{H}(t_{2})\cdot\cdot\cdot\hat{H}(t_{n}),$
(60)

known as the Dyson series expansion for unitary operators.

Let $\hat{E}$ denote the Hermitian operator representing the energy observable
that commutes with the Hamiltonian, i.e, $[\hat{H},\hat{E}]=0$. In the basis
of the energy eingenvectors, one can write the initial state of the system as
$|\psi_{0}\rangle$ $={\hbox{1\kern-1.49994pt\vrule
width=0.8pt,depth=-0.5pt}}|\psi_{0}\rangle$
$=\sum_{j=1}^{d}|E_{j}\rangle\langle
E_{j}|\psi_{0}\rangle=\sum_{j=1}^{d}c_{j}|E_{j}\rangle$. And for a time-
independent hamiltonian the evolution becomes:

$\displaystyle|\psi_{t}\rangle$ $\displaystyle=$ $\displaystyle
e^{-i\hat{H}t/{\hbar}}|\psi_{0}\rangle=e^{-i\hat{H}t/{\hbar}}\sum_{j=1}^{d}c_{j}|E_{j}\rangle$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{d}c_{j}e^{-iE_{j}t/{\hbar}}|E_{j}\rangle=\sum_{j=1}^{d}c_{j}(t)|E_{j}\rangle,$

given the linearity property of linear operators (see subsection 2.4) and the
fact that

$\displaystyle f(\hat{\mathcal{O}})|o_{j}\rangle=f(o_{j})|o_{j}\rangle.$ (61)

####  3.2.3 Postulate 3 (General measurements)

Measurements of a quantum system are described by a collection {$M_{m}$} of
measurement operators acting on the state space of the system with $m$
possible measurement outcomes. If the quantum system is prepared in a general
state $|\psi\rangle$, the probability associated with a measurement outcome
$o_{m}$ is:

$Pr(o_{m})\doteq\langle\psi|M^{\dagger}_{m}M_{m}|\psi\rangle,$ (62)

and the state of the system immediately after the measurement of the
eigenvalue $o_{m}$ will be:

$\displaystyle|\psi_{o_{m}}\rangle=\frac{M_{m}|\psi\rangle}{\sqrt{\langle\psi|M^{\dagger}_{m}M_{m}|\psi\rangle}}.$
(63)

The measurement operators satisfy the completeness relation:

$\sum_{m}M_{m}^{\dagger}M_{m}=I,$ (64)

meaning that probabilities must sum to one and $|\psi\rangle$ is normalized,
i.e,

$\sum_{m}Pr(o_{m})=\sum_{m}\langle\psi|M^{\dagger}_{m}M_{m}|\psi\rangle=1.$
(65)

####  3.2.4 Projective measurements

A special case of postulate 3 is the protective measurement (a.k.a von Neumann
measurement) described by a physical observable represented by a Hermitian
($\hat{\mathcal{O}}=\hat{\mathcal{O}}^{\dagger}$) and, therefore, Normal
($\hat{\mathcal{O}}\hat{\mathcal{O}}^{\dagger}=\hat{\mathcal{O}}^{\dagger}\hat{\mathcal{O}}$)
matrix with a diagonal representation (see Theorem (1)) of the form:

$\displaystyle\hat{\mathcal{O}}=\sum_{j=1}^{d}o_{j}P_{o_{j}}=\sum_{j=1}^{d}o_{j}|o_{j}\rangle\langle
o_{j}|.$ (66)

Where $P_{o_{j}}=|o_{j}\rangle\langle o_{j}|$ is the projector onto the
eigenspace of the observable $\hat{\mathcal{O}}$ in some $d$-dimensional
orthonormal basis set $\\{|o_{j}\rangle\\}_{j=1}^{d}$ ($\langle
o_{j}|o_{k}\rangle=\delta_{jk}$) of eigenvectors $|o_{j}\rangle$ with
eigenvalues $o_{j}$. A feature of such a linear operator, as it is hermitian,
is to have real eigenvalues ($o_{j}\in\mathbb{R}$). If the system is prepared
in the state $|\psi\rangle\doteq\sum_{j=1}^{d}c_{j}|o_{j}\rangle$, a
projective measurement (Born’s rule) entails the following conditional
probability for obtaining an outcome $o_{j}$ of $\hat{\mathcal{O}}$:

$\displaystyle Pr(o_{j}|\psi\rangle)$ $\displaystyle=$
$\displaystyle\langle\psi|P^{\dagger}_{o_{j}}P_{o_{j}}|\psi\rangle$ (67)
$\displaystyle=$ $\displaystyle\langle\psi|P_{o_{j}}^{2}|\psi\rangle$ (68)
$\displaystyle=$ $\displaystyle\langle\psi|(|o_{j}\rangle\langle
o_{j}|o_{j}\rangle\langle o_{j}|)|\psi\rangle$ (69) $\displaystyle=$
$\displaystyle\delta_{jj}\langle\psi|o_{j}\rangle\langle o_{j}|\psi\rangle$
(70) $\displaystyle=$ $\displaystyle|\langle\psi|o_{j}\rangle|^{2}=|\langle
o_{j}|\psi\rangle|^{2}$ (71) $\displaystyle=$ $\displaystyle\left|\langle
o_{j}|\sum_{k=1}^{d}c_{k}|o_{k}\rangle\right|^{2}$ (72) $\displaystyle=$
$\displaystyle\left|\sum_{k=1}^{d}c_{k}\langle o_{j}|o_{k}\rangle\right|^{2}$
(73) $\displaystyle=$
$\displaystyle\left|\sum_{k=1}^{d}c_{k}\delta_{jk}\right|^{2}=|c_{j}|^{2}.$
(74)

And the state of the system after the measurement shall collapse to

$\displaystyle\frac{P_{o_{j}}|\psi\rangle}{\sqrt{Pr(o_{j}|\psi\rangle)}}$
$\displaystyle=$ $\displaystyle\frac{P_{o_{j}}|\psi\rangle}{|\langle
o_{j}|\psi\rangle|^{2}}.$ (75)

The aforementioned result of the probability is consistent with the
frequentist probability interpretation of statistical mechanics stated as the
limit of the frequency of occurrence of a given outcome $o_{j}$ as the number
of measurement trials goes to infinity. In a non-rigorous way, the frequentist
probability reads:

$\displaystyle Pr(o_{j}|\psi\rangle)\doteq
lim_{n\rightarrow\infty}\frac{n_{j}}{n},$ (76)

where $n_{j}$ denotes the number of outcomes of $o_{j}$, and $n$ denotes the
number of measurement trials. In this formalism, the expectation value (a.k.a
mean value or average value) of the observable $\hat{\mathcal{O}}$ is defined
as:

$\displaystyle\langle\hat{\mathcal{O}}\rangle$ $\displaystyle\doteq
lim_{n\rightarrow\infty}\sum_{j=1}^{d}\frac{n_{j}}{n}o_{j}$ (77)
$\displaystyle=\sum_{j=1}^{d}o_{j}Pr(o_{j}|\psi\rangle),$ (78)

and using Eq. (70) one then has

$\displaystyle\langle\hat{\mathcal{O}}\rangle$
$\displaystyle=\sum_{j=1}^{d}o_{j}\langle\psi|o_{j}\rangle\langle
o_{j}|\psi\rangle$ (79)
$\displaystyle=\langle\psi|\left(\sum_{j=1}^{d}o_{j}|o_{j}\rangle\langle
o_{j}|\right)|\psi\rangle$ (80)
$\displaystyle=\langle\psi|\left(\sum_{j=1}^{d}o_{j}P_{o_{j}}\right)|\psi\rangle$
(81) $\displaystyle=\langle\psi|\hat{\mathcal{O}}|\psi\rangle.$ (82)

It is straightforward to extend this result to any function of the observable
$\hat{\mathcal{O}}$ as follows:

$\displaystyle\langle f(\hat{\mathcal{O})}\rangle$
$\displaystyle=\sum_{j=1}^{d}Pr(o_{j}|\psi\rangle)f(o_{j})$ (83)
$\displaystyle=\langle\psi|f(\hat{\mathcal{O}})|\psi\rangle,$ (84)

such that

$\displaystyle\langle\hat{\mathcal{O}}^{2}\rangle=\langle\psi|\hat{\mathcal{O}}^{2}|\psi\rangle.$
(85)

Moreover, for any experiment consisting of $N$ measurements, the amount of
dispersion of the dataset relative to its mean is quantified by the standard
deviation. In quantum mechanics, the more interesting quantity is the
uncertainty (a.k.a variance) associated with the observable and defined as the
square of the standard deviation. From the definitions of probability and
expectation value, the uncertainty can be derived according to:

$\displaystyle(\Delta\hat{\mathcal{O}})^{2}=lim_{n\rightarrow\infty}\sum_{j=1}^{d}\frac{n_{j}}{n}(o_{j}-\langle\hat{\mathcal{O}}\rangle)^{2}$
(86)
$\displaystyle=\sum_{j=1}^{d}Pr(o_{j}|\psi\rangle)(o_{j}-\langle\hat{\mathcal{O}}\rangle)^{2}$
(87) $\displaystyle=\sum_{j=1}^{d}\langle\psi|o_{j}\rangle\langle
o_{j}|\psi\rangle(o_{j}-\langle\hat{\mathcal{O}}\rangle)^{2}$ (88)
$\displaystyle=\sum_{j=1}^{d}\langle\psi|o_{j}\rangle\langle
o_{j}|\psi\rangle(o_{j}^{2}-2o_{j}\langle\hat{\mathcal{O}}\rangle+\langle\hat{\mathcal{O}}\rangle^{2})$
(89) $\displaystyle=\sum_{j=1}^{d}\langle\psi|o_{j}\rangle\langle
o_{j}|\psi\rangle o_{j}^{2}$ (90)
$\displaystyle+\langle\hat{\mathcal{O}}\rangle^{2}\sum_{j=1}^{d}\langle\psi|o_{j}\rangle\langle
o_{j}|\psi\rangle$ (91)
$\displaystyle-2\langle\hat{\mathcal{O}}\rangle\sum_{j=1}^{d}\langle\psi|o_{j}\rangle\langle
o_{j}|\psi\rangle o_{j},$ (92)

which after rewriting Eq. (90) according to
$f(\hat{\mathcal{O}})|o_{j}\rangle=f(o_{j})|o_{j}\rangle$, and bringing the
summation sign close to elements of the same index, becomes

$\displaystyle(\Delta\hat{\mathcal{O}})^{2}=\sum_{j=1}^{d}\langle\psi|\hat{\mathcal{O}}^{2}|o_{j}\rangle\langle
o_{j}|\psi\rangle$ (93)
$\displaystyle+\langle\hat{\mathcal{O}}\rangle^{2}\langle\psi|\left(\sum_{j=1}^{d}|o_{j}\rangle\langle
o_{j}|\right)|\psi\rangle$ (94)
$\displaystyle-2\langle\hat{\mathcal{O}}\rangle\sum_{j=1}^{d}\langle\psi|\hat{\mathcal{O}}|o_{j}\rangle\langle
o_{j}|\psi\rangle$ (95)
$\displaystyle=\langle\psi|\hat{\mathcal{O}}^{2}\left(\sum_{j=1}^{d}|o_{j}\rangle\langle
o_{j}|\right)\psi\rangle$ (96)
$\displaystyle+\langle\hat{\mathcal{O}}\rangle^{2}\langle\psi|\left(\sum_{j=1}^{d}|o_{j}\rangle\langle
o_{j}|\right)|\psi\rangle$ (97)
$\displaystyle-2\langle\hat{\mathcal{O}}\rangle\langle\psi|\hat{\mathcal{O}}\left(\sum_{j=1}^{d}|o_{j}\rangle\langle
o_{j}|\right)|\psi\rangle$ (98)
$\displaystyle=\langle\psi|\hat{\mathcal{O}}^{2}|\psi\rangle-\langle\hat{\mathcal{O}}\rangle^{2}-2\langle\hat{\mathcal{O}}\rangle^{2}$
(99)
$\displaystyle=\langle\hat{\mathcal{O}}^{2}\rangle-\langle\hat{\mathcal{O}}\rangle^{2}.$
(100)

An equivalent result is obtained from

$\displaystyle(\Delta\hat{\mathcal{O}})^{2}$
$\displaystyle=\sum_{j}\langle\psi|o_{j}\rangle\langle
o_{j}|\psi\rangle(o_{j}-\langle\hat{\mathcal{O}}\rangle)^{2}$ (101)
$\displaystyle=\sum_{j}\langle\psi|o_{j}\rangle(o_{j}-\langle\hat{\mathcal{O}}\rangle)^{2}\langle
o_{j}|\psi\rangle$ (102)
$\displaystyle=\sum_{j}\langle\psi|(\hat{\mathcal{O}}-\langle\hat{\mathcal{O}}\rangle)^{2}|o_{j}\rangle\langle
o_{j}|\psi\rangle$ (103)
$\displaystyle=\langle\psi|(\hat{\mathcal{O}}-\langle\hat{\mathcal{O}}\rangle)^{2}\left(\sum_{j}|o_{j}\rangle\langle
o_{j}|\right)|\psi\rangle$ (104)
$\displaystyle=\langle\psi|(\hat{\mathcal{O}}-\langle\hat{\mathcal{O}}\rangle)^{2}|\psi\rangle$
(105)
$\displaystyle=\langle(\hat{\mathcal{O}}-\langle\hat{\mathcal{O}}\rangle)^{2}\rangle.$
(106)

Therefore, the uncertainty (variance) in the measurement of the observable
$\hat{\mathcal{O}}$ is:

$\displaystyle\langle(\hat{\mathcal{O}}-\langle\hat{\mathcal{O}}\rangle)^{2}\rangle=\langle\hat{\mathcal{O}}^{2}\rangle-\langle\hat{\mathcal{O}}\rangle^{2}.$
(107)

A generalization of Born’s rule to realistic experiments is another special
case of postulate 3, and known as a positive operator valued measure (POVM)
[44]. Such a measurement accounts for open systems with intrinsic losses.

