# An inverter-chain link implementation of quantum teleportation and
superdense coding

Felix A. Buot,1,2 Roland E.S. Otadoy,1and Xavier L. Bacalla1
1LCFMNN, TCSE Group, Department of Physics,
University of San Carlos, Cebu City, Philippines, and
2C&LB Research Institute, Carmen, Cebu 6005, Philippines

###### Abstract

A new perspective in terms of inverter-chain link (ICL) diagrams of quantum
entanglement faithfully captures the fundamental concept of quantum
teleportation and superdense coding. The ICL may be considered a series of
$\sigma_{x}$ Pauli-matrix operations, where a physical/geometric
representation provides the mysterious link raised by EPR. Here, we employ
discrete phase space and ICL analyses of quantum entanglement as a resource
for quantum teleportation and superdense coding. We underscore the quantum
superposition principle and Hadamard transformation under a local single-qubit
operation. On the fundamental question posed by EPR, our result seems to lend
support to the geometric nature of quantum entanglement. In concluding
remarks, we discuss very briefly a bold conjecture in physics aiming to unify
general relativity with quantum mechanics, namely, ER=EPR.

##  1 Introduction

Quantum entanglement has developed from a mere intellectual curiosity [1] of
the fundamental structure of quantum mechanics111Note that although violation
of Bell’s inequality theorem asserts the nonlocality of quantum mechanics, the
EPR inquiry is still not resolved, i.e., what is still left unanswered is the
mysterious ’link’ between qubits corresponding to our model via a ”see-saw” or
mechanical inverter representation of the Pauli-matrix inverter, $\sigma_{x}$.
to become an important and practical resource for quantum information
processing in the evolving theory of quantum information and ultra-fast
massively-parallel computing. Here, we give a new perspective of quantum
teleportation and superdense coding employing discrete phase-space physics,
superposition principle, and Hadamard transform, coupled with ICL diagrammatic
techniques [2, 3].

We first discuss the discrete phase-space formalism for entangled qubits. In
the discrete phase space formalism, entangled qubits are Bloch function states
[4, 5]. From the general relation, we see that we can construct prime-number
dimensional spaces, besides the entangled basis states, that are connected by
Hadamard transform. In conventional superdense coding quantum circuits, using
controlled-not gates and Hadamard transforms, these other Bloch function
states has been useful. In our ICL diagrammatic techniques, we find no need to
use these other Bloch function states besides the entangled Bell basis states.

##  2 Computational basis or Wannier states

Consider the four Wannier states,

$\displaystyle\left|R_{0}\right\rangle$ $\displaystyle=$
$\displaystyle\left|0\right\rangle_{1}\left|0\right\rangle_{2}$ (1)
$\displaystyle\left|R_{1}\right\rangle$ $\displaystyle=$
$\displaystyle\left|0\right\rangle_{1}\left|1\right\rangle_{2}$ (2)
$\displaystyle\left|R_{2}\right\rangle$ $\displaystyle=$
$\displaystyle\left|1\right\rangle_{1}\left|0\right\rangle_{2}$ (3)
$\displaystyle\left|R_{3}\right\rangle$ $\displaystyle=$
$\displaystyle\left|1\right\rangle_{1}\left|1\right\rangle_{2}$ (4)
$\displaystyle\left|R_{4}\right\rangle$ $\displaystyle=$
$\displaystyle\left|R_{0}\right\rangle=\left|0\right\rangle_{1}\left|0\right\rangle_{2}$
(