####  3.2.5 Postulate 4 (Composite Space of multipartite systems)

Consider a quantum system composed of $N$ arbitrary subsystems, where
subsystem $s$ is prepared in a $d_{s}$-dimensional qudit state
$|\psi_{s}\rangle_{qd}=\sum_{j=1}^{d_{s}}c_{j_{s}}|o_{j}\rangle_{s}$ of a
$d_{s}$-dimensional Hilbert space $\mathcal{H}_{s}^{d_{s}}$ with orthonormal
basis set $\\{|o_{j}\rangle_{s}\\}_{j=1}^{d_{s}}$. The Hilbert state space
$\mathcal{H}^{d}_{1\cdots N}$ of the composite physical system is given by the
tensor product of its constituent Hilbert spaces (the state space of its
subsystems a.k.a component physical systems): $\mathcal{H}^{d}_{1\cdots
N}=\otimes_{s=1}^{N}\mathcal{H}_{s}^{d_{s}}=\mathcal{H}_{1}^{d_{1}}\otimes\mathcal{H}_{2}^{d_{2}}\otimes\cdots\otimes\mathcal{H}_{N}^{d_{N}}$,
where $d=\prod_{s=1}^{N}d_{s}$ is the dimension of the composite space. By its
turn, the corresponding orthonormal basis set of the composite state space is
obtained from the tensor product between the basis of each constituent state
space:
$\\{|o_{j}\rangle\\}_{j=1}^{d}=\\{|o_{j}\rangle_{s}\otimes\cdot\cdot\cdot\otimes|o_{k}\rangle_{N}\\}_{j,\cdots,k=1}^{d_{s},\cdots,d_{N}}$.

###  3.3 Quantum nondemolition measurements

In the context of conjugate observables, when a given measured observable
$\hat{\mathcal{O}}$ does not commute with the system Hamiltonian $\hat{H_{s}}$
(see Theorem (2)), a measurement of $\hat{\mathcal{O}}$ disturbs the system
and, as a consequence, the state of the system after the measurement is no
longer constrained to an eigenstate of the Hamiltonian (see Fig. 1(a)).
Conversely, when the Hamiltonian and the measured observable commute,
$[\hat{\mathcal{O}},\hat{H_{s}}]=0$, the state of the system after the
measurement is an eigenstate of the Hamiltonian, and repeated measurements
yield the same result (see Fig. 1(b)) thus minimally disturbing the system.

Figure 1: Disturbed and nondisturbed systems. Each square box represents a
measure of the indicated observable. The arrows represent the evolution of the
system state vector. (a) When the Hamiltonian contains a non-commuting
observable $\hat{B}$, which does not commute with a measured observable
$\hat{A}$, a measurement of $\hat{A}$ collapses the system initially in the
state $|\psi\rangle$ to a state $|\psi_{1}\rangle$ which is not an eigenstate
of $\hat{B}$. As a consequence, the system is disturbed by an increasing
uncertainty in the non-commuting observable $\hat{B}$ due to a measurement of
$\hat{A}$, such that the prediction of the measurement of $\hat{B}$ is
uncertain. (b) When the system Hamiltonian $\hat{H}_{s}$ is given in terms of
a commuting observable $\hat{B}$, which do commute with the measured
observable $\hat{A}$, predictability of the measurement outcome of $\hat{B}$
is preserved and certain, and the system is minimally disturbed.

A minimally disturbed quantum system, where
$[\hat{\mathcal{O}},\hat{H_{s}}]=0$, has a sequence of quantum nondemolition
(QND) measurements [52][53] of a continuous QND observable $\hat{\mathcal{O}}$
if and only if $[\hat{\mathcal{O}}(t_{1}),\hat{\mathcal{O}}(t_{n})]=0$ is
satisfied for any time $t_{n}$. The free evolution of the system observables
then becomes:

$\displaystyle
i\hbar\frac{d\hat{\mathcal{O}}}{dt}=[\hat{\mathcal{O}},\hat{H}_{s}]=0.$ (108)

A QND measurement enables the realization of multiple measurements on the same
physical system without introducing quantum uncertainty into the observable
being measured during each subsequent normal evolution that takes place after
each measurement trial. In this sense, the predictability of a subsequent
measurement outcome of a precisely measured observable is preserved, and any
random back-action caused by the quantum uncertainty in a noncommuting
observable is avoided. For this reason, a QND measurement is also called a
back-action evading (BAE) measurement. One should be wary that a QND
measurement, alike any ideal classical measurement, will also cause the wave-
function to collapse, i.e, a QND measurement is not a collapse-evading
measurement.

##  4 Quantum information road to quantum computing

The field of quantum information is primarily concerned with the transmission
of quantum data through quantum channels [21] [23] [24] [25]. In this section,
we focus on the backbones of the quantum information theory as we introduce
the state vector formalism of quantum superposition in multipartite systems.
The special case of quantum superposition, namely quantum entanglement, is
addressed in subsection 4.3.

###  4.1 Quantum superposition

Consider a quantum system in a $d$-dimensional state space $\mathcal{H}^{d}$.
The eigenvectors of some observable $\hat{\mathcal{O}}$ form a set
$\\{|o_{j}\rangle\\}_{j=1}^{d}$ of orthonormal basis vectors for the space,
given that physical observables in quantum mechanics are represented by
Hermitian operators. If the quantum system admits more than one independent
eigenstate $|o_{j}\rangle$ of the observable then the global state vector of
the system is termed a coherent state of quantum superposition [54]
[55][56][57]. According to postulate 1, the system’s global state vector is
described by a linear combination of the basis vectors as follows:

$\displaystyle|\psi\rangle\doteq\sum_{j=1}^{d}c_{j}|o_{j}\rangle,$ (109)

where $c_{j}=\langle o_{j}|\psi\rangle$ denotes the complex probability
amplitude associated with the relative frequency of occurrence (probability
density) $|c_{j}|^{2}$ of a measurement outcome of $\hat{\mathcal{O}}$. In
this sense, quantum superposition represents the uncertainty in the
measurement outcome of a physical observable. Henceforth, the information
concerning the preparation of the global state $|\psi\rangle$ of the system is
then obtained in a non-deterministically way according to a probability
distribution. And the associated probabilities ($|c_{j}|^{2}$) are given
according to the frequentist probability interpretation defined in Eq. (76).

The complex coefficients $c_{j}$ are also regarded as probability amplitudes
of the complex-valued wave-function $\psi(x)$ describing the system. To see
this is indeed so, one should recall that the wave-function in the position
space is the projection of the system state vector onto the “position stat”
according to

$\displaystyle\psi(x)=\langle x|\psi\rangle=\sum_{j=1}^{d}c_{j}\langle
x|o_{j}\rangle=c_{j}\psi_{j}(x).$ (110)

As a word of caution, $|x\rangle$ does not denote a basis in Hilbert space, as
it belongs to an uncountable set (a.k.a uncountably infinite set), whereas the
inner product Hilbert space is a finite-dimensional space by definition. The
probabilistic interpretation of the wave-function is attributed to Max Born,
who proposed that the square of the absolute value of the wave-function should
represent the probability density associated with a measurement outcome.
Furthermore, any ideal classical measurement causes the collapse of the wave-
function, consequently updating the global state $|\psi\rangle$ of coherent
superposition to one of the eigenstates $|o_{j}\rangle$ of the measured
observable $\hat{\mathcal{O}}$. That is to say, for a quantum system with $N$
interacting 2-level subsystems (a.k.a qubits), the act of measurement yields a
single value (a real number) out of $2^{N}$ possible classical measurement
outcomes.

Quantum computing harnesses quantum superposition to perform a computation in
all states simultaneously during a time window limited by the decoherence time
(see section 7). To help grasp understanding, consider a system in a
superposition of $2^{N}$ possible configurations (states) from $|000\cdots
0\rangle$ to $|111\cdots 1\rangle$ (see postulate 4 in subsection 3.2 and
subsubsection 4.1.4), where each combination in a classical computing is
representing a particular function to be computed. A classical computer would
require an exponential amount of resources to process each state sequentially
or in parallel as the number of bits increases. Quantum computers, however,
can encode all $2^{N}$ combinations in a single state and perform the
computation in a single go. This feature known as quantum parallelism backed
by quantum superposition does not require a cluster of quantum computers.
Superposition leads to a computational speed-up for certain classes of
computational problems which, in turn, rely on optimized algorithms that
provide speedups over the best classical algorithms for the same computational
tasks. In other terms, quantum computers do not uniformly speed up all
computational problems.

####  4.1.1 The d-level quantum system: qudit

A quantum system consisting of $d$ levels is termed a $d$-dimensional qudit.
Within the state vector formalism, $d$ denotes both the dimension of the qudit
and the dimension of the system’s state space, i.e, the number of linearly
independent vectors in the orthonormal basis set. Particular quantum systems
of $d=2$ and $d=3$ levels are commonly known as qubit and qutrit,
respectively. Is a common misconception to suppose that a qudit is a system of
$N$ qubits. While the former is a system of $d$ levels, the latter is a
composite system of $N$ 2-level subsystems. In this sense, an $N$-qudit system
is a quantum system of $N$ particles or artificial atoms each representing a
subsystem $s$ of $d_{s}$ levels.

In section 8, we describe an $N$-qudit system within the density operator
formalism, where the corresponding $N$-qudit state is represented by a density
operator (a.k.a density matrix) denoted $\rho_{Nqd}$ that is written in a
orthonormal basis formed by $d$x$d$ matrices instead of state vectors (column
vectors).

####  4.1.2 The 2-level quantum system: qubit

Generally, in quantum computing, and specially superconducting quantum
computing, only systems of $d=2$ levels (qubits) are of interest. The main
reason lies in the fact that higher order level systems are hard to control
and manipulate. In such a case, to put it bluntly, the main difference between
a classical (or conventional) computer and a quantum computer is that the
former encodes data into binary digits (a.k.a bits) with their classical
information measured by Shannon entropy [21], whereas the latter encodes data
in quantum bits (a.k.a qubits) with quantum information measured by the Von
Neumann entropy [58]. In the field of quantum information the term “qubit”,
coined by Benjamin Schumacher [22], refers to a quantum system in a coherent
superposition of two possible states of a single physical observable.
Therefore, a qubit is any two-level quantum system in a state of coherent
superposition. Those of skill in the art would appreciate that the term qubit
can refer to the actual physical device in which information is stored, and it
can also refer to the unit of information itself, abstracted away from its
physical device.

According to Postulate 1 (see subsection 3.2), the state vector of an
arbitrary $d$-level quantum system (qudit) in coherent superposition can be
written in the form of Eq. (109). For the particular case of a single 2-level
quantum system in the orthonormal basis set known as the computational basis
(or canonical basis), the 1-qubit state vector becomes:

$\displaystyle|\psi\rangle_{1qb}$
$\displaystyle=\sum_{j=1}^{d=2}c_{j}|j\rangle=\sum_{j=0}^{d-1=1}c_{j}|j\rangle$
(111) $\displaystyle=c_{0}|0\rangle+c_{1}|1\rangle=\begin{pmatrix}c_{0}\\\
c_{1}\end{pmatrix}.$ (112)

Where $c_{0}$ and $c_{1}$ are the alluded complex probability amplitudes, i.e,
the coefficients of the complex-valued wave-function. This state is often
regarded to be in the $|Z_{\pm}\rangle$ basis:

$\displaystyle|0\rangle\doteq|Z_{+}\rangle$ $\displaystyle\doteq$
$\displaystyle\begin{pmatrix}1\\\ 0\end{pmatrix},$ (113)
$\displaystyle|1\rangle\doteq|Z_{-}\rangle$ $\displaystyle\doteq$
$\displaystyle\begin{pmatrix}0\\\ 1\end{pmatrix},$ (114)

From Born’s rule (see Eq. (74)), a projective measurement yields either
outcome $|0\rangle$ with probability $|c_{0}|^{2}$ or $|1\rangle$ with
probability $|c_{1}|^{2}$. Normalization constraint is necessary in order to
satisfy the law of total probability: $|c_{0}|^{2}+|c_{1}|^{2}=1$.

It is always possible to write a qubit state vector in a different basis. One
possible choice is the following orthonormal basis set:

$\displaystyle|+\rangle\doteq|X_{+}\rangle$ $\displaystyle\doteq$
$\displaystyle\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$ (115) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\ 1\end{pmatrix},$ (116)
$\displaystyle|-\rangle\doteq|X_{-}\rangle$ $\displaystyle\doteq$
$\displaystyle\frac{1}{\sqrt{2}}(|0\rangle-|1\rangle)$ (117) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\ -1\end{pmatrix},$ (118)

such that

$\displaystyle|0\rangle$ $\displaystyle=$
$\displaystyle\frac{|+\rangle+|-\rangle}{\sqrt{2}},$ (119)
$\displaystyle|1\rangle$ $\displaystyle=$
$\displaystyle\frac{|+\rangle-|-\rangle}{\sqrt{2}}.$ (120)

The qubit state vector in this new basis now reads

$\displaystyle|\psi\rangle_{1qb}$ $\displaystyle=$ $\displaystyle
c_{0}|0\rangle+c_{1}|1\rangle$ (121) $\displaystyle=$
$\displaystyle\frac{c_{0}+c_{1}}{\sqrt{2}}|+\rangle+\frac{c_{0}-c_{1}}{\sqrt{2}}|-\rangle.$
(122)

A projective measurement in the $|\pm\rangle$ basis yields a probability

$\displaystyle Pr(o_{\pm}|\psi\rangle)$
$\displaystyle=|\langle\pm|\psi\rangle|^{2}$ (123)
$\displaystyle=\left|\langle\pm|\left(\frac{c_{0}+c_{1}}{\sqrt{2}}|+\rangle+\frac{c_{0}-c_{1}}{\sqrt{2}}|-\rangle\right)\right|^{2}$
(124) $\displaystyle=\left|\langle\pm|\pm\rangle\frac{c_{0}\pm
c_{1}}{\sqrt{2}}\right|^{2}=\left|\frac{c_{0}\pm c_{1}}{\sqrt{2}}\right|^{2}$
(125) $\displaystyle=\frac{\left|c_{0}\pm c_{1}\right|^{2}}{2},$ (126)

given the orthonormality condition $\langle\pm|\pm\rangle=1$ as defined in Eq.
(6). That is, measuring with respect to (w.r.t) the $|+\rangle$ basis yields a
probability $|c_{0}+c_{1}|^{2}/2$ collapsing the system to the state
$|+\rangle$ immediately after the measurement. On the other hand, measuring
w.r.t to $|-\rangle$ yields the probability $|c_{0}-c_{1}|^{2}/2$.

Another commonly basis for a qubit state vector is the following orthonormal
basis set:

$\displaystyle|+i\rangle$
$\displaystyle\doteq|Y_{+}\rangle\doteq\frac{1}{\sqrt{2}}(|0\rangle+i|1\rangle)$
(127) $\displaystyle=\frac{1}{\sqrt{2}}\left(\begin{pmatrix}1\\\
0\end{pmatrix}+i\begin{pmatrix}0\\\
1\end{pmatrix}\right)=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\ i\end{pmatrix},$
(128) $\displaystyle|-i\rangle$
$\displaystyle\doteq|Y_{-}\rangle\doteq\frac{1}{\sqrt{2}}(|0\rangle-i|1\rangle)$
(129) $\displaystyle=\frac{1}{\sqrt{2}}\left(\begin{pmatrix}1\\\
0\end{pmatrix}-i\begin{pmatrix}0\\\
1\end{pmatrix}\right)=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\ -i\end{pmatrix}.$
(130)

####  4.1.3 The 3-level quantum system: qutrit

A single three-level ($d$ $=$ $3$) quantum system known as qutrit has the
following state vector representation in the canonical basis:

$\displaystyle|\psi\rangle_{1qt}$
$\displaystyle=\sum_{j=0}^{d-1=2}c_{j}|j\rangle=c_{0}|0\rangle+c_{1}|1\rangle+c_{2}|1\rangle$
(131) $\displaystyle=c_{0}\begin{pmatrix}1\\\ 0\\\
0\end{pmatrix}+c_{1}\begin{pmatrix}0\\\ 1\\\
0\end{pmatrix}+c_{2}\begin{pmatrix}0\\\ 0\\\
1\end{pmatrix}=\begin{pmatrix}c_{0}\\\ c_{1}\\\ c_{2}\end{pmatrix}.$ (132)

In subsection 8.3, we obtain the 1-qutrit coherent and incoherent density
operator from the generalized Bloch’s representation for $d$-dimensional
$N$-qudit states.

####  4.1.4 Composite systems with separable states

From postulate 4 (see subsubsection 3.2.5), the resulting separable global
state vector of a composite physical system with $N$ qudit subsystems, where
subsystem $s$ is prepared in the qudit state vector $|\psi_{s}\rangle_{qd}$,
is given by the tensor product between the qudit states of each subsystem, as
follows:

$\displaystyle|{\psi_{1,2,\cdots,N}^{sep}}\rangle_{Nqd}$
$\displaystyle\doteq\otimes_{s=1}^{N}|\psi_{s}\rangle_{qd}$ (133)
$\displaystyle=|\psi_{1}\rangle_{qd}\otimes|\psi_{2}\rangle_{qd}\otimes\cdot\cdot\cdot\otimes|\psi_{N}\rangle_{qd}$
(134)
$\displaystyle=\bigg{(}c_{1_{1}}|o_{1}\rangle_{1}+c_{2_{1}}|o_{2}\rangle_{1}$
(135) $\displaystyle+\cdots+c_{{d}_{1}}|o_{{d}}\rangle_{1}\bigg{)}$ (136)
$\displaystyle\otimes\bigg{(}c_{1_{2}}|o_{1}\rangle_{2}+c_{2_{2}}|o_{2}\rangle_{2}$
(137) $\displaystyle+\cdots+c_{d_{2}}|o_{{d}}\rangle_{2}\bigg{)}$ (138)
$\displaystyle\otimes\cdot\cdot\cdot\otimes\bigg{(}c_{1_{N}}|o_{1}\rangle_{N}+c_{2_{N}}|o_{2}\rangle_{N}$
(139) $\displaystyle+\cdots+c_{d_{N}}|o_{{d}}\rangle_{N}\bigg{)},$ (140)

with the normalization condition

$\displaystyle\sum_{j=1}^{d_{s}^{N}}|\gamma_{j}|^{2}=1.$ (141)

Here, the coefficient $\gamma_{j}$ represents a complex probability amplitude
from the sample space of $d_{s}^{N}$ possible measurement outcomes associated
with each state vector of the composite system. Resulting states of this
fashion are called separable states (a.k.a product states), and they are never
entangled. A separable state is one that can be created by quantum local
operations and classical communication (LOCC) acting on its subsystems. By
local operations [59] we mean operations that act individually on each
subsystem (each single qudit) of a composite system.

For the particular case of a bipartite quantum system ($N=2$ qudits) there is
an arbitrary subsystem $s$$=$$1$ in the state $|\psi_{1}\rangle$ and subsystem
$s$$=$$2$ in the state $|\psi_{2}\rangle$ with Hilbert spaces
$\mathcal{H}_{1}^{d_{1}}$ and $\mathcal{H}_{2}^{d_{2}}$, respectively. The
Hilbert space of the composite system is denoted by
$\mathcal{H}^{d_{1}d_{2}}_{12}\doteq\mathcal{H}_{1}^{d_{1}}\otimes\mathcal{H}_{2}^{d_{2}}$.
Supposing that each subsystem is a $2$-level quantum system (qubit), the
resulting composite state vector in the canonical basis is the following
separable state (product state) in quantum superposition:

$\displaystyle|\psi_{12}^{sep}\rangle_{2qb}$
$\displaystyle=|\psi_{1}\rangle_{qb}\otimes|\psi_{2}\rangle_{qb}$ (142)
$\displaystyle=\left(\alpha_{1}|0\rangle_{1}+\beta_{1}|1\rangle_{1}\right)$
(143)
$\displaystyle\otimes\left(\alpha_{2}|0\rangle_{2}+\beta_{2}|1\rangle_{2}\right)$
(144)
$\displaystyle=\alpha_{1}\alpha_{2}|0\rangle_{1}\otimes|0\rangle_{2}+\alpha_{1}\beta_{2}|0\rangle_{1}\otimes|1\rangle_{2}$
(145)
$\displaystyle+\beta_{1}\alpha_{2}|1\rangle_{1}\otimes|0\rangle_{2}+\beta_{1}\beta_{2}|1\rangle_{1}\otimes|1\rangle_{2}$
(146) $\displaystyle=\gamma_{00}|00\rangle_{12}+\gamma_{01}|01\rangle_{12}$
(147) $\displaystyle+\gamma_{10}|10\rangle_{12}+\gamma_{11}|11\rangle_{12},$
(148)

with the corresponding matrix representation

$\displaystyle|\psi_{12}^{sep}\rangle_{2qb}$
$\displaystyle=\gamma_{00}\begin{pmatrix}1\\\ 0\\\ 0\\\
0\end{pmatrix}+\gamma_{01}\begin{pmatrix}0\\\ 1\\\ 0\\\
0\end{pmatrix}+\gamma_{10}\begin{pmatrix}0\\\ 0\\\ 1\\\ 0\end{pmatrix}$ (149)
$\displaystyle+\gamma_{11}\begin{pmatrix}0\\\ 0\\\ 0\\\
1\end{pmatrix}=\begin{pmatrix}\gamma_{00}\\\ \gamma_{01}\\\ \gamma_{10}\\\
\gamma_{11}\end{pmatrix}.$ (150)

Here, we have defined: $\gamma_{00}\doteq\alpha_{1}\alpha_{2}$,
$\gamma_{01}\doteq\alpha_{1}\beta_{2}$,
$\gamma_{10}\doteq\beta_{1}\alpha_{2}$, $\gamma_{11}\doteq\beta_{1}\beta_{2}$.
Finally, normality condition for state vectors entails

$\displaystyle\sum_{x\in\\{0,1\\}^{N}=1}^{4}|\gamma_{x}|^{2}$
$\displaystyle=|\gamma_{00}|^{2}+|\gamma_{01}|^{2}$ (151)
$\displaystyle+|\gamma_{10}|^{2}+|\gamma_{11}|^{2}=1,$ (152)

where $x$ denotes an element from the set containing all $2^{N}=4$ possible
permutations of strings of length $N=2$. And for a bipartite three-level
system ($N$$=$$2$ qutrits), the composite state of the system becomes:

$\displaystyle|\psi_{12}^{sep}\rangle_{2qt}$
$\displaystyle=|\psi_{1}\rangle_{qt}\otimes|\psi_{2}\rangle_{qt}$ (153)
$\displaystyle=\left(a_{1}|o_{1}\rangle_{1}+b_{1}|o_{2}\rangle_{1}+c_{1}|o_{3}\rangle_{1}\right)$
(154)
$\displaystyle\otimes\left(a_{2}|o_{1}\rangle_{2}+b_{2}|o_{2}\rangle_{2}+c_{2}|o_{3}\rangle_{2}\right)$
(155)
$\displaystyle=a_{1}a_{2}|o_{1}o_{1}\rangle_{12}+a_{1}b_{2}|o_{1}o_{2}\rangle_{12}$
(156)
$\displaystyle+a_{1}c_{2}|o_{1}o_{3}\rangle_{12}+b_{1}a_{2}|o_{2}o_{1}\rangle_{12}$
(157)
$\displaystyle+b_{1}b_{2}|o_{2}o_{2}\rangle_{12}+b_{1}c_{2}|o_{2}o_{3}\rangle_{12}$
(158)
$\displaystyle+c_{1}a_{2}|o_{3}o_{1}\rangle_{12}+c_{1}b_{2}|o_{3}o_{2}\rangle_{12}$
(159) $\displaystyle+c_{1}c_{2}|o_{3}o_{3}\rangle_{12}.$ (160)

In the context of computer hardware, a classical register of $N$ qubits has
$2^{N}$ possible configurations, however it can store only one number of
information at a time, while a quantum register of $N$ qubits will have a
composite system with $2^{N}$ complex coefficients (amplitudes). Therefore, a
quantum register can store a superposition of $2^{N}$ classical bits of
information simultaneously, i.e, it can store all numbers from 1 to $2^{N}$.
In this sense a quantum register of $N=11$ qubits will store all numbers from
1 to 2048, whereas a classical register with a sequence 11111011011 will store
only the number 2011. Moreover, if each complex amplitude is stored to 128
bits of precision, then each requires 256 bits (or 32 bytes) of information,
and $2^{N}$ qubits will then require $32$x$2^{N}$ bytes of information for
storage.

###  4.2 Bloch Sphere

A generic 1-qubit state vector can be written as

$\displaystyle|\psi\rangle_{1qb}$
$\displaystyle=c_{0}|0\rangle+c_{1}|1\rangle$ (161)
$\displaystyle=(\alpha_{1}+i\beta_{1})|0\rangle+(\alpha_{2}+i\beta_{2})|1\rangle,$
(162)

where we have expanded the complex coefficients $c_{j}$’s, with $\alpha$’s and
$\beta$’s as real numbers. Normalization condition
$\sqrt{c_{0}^{2}+c_{1}^{2}}=1$ then implies in the reduction of four degrees
of freedom to only three. Choosing the Hopf coordinate system, one can write
the complex coefficients as:

$\displaystyle c_{0}$ $\displaystyle=e^{i\omega}cos(\theta/2),$ (163)
$\displaystyle c_{1}$ $\displaystyle=e^{i\omega+\phi}sin(\theta/2),$ (164)

and the 1-qubit state vector becomes

$\displaystyle|\psi\rangle_{1qb}$
$\displaystyle=e^{i\omega}cos(\theta/2)|0\rangle+e^{i\omega+\phi}sin(\theta/2)|1\rangle$
(165)
$\displaystyle=e^{i\omega}\left(cos(\theta/2)|0\rangle+e^{i\phi}sin(\theta/2)|1\rangle\right).$
(166)

However, qubit states that differ only by a global factor $e^{i\omega}$ are
physically indistinguishable, i.e, a global factor does not change the
measurement outcome:

$\displaystyle|\langle x|(e^{i\omega}|\psi\rangle)|^{2}=|e^{i\omega}(\langle
x|\psi\rangle)|^{2}=|\langle x|\psi\rangle|^{2}.$ (167)

With that, one can adopt the following convenient representation for a generic
single qubit state vector:

$\displaystyle|\psi\rangle_{1qb}$
$\displaystyle=cos(\theta/2)|0\rangle+e^{i\phi}sin(\theta/2)|1\rangle$ (168)
$\displaystyle=\alpha|0\rangle+\gamma e^{i\phi}|1\rangle,$ (169)

with $\alpha,\gamma,\theta\in\mathbb{R}$, and $0\leq\theta\leq\pi$, and
$0\leq\phi\leq 2\pi$. The factor $e^{i\phi}$ is a complex number with absolute
value equal to one ($|e^{i\phi}|=1$) that can be decomposed in terms of sines
and cosines using Euler’s formula $e^{i\phi}=cos(\phi)+isin(\phi)$. Because of
that, the real number $\phi$ is regarded as the relative phase of the wave-
function in analogy to the phase appearing in the argument of sinusoidal
functions describing oscillations, even though no time-dependence and,
therefore, no oscillation exists in this case. Whereas $\theta$ gives the
probability of each possible measurement outcome:

$\displaystyle p(|0\rangle)$ $\displaystyle=|\alpha|^{2}=cos^{2}(\theta/2),$
(170) $\displaystyle p(|1\rangle)$ $\displaystyle=|\gamma
e^{i\phi}|^{2}=\gamma e^{i\phi}\gamma
e^{-i\phi}=\gamma^{2}=sin^{2}(\theta/2).$ (171)

Since the magnitude of this normalized 1-qubit state vector is equal to one
($\langle\psi|\psi\rangle=1$), the coordinates $\theta$ and $\phi$ can be
regarded as spherical coordinates of a unit sphere of radius $||\vec{r}||=1$
and vector:

$\displaystyle\vec{r}$
$\displaystyle=||\vec{r}||\hat{r}=||\vec{r}||(x\hat{x}+y\hat{y}+z\hat{z})$
(172) $\displaystyle=||\vec{r}||sin(\theta)cos(\phi)\hat{x}$ (173)
$\displaystyle+||\vec{r}||sin(\theta)sin(\phi)\hat{y}$ (174)
$\displaystyle+||\vec{r}||cos(\theta)\hat{z}$ (175)
$\displaystyle=||\vec{r}||\begin{pmatrix}sin(\theta)cos(\phi)\\\
sin(\theta)sin(\phi)\\\ cos(\theta)\end{pmatrix}.$ (176)

In this scenario, a generic 1-qubit state vector in the form of Eq. (168) can
be associated with a so-called Bloch’s vector whose coordinates are of a point
on the surface of a three-dimensional unit sphere known as the Bloch’s sphere
(see Fig. 2), even though a 2-level system is embedded in a two-dimensional
Hilbert space. For historical reasons, the Bloch’s sphere is also referred to
as the Poincaré ball when describing classical polarization states.

Figure 2: Bloch sphere representation for a generic 1-qubit system. The basis,
$|0\rangle$ and $|1\rangle$, are orthonormal column vectors in the Hilbert
space, however, they are represented by antiparallel (antipodal) vectors on
the Bloch sphere. In this case $\theta/2$ is the actual angle in the Hilbert
space, while $\theta$ is the angle in the Bloch sphere. In the context of
density operators (see section 5), single 1-qubit state vectors whose
coordinates of their Bloch’s vectors are of points on the surface of the
sphere ($||\vec{r}||=1$) have each a corresponding pure density operator,
while coordinates of points located inside the sphere ($||\vec{r}||<1$) have
corresponding mixed density operators that cannot be associated with a single
state vector.

In the Bloch’s sphere representation, the basis states can be obtained from
the following choice of coordinates:

$\displaystyle\theta$
$\displaystyle=0,\forall\phi\implies\vec{r}=\begin{pmatrix}0\\\ 0\\\
1\end{pmatrix}\text{and }|\psi\rangle=|0\rangle,$ $\displaystyle\theta$
$\displaystyle=\pi,\forall\phi\implies\vec{r}=\begin{pmatrix}0\\\ 0\\\
-1\end{pmatrix}\text{and }|\psi\rangle=e^{i\phi}|1\rangle,$

and since the factor $e^{i\phi}$ now acts as a global phase it can be
neglected. The remaining basis are:

$\displaystyle\theta$
$\displaystyle=\frac{\pi}{2},\phi=0\implies\vec{r}=\begin{pmatrix}1\\\ 0\\\
0\end{pmatrix}\text{and }|\psi\rangle=|+\rangle,$ $\displaystyle\theta$
$\displaystyle=\frac{\pi}{2},\phi=\pi\implies\vec{r}=\begin{pmatrix}-1\\\ 0\\\
0\end{pmatrix}\text{and }|\psi\rangle=|-\rangle,$ $\displaystyle\theta$
$\displaystyle=\frac{\pi}{2},\phi=\frac{\pi}{2}\implies\vec{r}=\begin{pmatrix}0\\\
1\\\ 0\end{pmatrix}\text{and }|\psi\rangle=|+i\rangle,$ $\displaystyle\theta$
$\displaystyle=\frac{\pi}{2},\phi=\frac{3\pi}{2}\implies\vec{r}=\begin{pmatrix}0\\\
0\\\ 1\end{pmatrix}\text{and }|\psi\rangle=|-i\rangle.$

In subsection 8.3, we derive the conditions to describe a generic 1-qubit
state as a pure or mixed density operator whose Bloch’s vectors have
coordinates of points located inside the Bloch’s sphere or on its surface,
respectively.

###  4.3 Quantum entanglement

Quantum entanglement is a special kind of quantum superposition that quantum
computers rely on to speed up calculations [31][32], protect communication
against eavesdropping [25][60], and double the classical capacity of a
communication channel [23]. More generally, entanglement is a feature where
two or many subsystems of a multipartite quantum system are strongly
correlated to one another that the state of each subsystem cannot be described
independently of the state of the other without taking into account the global
state of the system: a holistic feature of entangled states. Whereas the
quantum correlation between two subsystems is such that the state of one
subsystem (its physical reality) is immediately influenced by a measurement of
the state of the other, regardless of their spacelike separation. This
peculiar phenomenon dubbed “quantum nonlocality”, whereby quantum information
appears to travel faster than the speed of light, sparked a heated debate
culminating in the famous article co-authored by Einstein, Podolsky, and Rosen
(EPR) in 1935 [15]. For a more in depth review on quantum entanglement, the
reader may resort to Horodecki et al. [61].

####  4.3.1 The EPR hypothesis

In considering Max Born’s probabilistic interpretation of the wave-function,
Albert Einstein addressed the mechanism of the wave-function collapse during
discussions in the 1927 Solvay conference. Einstein’s gedanken (thought
experiment) considered the diffraction of a beam of a single electron passing
through a pinhole. According to the principle of superposition, before the
electron hits a screen, the wave-function in the position space represents the
superposition of all the eigenstates associated with the positions available
to the electron on that screen. The act of measurement would then cause the
wave-function to collapse to a single eingestate of the position observable.
Einstein referred to this mechanism by which the wave-function is instantly
updated as “action at a distanc”. The famous phrase “spooky action at a
distance” is believed to had its origin only twenty years later, in the 1947
letter to Max Born [62]: “Ich kann aber deshalb nicht ernsthaft daran glauben,
weil die Theorie mit dem Grundsatz unvereinbar ist, daß die Physik eine
Wirklichkeit in Zeit und Raum darstellen soll, ohne spukhafte Fernwirkungen”
(I cannot seriously believe in it (quantum mechanics) because the theory is
incompatible with the principle that physics should represent reality in space
and time, without spooky action at a distance).

Einstein went further with his hypothesis that quantum mechanics was
incomplete and published, among three other physicists, the so-called EPR
paper in 1935 [15]. The paper devised a thought experiment considering a
system of two entangled particles to assess whether a measurement in one
particle would have an instantaneous effect on the other. The starting point
of the EPR paper adopts the following definitions regarding the quantum
mechanical theory:

  * •

Realism: every element of reality in nature (the so-called observables) must
be predicted within the quantum theory without causing any disturbance to the
quantum system. That is, it should be possible to predict the outcome of a
measurement with 100% certainty before the measurement is even realized.

  * •

Self-consistency: a quantum theory shall be able to predict and explain
experimental data in a consistent manner, i.e, free of contradictions.

  * •

Completeness: a complete quantum theory is such that there is a theoretical
analog for every element of reality in nature.

  * •

Locality: a measurement in one subsystem should not influence the outcome of a
measurement in another subsystem.

Then, for two conjugate observables $\hat{A}$ and $\hat{B}$ satisfying Eq.
(47) (e.g. spin $\hat{\sigma}_{x}$ and $\hat{\sigma}_{z}$), the EPR paper
states that only one of the following two hypotheses is a correct assumption
about quantum mechanics:

  1. 1.

$\hat{A}$ and $\hat{B}$ cannot simultaneously satisfy the criterion of
realism;

  2. 2.

The theory of quantum mechanics is incomplete.

For the purpose of explaining the thought experiment, consider the following
scenario proposed by David Bohm where a subatomic particle of total spin zero
decays in two new particles denoted $q_{1}$ and $q_{2}$ of subsystems $s=1$
and $s=2$, respectively. (It is worth mentioning that entangled systems have
already been produced in the laboratory with spin-1 particles, particularly
photons, via a process known as parametric down-conversion [24], and with
Cooper pairs of electrons in a circuit-based superconducting quantum processor
driven by a series of microwave pulses.) If the process of decay occurs in an
isolated system then the total spin is a conserved quantity, such that the
resulting entangled particles have opposite spins and the total spin of the
bipartite system adds up to zero as expected. Moreover, if the experiment is
controlled in a way that the spins are only allowed to have vertical
directions, the two possible configurations for the outcome of the decay are:
particle $q_{1}$ with spin up ($|1\rangle_{1}$) and particle $q_{2}$ with spin
down ($|0\rangle_{2}$) or, conversely, particle $q_{1}$ with spin down and
$q_{2}$ with spin up. The global state of the system is then regarded as a
superposition of the aforementioned two possible outcomes of the decay with
both particles described by a single wave-function and, therefore, a single
Schrödinger equation. Such an entangled state is refereed to as a singlet
state or Bell-like state. It is then possible to write the state of the system
in two conjugate basis:

$\displaystyle|\psi_{x}^{Bell}\rangle$
$\displaystyle\doteq\frac{|X+\rangle_{1}\otimes|X-\rangle_{2}-|X-\rangle_{1}\otimes|X+\rangle_{2}}{\sqrt{2}},$
(177) $\displaystyle|\psi_{z}^{Bell}\rangle$
$\displaystyle\doteq\frac{|Z+\rangle_{1}\otimes|Z-\rangle_{2}-|Z-\rangle_{1}\otimes|Z+\rangle_{2}}{\sqrt{2}}.$
(178)

While the spin observables are given according to the spectral decomposition
defined in Eq. (34), as follows:

$\displaystyle\hat{\sigma}_{x}=\sum_{j=1}^{2}o_{j}P_{o_{j}}$
$\displaystyle=1P_{+}+(-1)P_{-}$ (179)
$\displaystyle=|+\rangle\langle+|-|-\rangle\langle-|$ (180)
$\displaystyle=|0\rangle\langle 1|+|1\rangle\langle 0|,$ (181)

and

$\displaystyle\hat{\sigma}_{z}$
$\displaystyle=\sum_{j=1}^{2}o_{j}P_{o_{j}}=|0\rangle\langle
0|-|1\rangle\langle 1|.$ (182)

EPR then assumes locality as a true assumption about quantum mechanics,
ensuring both particles to be so far apart that no information traveling at
the speed of light can influence the result of a measurement. Within the
constraints set by the experiment, in which the spins in each direction are
antiparallel to each other, the two particles are regarded as maximally
correlated, as are their measurement outcomes. That being so, from the
measurement postulate: if a measurement of $\hat{\sigma}_{z}$ on particle
$q_{1}$ results in spin up, then particle $q_{2}$ must have spin down with
100% certainty. This means that particle $q_{2}$ can be measured without
disturbing the system, hence, there should be an element of reality associated
with $\hat{\sigma}_{z}$. The same thought process applies for a measurement of
$\hat{\sigma}_{x}$. The logic inference is that assumption 1 must then be
false, i.e, the aforementioned Bell state seems to allow simultaneous reality
for $\hat{\sigma}_{x}$ and $\hat{\sigma}_{z}$. This led EPR to the erroneous
conclusion in which the theory of quantum mechanics is incomplete. Advocating
on behalf of EPR, one must recognize the sheer lack of experiments available
at the time. As a matter of fact, the first loophole-free experiment
demonstrating the non-local phenomenon of quantum entanglement was carried out
only 80 years later, in 2015, by Hensen et al. [18].

The key conundrum of quantum entanglement lies in the fact that quantum
correlations may actually be stronger than non-quantum (classical)
correlations. This result becomes evident upon quantification of the degree of
correlation of measurement outcomes from conjugate observables, where the
prime example is the measurement of the spin in different directions. Once
again, the nonlocal quantum correlation of the bipartite system allows the
state of one subsystem to be simultaneously determined upon a measurement of
the state of the other. Given that such a non-local phenomenon happens
seemingly faster than the speed of light, quantum mechanics appears to violate
Einstein’s principle of special relativity in which the speed of any physical
particle (or wave) is upper bounded by the speed of light. However, as
measurement outcomes must be compared, and such information travels
classically, bounded by the speed of light, Einstein’s principle is salved. As
a consequence, faster-than-light communication becomes impossible.

####  4.3.2 The CHSH inequality

Driven by the misleading conclusion in which the theory of quantum mechanics
was incomplete, EPR suggested the existence of a theory of hidden variables to
account for the classical ignorance related to the initial conditions of the
experiment. A mindset long espoused by Einstein and conveyed through his
famous 1926 quote “God doesn’t play dice”, believing that quantum mechanics
should not be framed as a probabilistic theory, but rather as a deterministic
one. The first deterministic theory of this kind, where the so-called hidden
variables represent the position of a given particle, was elaborated in 1952
by David Bohm [63] in a similar way to an earlier but less famous theory
pioneered by Louis de Broglie in 1927. The theory known as Bohmian mechanics
or “pilot wave interpretation” was devised as an alternative approach to the
standard notion of quantum mechanics formulated by Bohr, Heisenberg, and
Schrödinger, namely, “the Copenhagen interpretation” whose postulates we have
described in subsection 3.2. In a nutshell, Bohmian mechanics is based on a
so-called guiding field whose definition renders the theory more non-local
than the original probabilistic theory of quantum mechanics is. And for that
reason, it does not reproduce the same successful predictions as the standard
Copenhagen interpretation.

Then, in 1964, John Bell [16] proposed a deterministic local hidden variable
model (LHVM) to assess the claims put forward by the EPR paper. Bell’s
approach considered a generic statistical theory (that is, not necessarily
related to quantum mechanics) with notions of a probability distribution
$p(\lambda)$ of a sample space $\Lambda$ with hidden variables $\lambda$.
Restricting the case for a bipartite system, the model allowed the possibility
of measuring variables (observables) $A_{1}$ and $A_{2}$ in subsystem $A$, and
variables $B_{1}$ and $B_{2}$ in subsystem $B$ with probability outcomes
$a_{1},a_{2}$ and $b_{1},b_{2}$, respectively, sharing the following relation:

$\displaystyle C(\lambda)$ $\displaystyle\doteq
A_{1}(\lambda)B_{1}(\lambda)+A_{1}(\lambda)B_{2}(\lambda)$ (183)
$\displaystyle+A_{2}(\lambda)B_{1}(\lambda)-A_{2}(\lambda)B_{2}(\lambda),$
(184)

where the variables take on two dichotomic measurements whose spectrum of
values range between $\pm 1$. Then, for an ensemble of particles in a perfect
entangled state (e.g. singlet state), the joint probability of the ensemble
statistics yields:

$\displaystyle p(ab|AB)\neq p(a|A)p(b|B).$ (185)

If locality is assumed, one then has

$\displaystyle C(\lambda)$ $\displaystyle\doteq
A_{1}(\lambda)[B_{1}(\lambda)+B_{2}(\lambda)]$ (186)
$\displaystyle+A_{2}(\lambda)[B_{1}(\lambda)-B_{2}(\lambda)]=\pm 2.$ (187)

and also

$\displaystyle
p(ab|AB)=\bigints_{\Lambda}p(a|A\lambda)p(b|B\lambda)p(\lambda)d\lambda.$
(188)

Building on Bell’s paper, and assuming locality, Clauser, Horne, Shimony, and
Holt devised the following correlation inequality known as the CHSH inequality
[17] for bipartite systems:

$\displaystyle|\langle C(\lambda)\rangle|$
$\displaystyle\doteq\bigg{|}\bigints_{\Lambda}\bigg{\\{}A_{1}(\lambda)[B_{1}(\lambda)+B_{2}(\lambda)]$
$\displaystyle+A_{2}(\lambda)[B_{1}(\lambda)-B_{2}(\lambda)]\bigg{\\}}p(\lambda)d\lambda\bigg{|}\leq
2,$

corresponding to a bound on any LHVM. The CHSH inequality is regarded as more
feasible inequality for an experimental test of the LHVM as an independent
hypothesis from quantum formalism. To name a few, Ursin et al [64], in 2006,
reported a experimental violation of the CHSH inequality measured by two
observers separated by a distance of $144km$. Aspect et al. [19][20] were the
first to verify experimentally the violation of Bell’s inequality. However, to
ensure that the state of particle $q_{2}$ is actually instantly determined
once the state of particle $q_{1}$ is measured, the experiment must be carried
out during a time window shorter than the time taken by any possible
contaminating information traveling from one particle to the other at the
speed of light and which could influence the result of a measurement. The
first loophole-free experiment of this kind was only carried out by Hensen et
al., in 2015 [18], reporting a violation of the CHSH inequality. The
experiment consisted of spin-like particles separated by a distance of
$1,280m$ setting a $4.27$-$ms$ time window during which two local events are
space-like separated. The experimental evidence that certain entangled states
do violate Bell’s inequality confirms that locality and realism known together
as local-realism [19][20] cannot be simultaneously satisfied.

Theoretically, one can show the maximum violation of the CHSH inequality for
the following quantum observables:

$\displaystyle\hat{A}_{1}$ $\displaystyle\doteq
R\doteq\vec{r}\cdot\vec{\sigma},$ (189) $\displaystyle\hat{A}_{2}$
$\displaystyle\doteq Q\doteq\vec{q}\cdot\vec{\sigma},$ (190)
$\displaystyle\hat{B}_{1}$ $\displaystyle\doteq
S\doteq\vec{s}\cdot\vec{\sigma},$ (191) $\displaystyle\hat{B}_{2}$
$\displaystyle\doteq T\doteq\vec{t}\cdot\vec{\sigma},$ (192)

where $\vec{r}$, $\vec{q}$, $\vec{s}$, and $\vec{t}$ are 3-dimensional real
unit vectors describing the measurements performed by each parties, and
$\vec{\sigma}=\sigma_{x}\hat{x}+\sigma_{y}\hat{y}+\sigma_{z}\hat{z}$ is a
3-dimensional vector with the Pauli-$\sigma$ matrices defined in section 8.
One then has

$\displaystyle\hat{c}$ $\displaystyle=R\otimes S+R\otimes T+Q\otimes
S-Q\otimes T,$ (193)

so that

$\displaystyle\langle\hat{c}\rangle$ $\displaystyle=\langle\psi_{ab}|R\otimes
S|\psi_{ab}\rangle+\langle\psi_{ab}|R\otimes T|\psi_{ab}\rangle$ (194)
$\displaystyle+\langle\psi_{ab}|Q\otimes
S|\psi_{ab}\rangle-\langle\psi_{ab}|Q\otimes T|\psi_{ab}\rangle.$ (195)

Then, using Tsirelson’s inequality [65] defined as

$\displaystyle 4{\hbox{1\kern-1.49994pt\vrule
width=0.8pt,depth=-0.5pt}}_{2}+[Q,R]\otimes[S,T]$
$\displaystyle\doteq\bigg{(}Q\otimes S+R\otimes S$ $\displaystyle+R\otimes
T-Q\otimes T\bigg{)}^{2},$

one has

$\displaystyle\langle\hat{c}^{2}\rangle$ $\displaystyle=\langle
4{\hbox{1\kern-1.49994pt\vrule
width=0.8pt,depth=-0.5pt}}_{2}\rangle+\langle[Q,R]\otimes[S,T]\rangle$ (196)
$\displaystyle=\langle 4{\hbox{1\kern-1.49994pt\vrule
width=0.8pt,depth=-0.5pt}}_{2}\rangle+\langle[Q,R]\rangle\langle[S,T]\rangle,$
(197)

given that $\langle a+b\rangle=\langle a\rangle+\langle b\rangle$. Cauchy-
Schwarz inequality then yields

$\displaystyle|\langle[Q,R]\rangle|\leq 2\sqrt{\langle Q^{2}\rangle\langle
R^{2}\rangle}=2.$ (198)

Moreover,

$\displaystyle\langle\hat{c}^{2}\rangle-\langle\hat{c}\rangle^{2}\geq
0\implies\langle\hat{c}\rangle\leq\sqrt{\langle\hat{c}^{2}\rangle},$ (199)

and since $\langle 4{\hbox{1\kern-1.49994pt\vrule
width=0.8pt,depth=-0.5pt}}_{2}\rangle=\langle\psi|4{\hbox{1\kern-1.49994pt\vrule
width=0.8pt,depth=-0.5pt}}_{2}|\psi\rangle=4\langle\psi|\psi\rangle=4|\psi|^{2}=4$
one finally has:

$\displaystyle\langle\hat{c}\rangle\leq\sqrt{4+2\cdot 2}=2\sqrt{2}.$ (200)

This result is known as the Tsirelson bound that violates the CHSH inequality.
It is worth mentioning that within the density operator formalism, any
entangled pure state violates Bell’s inequalities, with the exception of mixed
entangled states admitting a local model. In the case of entangled pure
states, the violation of Bell’s inequalities is solely related to the presence
of nonlocality.

####  4.3.3 Composite systems with entangled states

As already mentioned, an entangled state is a special kind of quantum
superposition state, so it is reasonable to write the global entangled state
vector of a composite system using the superposition principle. Let
$\mathcal{H}_{d}=\otimes_{s=1}^{N}\mathcal{H}_{s}^{d_{s}}=\mathcal{H}_{1}^{d_{1}}\otimes\mathcal{H}_{2}^{d_{2}}\otimes\cdots\otimes\mathcal{H}_{N}^{d_{N}}$
denote the composite Hilbert space of a multipartite quantum system with $N$
qudits, where qudit $s$ written in an orthonormal basis set
$\\{|j_{s}\rangle\\}_{j_{s}=1}^{d_{s}}$. From the superposition principle, the
$N$-qudit global entangled state vector of the multipartite system can be
written in the form:

$\displaystyle|\psi_{1\cdots
N}^{entg}\rangle_{Nqd}=\sum_{j_{1},\cdots,j_{N}=1}^{d_{1}\cdots
d_{N}}c_{j_{1}\cdots
j_{N}}|j_{1}\rangle_{1}\otimes\cdots\otimes|j_{N}\rangle_{N}.$ (201)

In general, a quantum system represented by a state vector $|\psi\rangle$ is
entangled if and only if it is not a product state (separable state) of its
subsystems, i.e, if it cannot be written in the form of Eq. (133).
Consequently, an entangled state is one to which it is not possible to assign
a single coherent state vector to any one of its subsystems, i.e, there is an
intrinsic classical ignorance regarding the state preparation of its
subsystems. This inability to write the subsystem of a composite system as a
state vector spurred up one of the motivations that led to the development of
a quantity known as the density operator (see section 5).

Consider the particular case of a bipartite system with some set of
orthonormal basis $\\{|j_{1}\rangle\\}_{j_{1}=1}^{d_{1}}$ $\in\mathcal{H}_{1}$
of subsystem $s$$=$$1$, and $\\{|j_{2}\rangle\\}_{j_{2}=1}^{d_{2}}$
$\in\mathcal{H}_{2}$ of subsystem $s$$=$$2$. From Eq. (201), one can always
write the bipartite global state $|\psi_{12}\rangle\in$
$\mathcal{H}_{1}\otimes\mathcal{H}_{2}$ of the composite system in the form:

$\displaystyle|\psi_{12}^{entg}\rangle_{2qd}=\sum_{j_{1}=1}^{d_{1}}\sum_{j_{2}=1}^{d_{2}}c_{j_{1}j_{2}}|j_{1}\rangle_{1}\otimes|j_{2}\rangle_{2},$
(202)

with $c_{j_{1}j_{2}}=\langle j_{1}|\otimes\langle j_{2}|\psi_{12}\rangle$ (we
momentarily drop the subscripts to avoid notation overload). The above global
state is separable if and only if the state of its subsystems can be written
as state vectors of the form:

$\displaystyle|\psi_{1}\rangle_{1qd}=\sum_{j_{1}=1}^{d_{1}}c_{j_{1}}|j_{1}\rangle_{1},$
(203)

and

$\displaystyle|\psi_{2}\rangle_{1qd}=\sum_{j_{2}=1}^{d_{2}}c_{j_{2}}|j_{2}\rangle_{2},$
(204)

with $c_{j_{1}j_{2}}=c_{j_{1}}c_{j_{2}}$. Otherwise, the global state is
entangled (not separable) and one would not be able to write it as a tensor
product of the state vectors of its subsystems such as in Eq. (133). Special
bipartite entangled states of $2$-level subsystems ($d_{1}=d_{2}=2$) are the
following four maximally entangled two-qubit Bell states [16][66], commonly
known as EPR states. In the canonical basis and within the state vector
formalism, they write:

$\displaystyle|\psi_{12}^{Bell\pm}\rangle$
$\displaystyle\doteq\frac{1}{\sqrt{2}}\bigg{(}|0\rangle_{1}\otimes|1\rangle_{2}\pm|1\rangle_{1}\otimes|0\rangle_{2}\bigg{)}$
(205)
$\displaystyle\doteq\frac{1}{\sqrt{2}}\bigg{(}|01\rangle_{12}\pm|10\rangle_{12}\bigg{)},$
(206)

and

$\displaystyle|\phi_{12}^{Bell\pm}\rangle$
$\displaystyle\doteq\frac{1}{\sqrt{2}}\bigg{(}|0\rangle_{1}\otimes|0\rangle_{2}\pm|1\rangle_{1}\otimes|1\rangle_{2}\bigg{)}$
(207)
$\displaystyle\doteq\frac{1}{\sqrt{2}}\bigg{(}|00\rangle_{12}\pm|11\rangle_{12}\bigg{)}.$
(208)

Here, $|\psi_{12}^{Bell-}\rangle$ is termed a singlet state, and
$|\psi_{12}^{Bell+}\rangle$ a triplet state. Explicitly, for the singlet
state:

$\displaystyle|\psi_{12}^{Bell-}\rangle$
$\displaystyle\doteq\frac{1}{\sqrt{2}}\Bigg{\\{}\begin{pmatrix}1\\\
0\end{pmatrix}_{1}\otimes\begin{pmatrix}0\\\
1\end{pmatrix}_{2}-\begin{pmatrix}0\\\
1\end{pmatrix}_{1}\otimes\begin{pmatrix}1\\\ 0\end{pmatrix}_{2}\Bigg{\\}}$
(209) $\displaystyle=\frac{1}{\sqrt{8}}\Bigg{\\{}\begin{pmatrix}1\\\
i\end{pmatrix}_{1}\otimes\begin{pmatrix}i\\\
1\end{pmatrix}_{2}-\begin{pmatrix}i\\\
1\end{pmatrix}_{1}\otimes\begin{pmatrix}1\\\ i\end{pmatrix}_{2}\Bigg{\\}}.$
(210)

Any attempt of writing an entangled state as a tensor product of well defined
coherent state vectors yields an absurd. To demonstrate that we equate the
Bell state $|\phi_{12}^{Bell+}\rangle$ to a tensor product of state vectors
$|\psi_{1}\rangle\doteq\alpha_{1}|0\rangle+\beta_{1}|1\rangle$ and
$|\psi_{2}\rangle\doteq\alpha_{2}|0\rangle+\beta_{2}|1\rangle$ from subsystems
$s=1$ and $s=2$, respectively, as follows:

$\displaystyle|\psi_{12}^{sep}\rangle$
$\displaystyle=|\psi_{1}\rangle\otimes|\psi_{2}\rangle$ (211)
$\displaystyle=\alpha_{1}\alpha_{2}|00\rangle+\alpha_{1}\beta_{2}|01\rangle$
(212)
$\displaystyle+\beta_{1}\alpha_{2}|10\rangle+\beta_{1}\beta_{2}|11\rangle$
(213) $\displaystyle=\frac{1}{\sqrt{2}}\left(|00\rangle+|11\rangle\right)$
(214)
$\displaystyle\implies\alpha_{1}\alpha_{2}=\beta_{1}\beta_{2}=\frac{1}{\sqrt{2}}$
(215) $\displaystyle\implies\alpha_{1}\beta_{2}\wedge\beta_{1}\alpha_{2}\neq
0,$ (216)

which is an absurd since the last coefficients must vanish. Therefore, the
global state of the composite system is said to be entangled. It is worth
mentioning that, for Bell entangled states, while the state preparation of
their subsystems is unknown, the global state of the entangled system is
completely known. Global states that are completely known are commonly
referred to as pure states (whether entangled or not). In section section 5,
within the density operator formalism, we demonstrate the criteria used to
decide whether a generic qudit state is pure or mixed. In subsection 10.1, we
explore the characteristic of maximally entangled pure states whose subsystems
are described by maximally mixed reduced density operators.

Moreover, it is possible to go from one Bell state to another via simple local
operations, which do not change the degree of entanglement. For instance:
$\sigma_{x}|\phi_{12}^{Bell+}\rangle=|\psi_{12}^{Bell+}\rangle$, where
$\sigma_{x}$ is the Pauli-x matrix defined in Eq. (354). However, there are
multipartite entangled states that, in general, cannot be transformed into one
another by stochastic local operations and classical communications (SLOCC)
[67], with the exception of tripartite states ($N$$=$$3$ qudits) in the
asymptotic regime [68] such as the 3-qubit Greenberger-Horne-Zeilinger
(GHZ)-like [69] and the W-like [70] multipartite maximally entangled states:

$\displaystyle|GHZ\rangle_{3qb}$
$\displaystyle\doteq\frac{1}{\sqrt{2}}\left(|000\rangle+|111\rangle\right),$
(217) $\displaystyle|W\rangle_{3qb}$
$\displaystyle\doteq\frac{1}{\sqrt{3}}\left(|001\rangle+|010\rangle+|100\rangle\right).$
(218)

As a word of caution, entanglement cannot be created via LOCC acting on its
subsystems, it can either be preserved or destroyed.

##  5 The Density operator formalism

One can rephrase everything that has been said so far in the formalism of the
density operator (a.k.a density matrix) developed by John Neumann [58] and,
independently, by Lev Landau [71], in 1927. The density operator formalism has
two main motivations: first, to describe a quantum system whose state is not
completely known, i.e, when there is classical ignorance regarding the state
preparation; and second, to describe entangled systems since their underlying
subsystems cannot be assigned to a state vector, i.e, there is classical
ignorance about the preparation of the subsystems. One approach considers an
ensemble $\\{p_{j},|\psi_{j}\rangle\\}_{j=1}^{n}$ corresponding to a
statistical mixture of $n$ different state vectors denoted $|\psi_{j}\rangle$
with probability distribution $\\{p_{j}\\}$, meaning that the system can be
found in one of the aforementioned state vectors with probability $p_{j}$. In
either case, the global state of the system is fully described by a mixed
density operator defined as:

$\displaystyle\rho\doteq\sum_{j=1}^{n}p_{j}|\psi_{j}\rangle\langle\psi_{j}|.$
(219)

On the other hand, whenever the global state of the system is completely
known, it is a unique state vector $|\psi\rangle$ with the following
corresponding pure density operator:

$\displaystyle\rho\doteq|\psi\rangle\langle\psi|.$ (220)

The logical inference is that a pure density operator always corresponds to a
state vector. It is worth mentioning that a system whose global state is
represented by a pure or a mixed density operator may or may not be entangled,
since the nature of entanglement is related to the degree of purity (classical
ignorance) of the underlying subsystems, rather than the degree of purity of
the global state. For instance, whenever the subsystems of a multipartite
system are each described by a pure density operator, it is always possible to
assign to each one of them a single state vector, hence the global state of
the system is separable and not entangled. The alluded Bell states are
examples of global pure states that are maximally entangled for its subsystems
are each in a maximally mixed state. There are also global pure states that
are partially entangled, and global mixed states with no entanglement at all
(see section 10). In summary: the degree of purity of the global state is not
a metric of entanglement, however, the degree of purity of its subsystems can
be regarded as such a metric (see subsection 10.1).

In general, any density operator $\rho$ describing a physical observable
satisfies the following conditions:

  1. 1.

Unit trace: $tr(\rho)=1$.

  2. 2.

Positiveness: $\langle\phi|\rho|\phi\rangle\geq 0$.

  3. 3.

Hermiticity: $\rho=\rho^{\dagger}$.

Where $|\phi\rangle$ is any arbitrary state vector, and the unit trace
condition (a.k.a normalization condition) is the requirement of the law of
total probability. It is straightforward to verify conditions no. (1) and no.
(2) for the generic mixed density operator defined in Eq. (219). Let
$|\psi_{j}\rangle$ be any orthonormal basis of a $d$-dimensional Hilbert
space. Positiveness of the density operator follows directly from

$\displaystyle\langle\phi|\rho|\phi\rangle$
$\displaystyle=\langle\phi|\left(\sum_{j=1}^{d}p_{j}|\psi_{j}\rangle\langle\psi_{j}|\right)|\phi\rangle$
(221)
$\displaystyle=\sum_{j=1}^{d}p_{j}\langle\phi|\psi_{j}\rangle\langle\psi_{j}|\phi\rangle$
(222) $\displaystyle=\sum_{j=1}^{d}p_{j}|\langle\phi|\psi_{j}\rangle|^{2}\geq
0,$ (223)

given that the probabilities $p_{j}$ are non-negative real numbers. Then,
making use of trace identities no. (5) and no. (6) for the trace of a scalar
and trace of sums, respectively, one gets

$\displaystyle tr(\rho)$
$\displaystyle=tr\left(\sum_{j=1}^{d}p_{j}|\psi_{j}\rangle\langle\psi_{j}|\right)$
(224)
$\displaystyle=\sum_{j=1}^{d}tr\left(p_{j}|\psi_{j}\rangle\langle\psi_{j}|\right)$
(225)
$\displaystyle=\sum_{j=1}^{d}p_{j}tr\left(\langle\psi_{j}|\psi_{j}\rangle\right)$
(226) $\displaystyle=\sum_{j=1}^{d}p_{j}tr(\delta_{jj})$ (227)
$\displaystyle=\sum_{j=1}^{d}p_{j}=1,$ (228)

given the law of total probability and the orthogonality condition of the
basis set: $\langle\psi_{j}|\psi_{j}\rangle=\delta_{jj}=1$. The same result
could have been obtained from the definition of the trace function stated in
Eq. (45).

A bound can then be derived for $tr(\rho^{2})$ as follows:

$\displaystyle tr(\rho^{2})$
$\displaystyle=\sum_{j=1}^{d}\langle\psi_{j}|\rho^{2}|\psi_{j}\rangle,$ (229)

which after introducing the completeness relation stated in Eq. (20) becomes

$\displaystyle tr(\rho^{2})$
$\displaystyle=\sum_{j=1}^{d}\langle\psi_{j}|\rho\left(\sum_{k=1}^{d}|\psi_{k}\rangle\langle\psi_{k}|\right)\rho|\psi_{j}\rangle$
(230)
$\displaystyle=\sum_{j,k=1}^{d}\langle\psi_{j}|\rho|\psi_{k}\rangle\langle\psi_{k}|\rho|\psi_{j}\rangle,$
(231)

then, on replacing $\rho$ with Eq. (219) for a mixed density operator, one
gets

$\displaystyle tr(\rho^{2})$
$\displaystyle=\sum_{j,k=1}^{d}\bigg{[}\langle\psi_{j}|\left(\sum_{l=1}^{d}p_{l}|\phi_{l}\rangle\langle\phi_{l}|\right)|\psi_{k}\rangle\bigg{]}$
(232)
$\displaystyle\cdot\bigg{[}\langle\psi_{k}|\left(\sum_{m=1}^{d}p_{m}|\phi_{m}\rangle\langle\phi_{m}|\right)|\psi_{j}\rangle\bigg{]}$
(233)
$\displaystyle=\sum_{l,m,j,k=1}^{d}\bigg{\\{}p_{l}p_{m}\langle\psi_{j}|\phi_{l}\rangle\langle\phi_{l}|\psi_{k}\rangle$
(234)
$\displaystyle\cdot\langle\psi_{k}|\phi_{m}\rangle\langle\phi_{m}|\psi_{j}\rangle\bigg{\\}},$
(235)

then, shifting the summation in $k$ for the completeness relation
$\sum_{k=1}^{d}|\psi_{k}\rangle\langle\psi_{k}|={\hbox{1\kern-1.49994pt\vrule
width=0.8pt,depth=-0.5pt}}_{d}$ to reappear, and noting that
$\\{|\phi_{l}\rangle,|\phi_{m}\rangle\\}$ form a orthonormal basis set
($\langle\phi_{l}|\phi_{m}\rangle=\delta_{lm}$), leads to

$\displaystyle tr(\rho^{2})$
$\displaystyle=\sum_{l,m,j=1}^{d}\bigg{\\{}p_{l}p_{m}\langle\psi_{j}|\phi_{l}\rangle$
(236)
$\displaystyle\cdot\langle\phi_{l}|\bigg{(}\sum_{k=1}^{d}|\psi_{k}\rangle\langle\psi_{k}|\bigg{)}|\phi_{m}\rangle\langle\phi_{m}|\psi_{j}\rangle\bigg{\\}}$
(237)
$\displaystyle=\sum_{l,m,j=1}^{d}p_{l}p_{m}\langle\psi_{j}|\phi_{l}\rangle\delta_{lm}\langle\phi_{m}|\psi_{j}\rangle$
(238)
$\displaystyle=\sum_{l,j=1}^{d}p_{l}^{2}\langle\psi_{j}|\phi_{l}\rangle\langle\phi_{l}|\psi_{j}\rangle$
(239)
$\displaystyle=\sum_{l,j=1}^{d}p_{l}^{2}\langle\phi_{l}|\psi_{j}\rangle\langle\psi_{j}|\phi_{l}\rangle$
(240)
$\displaystyle=\sum_{l=1}^{d}p_{l}^{2}\langle\phi_{l}|\left(\sum_{j}|\psi_{j}\rangle\langle\psi_{j}|\right)|\phi_{l}\rangle$
(241) $\displaystyle=\sum_{l=1}^{d}p_{l}^{2}\langle\phi_{l}|\phi_{l}\rangle$
(242)
$\displaystyle=\sum_{l=1}^{d}p_{l}^{2}\leq\left(\sum_{l=1}^{d}p_{l}\right)^{2}=1.$
(243)

From that, one can devise the following rule-of-thumb for deciding whether a
density operator is mixed or pure:

  * •

Mixed density operator:

$\displaystyle tr(\rho^{\dagger}\rho)=tr(\rho^{2})<1,$ (244)

and $\rho^{2}\neq\rho$.

  * •

Pure density operator:

$\displaystyle tr(\rho^{\dagger}\rho)=tr(\rho^{2})=1,$ (245)

and $\rho^{2}=\rho$ (idempotent).

Furthermore, positiveness condition implies a spectral decomposition (see
Theorem (1)), so that in the canonical basis the density operator reads:

$\displaystyle\rho=\sum_{j=1}^{d}\lambda_{j}|j\rangle\langle j|,$ (246)

and

$\displaystyle\rho^{2}=\sum_{j=1}^{d}\lambda_{j}^{2}|j\rangle\langle j|,$
(247)

where the $\lambda_{j}$’s are the corresponding eigenvalues of the density
operator. Then, unit trace condition implies

$\displaystyle tr(\rho)$ $\displaystyle=\sum_{j=1}^{d}\lambda_{j}=1,$ (248)
$\displaystyle tr(\rho^{2})$ $\displaystyle=\sum_{j=1}^{d}\lambda_{j}^{2}.$
(249)

Thereafter, if only one eigenvalue is non-zero, whose value is equal to one,
the matrix represents a pure density operator. Conversely, a diagonalized and
normalized (unit trace) pure density operator has only one non-zero
eigenvalue, which is always equal to one. To see the converse, let
$\\{|\xi_{j}\rangle\\}$ denote a set that diagonalize $\rho$, then:

$\displaystyle\langle\xi_{j}|\rho^{2}|\xi_{k}\rangle=\langle\xi_{j}|\rho|\xi_{k}\rangle=\lambda_{j}\delta_{jk},$
(250)

and

$\displaystyle\langle\xi_{j}|\rho^{2}|\xi_{k}\rangle$
$\displaystyle=\sum_{l}\langle\xi_{j}|\rho|\xi_{l}\rangle\langle\xi_{l}|\rho|\xi_{k}\rangle,$
(251)
$\displaystyle=\lambda_{j}\delta_{jl}\lambda_{l}\delta_{lk}=\lambda_{j}^{2}\delta_{jk},$
(252)

so that

$\displaystyle\lambda_{j}^{2}\delta_{jk}=\lambda_{j}\delta_{jk}\implies\lambda_{j}(\lambda_{j}-1)\delta_{jk}=0.$
(253)

It is straightforward seeing that whenever $j=k$ ($\delta_{jj}=\delta_{kk}=1$)
the eigenvalue $\lambda_{j}$ is either $0$ or $1$, and since the matrix is now
diagonal it is sufficient to prove the converse statement.

###  5.1 Postulates within the density operator formalism

Within the framework of the density operator it is possible to reformulate the
postulates of quantum mechanics stated in subsection 3.2, as follows.

  * •

Postulate 1: Any isolated physical system (close system) is described within a
complex vector space $\mathcal{H}$ (Hilbert space) endowed with an inner
product $(\cdot,\cdot)\doteq\langle\cdot|\cdot\rangle$, and known as the state
space of the system. The system is completely described by a Positive and unit
trace operator $\rho=\sum_{j}p_{j}\rho_{j}$, known as the density operator,
which is a statistical mixture of states
$\rho_{j}=\sum_{k}p_{jk}|\psi_{jk}\rangle\langle\psi_{jk}|$ with probability
$p_{j}$ from some ensemble $\\{p_{jk},\psi_{jk}\\}$.

  * •

Postulate 2: In the Schrödinger picture formalism, a closed quantum system
prepared in the initial state $|\psi_{j,t}\rangle$ evolves according to a
unitary transformation $\hat{U}_{t}|\psi_{j,t}\rangle$, so that the initial
density operator $\rho=\sum_{j}p_{j}|\psi_{j,t}\rangle\langle\psi_{j,t}|$
evolves to

$\displaystyle\rho_{t}$
$\displaystyle=\sum_{j}p_{j}(\hat{U}_{t}|\psi_{j,t}\rangle)(\langle\psi_{j,t}|\hat{U}_{t}^{\dagger})$
(254)
$\displaystyle=\hat{U}_{t}(\sum_{j}p_{j}|\psi_{j,t}\rangle\langle\psi_{j,t}|)\hat{U}_{t}^{\dagger}$
(255) $\displaystyle=\hat{U}_{t}\rho\hat{U}_{t}^{\dagger},$ (256)

given that
$(\hat{U}_{t}|\psi_{j,t}\rangle)^{\dagger}=\langle\psi_{j,t}|\hat{U}_{t}^{\dagger}$.
This equation is valid whenever the Hamiltonian $\hat{H}$ of the system is
time-independent, and the unitary operator is
$\hat{U}=e^{\frac{-i\hat{H}t}{\hbar}}$.

When the Hamiltonian is time-dependent, the continuous time evolution of this
system is given by the Von Neumann equation (a.k.a Liouville–von Neumann
equation), as follows:

$\displaystyle\frac{\partial\rho}{\partial t}$
$\displaystyle=\frac{\partial}{\partial
t}\left(\sum_{j}p_{j}|\psi_{j,t}\rangle\langle\psi_{j,t}|\right)$ (257)
$\displaystyle=\sum_{j}p_{j}\Bigg{\\{}\left(\frac{\partial}{\partial
t}|\psi_{j,t}\rangle\right)\langle\psi_{j,t}|$ (258)
$\displaystyle+|\psi_{j,t}\rangle\left(\frac{\partial}{\partial
t}\langle\psi_{j,t}|\right)\Bigg{\\}}$ (259)
$\displaystyle=\sum_{j}p_{j}\left\\{\frac{\hat{H}}{i\hbar}|\psi_{j,t}\rangle\langle\psi_{j,t}|-|\psi_{j,t}\rangle\langle\psi_{j,t}|\frac{\hat{H}}{i\hbar}\right\\}$
(260) $\displaystyle=\frac{1}{i\hbar}\left[\hat{H},\rho\right],$ (261)

which is true given the product rule for derivatives and the fact that

$\displaystyle\frac{\partial}{\partial
t}|\psi_{j,t}\rangle=\frac{\hat{H}}{i\hbar}|\psi_{j,t}\rangle,$ (262)

and

$\displaystyle\frac{\partial}{\partial
t}\langle\psi_{j,t}|=-\frac{\hat{H}}{i\hbar}\langle\psi_{j,t}|.$ (263)

  * •

Postulate 3: Measurements of a quantum system are described by a collection
{$M_{m}$} of measurement operators acting on the state space of the system
with $m$ possible measurement outcomes. The measurement operators satisfy the
completeness relation

$\sum_{m}M_{m}^{\dagger}M_{m}=I.$ (264)

By the law of total probability, the probability of obtaining an outcome
$o_{m}$ associated to an observable $\hat{\mathcal{O}}$ of a quantum system
prepared in the state $|\psi_{j}\rangle$ with corresponding density operator
$\rho$ is

$\displaystyle Pr(o_{m}|\rho)$
$\displaystyle=\sum_{j}p_{j}\langle\psi_{j}|M^{\dagger}_{m}M_{m}|\psi_{j}\rangle,$
(265)

which after applying trace identities no. (2) and no. (6) for the trace of a
scalar and trace of sums, respectively, becomes

$\displaystyle Pr(o_{m}|\rho)$
$\displaystyle=\sum_{j}p_{j}tr(M^{\dagger}_{m}M_{m}|\psi_{j}\rangle\langle\psi_{j}|)$
(266)
$\displaystyle=tr\left(M^{\dagger}_{m}M_{m}\sum_{j}p_{j}|\psi_{j}\rangle\langle\psi_{j}|\right)$
(267) $\displaystyle=tr(M_{m}^{\dagger}M_{m}\rho).$ (268)

And the state of the system after the measurement shall collapse to

$\displaystyle|\psi_{j}^{o_{m}}\rangle$
$\displaystyle=\frac{M_{m}|\psi\rangle}{\sqrt{\langle\psi_{j}|M^{\dagger}_{m}M_{m}|\psi_{j}\rangle}},$
(269)

while the corresponding new density operator becomes:

$\displaystyle\rho_{o_{m}}$
$\displaystyle=\sum_{j}\frac{p_{j}\langle\psi_{j}|M^{\dagger}_{m}M_{m}|\psi_{j}\rangle}{P_{r}(o_{m})}|\psi_{j}^{o_{m}}\rangle\langle\psi_{j}^{o_{m}}|$
(270)
$\displaystyle=\sum_{j}\frac{p_{j}\langle\psi_{j}|M^{\dagger}_{m}M_{m}|\psi_{j}\rangle}{P_{r}(o_{m})}\frac{M_{m}|\psi_{j}\rangle\langle\psi_{j}|M_{m}^{\dagger}}{\langle\psi_{j}|M^{\dagger}_{m}M_{m}|\psi_{j}\rangle}$
(271)
$\displaystyle=\sum_{j}\frac{p_{j}}{P_{r}(o_{m})}M_{m}|\psi_{j}\rangle\langle\psi_{j}|M_{m}^{\dagger}$
(272)
$\displaystyle=\sum_{j}p_{j}\frac{M_{m}|\psi_{j}\rangle\langle\psi_{j}|M_{m}^{\dagger}}{tr(M_{m}^{\dagger}M_{m}\rho)}$
(273)
$\displaystyle=\frac{M_{m}\left(\sum_{j}p_{j}|\psi_{j}\rangle\langle\psi_{j}|\right)M_{m}^{\dagger}}{tr(M_{m}^{\dagger}M_{m}\rho)}$
(274) $\displaystyle=\frac{M_{m}\rho
M_{m}^{\dagger}}{tr(M_{m}^{\dagger}M_{m}\rho)}.$ (275)

Now, consider an observable $\hat{A}$ with an orthonormal basis set
$\\{a_{j}\\}$, and an observable $\hat{B}$ with an orthonormal basis set
$\\{b_{k}\\}$. The projective measurement (see subsubsection 3.2.4) for
obtaining an outcome $b_{k}$ of the observable $\hat{B}$ conditioned to a
measurement with outcome $a_{j}$ of the observable $\hat{A}$ in an statistical
sub-ensemble given by the mixed density operator $\rho$ with probabilities
$p_{j}$ shall be:

$\displaystyle Pr(b_{k}|\\{p_{j},|a_{j}\rangle\\})$
$\displaystyle=\sum_{j}Pr(b_{k}|a_{j}\rangle)p_{j}$ (276)
$\displaystyle=\sum_{j}|\langle b_{k}|a_{j}\rangle|^{2}p_{j}$ (277)
$\displaystyle=\sum_{j}\langle b_{k}|a_{j}\rangle\langle a_{j}|b_{k}\rangle
p_{j}$ (278) $\displaystyle=\langle
b_{k}\left\\{\sum_{j}p_{j}|a_{j}\rangle\langle a_{j}|\right\\}|b_{k}\rangle$
(279) $\displaystyle=\langle b_{k}|\rho|b_{k}\rangle,$ (280)

and with trace identity no. (2) of the trace of a scalar, we then have

$\displaystyle Pr(b_{k}|\\{p_{j},|a_{j}\rangle\\})$ $\displaystyle=tr(\langle
b_{k}|\rho|b_{k}\rangle)$ (281) $\displaystyle=tr(\rho|b_{k}\rangle\langle
b_{k}|).$ (282)

Finally, the expectation value of some arbitrary observable
$\hat{\mathcal{O}}$ will be given as

$\displaystyle\langle\hat{\mathcal{O}}\rangle_{\rho}$
$\displaystyle=\sum_{j}p_{j}\langle\psi_{j}|\hat{\mathcal{O}}|\psi_{j}\rangle$
(283)
$\displaystyle=\sum_{j}p_{j}tr\left(\langle\psi_{j}|\hat{\mathcal{O}}|\psi_{j}\rangle\right)$
(284)
$\displaystyle=\sum_{j}p_{j}tr\left(\hat{\mathcal{O}}|\psi_{j}\rangle\langle\psi_{j}|\right)$
(285)
$\displaystyle=tr\left(\hat{\mathcal{O}}\sum_{j}p_{j}|\psi_{j}\rangle\langle\psi_{j}|\right)$
(286) $\displaystyle=tr(\rho\hat{\mathcal{O}}).$ (287)

  * •

Postulate 4: For $N$ arbitrary quantum physical systems, the composite state
space of the system is given by the Kronecker product of the state space of
its subsystems: $\mathcal{H}_{12\cdots
N}=\mathcal{H}_{1}\otimes\mathcal{H}_{2}\otimes\cdots\otimes\mathcal{H}_{N}$.
And the global state of the composite system will be
$\rho_{\otimes^{N}}=\rho_{1}\otimes\rho_{2}\otimes\cdot\cdot\cdot\otimes\rho_{N}$.

##  6 The partial trace

The partial trace function [44][72] is one of the many functions used to
extract information about a composite quantum system. For instance, in the
context of local operations it is expedient to associate to each subsystem a
reduced density operator (a.k.a reduced density matrix) via the partial trace
function. In subsection 10.1, we demonstrate the use of the partial trace to
infer the degree of purity of subsystems from a bipartite system. Here, we
derive the analytical expression of the reduced density operator given the
definition of the partial trace for bipartite systems. The computation of the
partial trace for general multipartite systems requires a numerical approach,
and is beyond the scope of this review.

Let $\mathcal{H}_{ab}\doteq\mathcal{H}_{a}\otimes\mathcal{H}_{b}$ denote the
composite Hilbert space of a bipartite system, and $\\{b_{j}\\}_{j=1}^{d_{b}}$
denote any $d_{b}$-dimensional orthonormal basis set for $\mathcal{H}_{b}$.
The partial trace over subsystem $b$ is a map
$tr_{b}:\mathcal{L}(\mathcal{H}_{ab})\rightarrow\mathcal{L}(\mathcal{H}_{a})$
defined as [73]:

$\displaystyle
tr_{b}(\mathcal{O})\doteq\sum_{j=1}^{d_{b}}({\hbox{1\kern-1.49994pt\vrule
width=0.8pt,depth=-0.5pt}}_{a}\otimes\langle
b_{j}|)\mathcal{O}({\hbox{1\kern-1.49994pt\vrule
width=0.8pt,depth=-0.5pt}}_{a}\otimes|b_{j}|\rangle).$ (288)

Conversely, the partial trace over subsytem $a$ is a map
$tr_{a}:\mathcal{L}(\mathcal{H}_{ab})\rightarrow\mathcal{L}(\mathcal{H}_{b})$.
Furthermore, the partial trace is a function with the property:

$\displaystyle tr_{a}(\hat{A}f(\hat{O}))$
$\displaystyle=tr_{ab}((\hat{A}\otimes{\hbox{1\kern-1.49994pt\vrule
width=0.8pt,depth=-0.5pt}}_{b})\hat{O}),$ (289)

for generic linear operators $\hat{A}\in\mathcal{L}(\mathcal{H}_{a})$ and
$\hat{O}\in\mathcal{L}(\mathcal{H}_{ab})$.

One ubiquitous application of the partial trace is to compute the reduced
density matrix (a.k.a partial state or quantum marginal) of a subsystem. Let
$\rho_{ab}\doteq\rho_{a}\otimes\rho_{b}\in\mathcal{H}_{ab}$ denote a density
operator of a bipartite system. The partial trace of $\rho_{ab}$ over
subsystem $b$ yields a reduced density operator
$\rho_{a}\in\mathcal{L}(\mathcal{H}_{a})$ of subsystem $a$. This assertion can
be verified according to (288), as follows:

$\displaystyle tr_{b}(\rho_{ab})$
$\displaystyle=\sum_{j=1}^{d_{b}}({\hbox{1\kern-1.49994pt\vrule
width=0.8pt,depth=-0.5pt}}_{a}\otimes\langle
b_{j}|)(\rho_{a}\otimes\rho_{b})({\hbox{1\kern-1.49994pt\vrule
width=0.8pt,depth=-0.5pt}}_{a}\otimes|b_{j}|\rangle)$ (290)
$\displaystyle=\sum_{j=1}^{d_{b}}({\hbox{1\kern-1.49994pt\vrule
width=0.8pt,depth=-0.5pt}}_{a}\rho_{a}\otimes\langle
b_{j}|\rho_{b})({\hbox{1\kern-1.49994pt\vrule
width=0.8pt,depth=-0.5pt}}_{a}\otimes|b_{j}|\rangle)$ (291)
$\displaystyle=\sum_{j=1}^{d_{b}}{\hbox{1\kern-1.49994pt\vrule
width=0.8pt,depth=-0.5pt}}_{a}\rho_{a}{\hbox{1\kern-1.49994pt\vrule
width=0.8pt,depth=-0.5pt}}_{a}\otimes\langle b_{j}|\rho_{b}|b_{j}\rangle$
(292) $\displaystyle=\sum_{j=1}^{d_{b}}\rho_{a}\otimes\langle
b_{j}|\rho_{b}|b_{j}\rangle,$ (293)

and making use of trace identities no. (5) and no. (7), one gets

$\displaystyle tr_{b}(\rho_{ab})$
$\displaystyle=\sum_{j=1}^{d_{b}}\rho_{a}\otimes tr(\langle
b_{j}|\rho_{b}|b_{j}\rangle)$ (294)
$\displaystyle=\sum_{j=1}^{d_{b}}\rho_{a}\otimes tr(|b_{j}\rangle\langle
b_{j}|\rho_{b})$ (295) $\displaystyle=\rho_{a}\otimes
tr\left(\left(\sum_{j=1}^{d_{b}}|b_{j}\rangle\langle
b_{j}|\right)\rho_{b}\right)$ (296) $\displaystyle=\rho_{a}\otimes
tr(\rho_{b})=\rho_{a}\otimes 1=\rho_{a},$ (297)

given the completeness relation, and the unit trace condition
$tr(\rho_{b})=1$. Another equivalent definition of the partial trace for
bipartite systems is [44]:

$\displaystyle tr_{b}(|a_{j}\rangle\langle a_{k}|\otimes|b_{l}\rangle\langle
b_{m}|)$ $\displaystyle\doteq|a_{j}\rangle\langle a_{k}|\otimes
tr_{b}(|b_{l}\rangle\langle b_{m}|)$ (298) $\displaystyle=|a_{j}\rangle\langle
a_{k}|\cdot tr(|b_{l}\rangle\langle b_{m}|)$ (299)
$\displaystyle=|a_{j}\rangle\langle a_{k}|\cdot tr(\langle
b_{m}|b_{l}\rangle)$ (300) $\displaystyle=|a_{j}\rangle\langle
a_{k}|\cdot\langle b_{m}|b_{l}\rangle,$ (301)

for generic state vectors $|a_{j}\rangle,|a_{k}\rangle\in\mathcal{H}_{a}$ and
$|b_{l}\rangle,|b_{m}\rangle\in\mathcal{H}_{b}$.

From the definition of the mixed density operator stated in Eq. (219), the
bipartite mixed density operator takes the form

$\displaystyle\rho_{ab}$
$\displaystyle=\sum_{i=1}^{d}p_{i}|\psi_{i}^{ab}\rangle\langle\psi_{i}^{ab}|.$
(302)

The more general expression is obtained by writing the composite state
$|\psi_{ab}\rangle$ in the form of Eq. (202), such that

$\displaystyle\rho_{ab}=\sum_{i}p_{i}\Bigg{\\{}\left(\sum_{j,l}c_{ijl}|a_{j}\rangle\otimes|b_{l}\rangle\right)$
(303) $\displaystyle\cdot\left(\sum_{k,m}c_{ikm}\langle a_{k}|\otimes\langle
b_{m}|\right)\Bigg{\\}}$ (304)
$\displaystyle=\sum_{i,j,k,l,m}p_{i}c_{ijklm}|a_{j}\rangle\langle
a_{k}|\otimes|b_{l}\rangle\langle b_{m}|.$ (305)

Within this new framework, the reduced density operator $\rho_{a}$ becomes:

$\displaystyle\rho_{a}$ $\displaystyle\doteq
tr_{b}(\rho_{ab})=tr_{b}\left(\sum_{i}p_{i}|\psi_{i}^{ab}\rangle\langle\psi_{i}^{ab}|\right)$
(306)
$\displaystyle=tr_{b}\left(\sum_{i,j,k,l,m}p_{i}c_{ijklm}|a_{j}\rangle\langle
a_{k}|\otimes|b_{l}\rangle\langle b_{m}|\right)$ (307)
$\displaystyle=\sum_{i,j,k,l,m}p_{i}c_{ijklm}tr_{b}\Bigg{(}|a_{j}\rangle\langle
a_{k}|\otimes|b_{l}\rangle\langle b_{m}|\Bigg{)}$ (308)
$\displaystyle=\sum_{i,j,k,l,m}p_{i}c_{ijklm}(|a_{j}\rangle\langle
a_{k}|)\otimes tr_{b}(|b_{l}\rangle\langle b_{m}|),$ (309)

which after applying the base independence of the trace function

$\displaystyle tr_{b}(|b_{l}\rangle\langle b_{m}|)$
$\displaystyle=tr_{b}(\langle b_{m}|b_{l}\rangle)$ (310)
$\displaystyle=\langle b_{m}|b_{l}\rangle$ (311) $\displaystyle=\langle
b_{m}|\left(\sum_{k=1}^{d_{b}}|b_{k}\rangle\langle b_{k}|\right)|b_{l}\rangle$
(312) $\displaystyle=\sum_{k=1}^{d_{b}}\langle b_{m}|b_{k}\rangle\langle
b_{k}|b_{l}\rangle$ (313) $\displaystyle=\sum_{k=1}^{d_{b}}\langle
b_{k}|b_{l}\rangle\langle b_{m}|b_{k}\rangle=\delta_{kl}\delta_{mk}$ (314)

one then gets

$\displaystyle\rho_{a}$
$\displaystyle=\sum_{i,j,k,l,m}p_{i}c_{ijklm}|a_{j}\rangle\langle
a_{k}|\delta_{kl}\delta_{mk}$ (315)
$\displaystyle=\sum_{i,j,k,l,m}p_{i}c_{ijklm}|a_{j}\rangle\langle
a_{k}|\delta_{ml}.$ (316)

###  6.1 Measurement in a bipartite system via the partial trace function

A measurement operator in a composite Hilbert space
$\mathcal{H}_{ab}\doteq\mathcal{H}_{a}\otimes\mathcal{H}_{b}$ of a bipartite
system is a local projection operator defined as $M_{ab}\doteq
M\otimes{\hbox{1\kern-1.49994pt\vrule width=0.8pt,depth=-0.5pt}}_{d}$. For
instance, let $M\doteq M_{a}\doteq|a_{j}\rangle\langle
a_{j}|\in\mathcal{L}(\mathcal{H}_{a})$ denote a projector operator on
subsystem $a$, such that $M_{ab}=M_{a}\otimes{\hbox{1\kern-1.49994pt\vrule
width=0.8pt,depth=-0.5pt}}_{d}\in\mathcal{L}(\mathcal{H}_{ab})$. Recall that
any projector operator satisfies $M_{ab}^{\dagger}M_{ab}=M_{ab}^{2}=M_{ab}$.
From postulate 3 (see subsection 5.1), the corresponding probabilities
associated to the eigenvalue $a_{j}$ of an observable
$\hat{O}_{a}\in\mathcal{L}(\mathcal{H}_{a})$ on subsystem $a$ are given
according to Eq. (268), as follows:

$\displaystyle Pr(a_{j}|\rho_{ab})$
$\displaystyle=tr_{ab}((M_{ab}^{\dagger}M_{ab})\rho_{ab})$ (317)
$\displaystyle=tr_{ab}(M_{ab}\rho_{ab})$ (318)
$\displaystyle=tr_{ab}\left((M_{a}\otimes{\hbox{1\kern-1.49994pt\vrule
width=0.8pt,depth=-0.5pt}}_{d})p_{ab}\right).$ (319)

From Eq. (289) one has $tr_{ab}((M_{a}\otimes{\hbox{1\kern-1.49994pt\vrule
width=0.8pt,depth=-0.5pt}}_{d})\rho_{ab})=tr_{a}(M_{a}(tr_{b}(\rho_{ab}))$,
such that Eq. (319) becomes

$\displaystyle Pr(a_{j}|\rho_{ab})$ $\displaystyle=tr_{a}(M_{a}\rho_{a})$
(320) $\displaystyle=tr_{a}(\rho_{a}M_{a})$ (321)
$\displaystyle=tr_{a}(\rho_{a}(|a_{j}\rangle\langle a_{j}|))$ (322)
$\displaystyle=tr(\rho_{a}(|a_{j}\rangle\langle a_{j}|)),$ (323)

where we have dropped the sub-index of the trace function in Eq. (323) since
the argument only depends on terms of subsystem $a$, such that the partial
trace becomes the ordinary trace defined in subsection 2.12.

By its turn, the expectation value according to Eq. (283) becomes:

$\displaystyle\langle\hat{O}_{a}\rangle_{\rho_{ab}}$
$\displaystyle=\sum_{i}p_{i}\langle\psi_{ab}|(\hat{O}_{a}\otimes{\hbox{1\kern-1.49994pt\vrule
width=0.8pt,depth=-0.5pt}}_{b})|\psi_{ab}\rangle$ (324)
$\displaystyle=\sum_{i}p_{i}tr_{ab}\left(\langle\psi_{ab}|(\hat{O}_{a}\otimes{\hbox{1\kern-1.49994pt\vrule
width=0.8pt,depth=-0.5pt}}_{b})|\psi_{ab}\rangle\right)$ (325)
$\displaystyle=tr_{ab}\left(\sum_{i}p_{i}(\hat{O}_{a}\otimes{\hbox{1\kern-1.49994pt\vrule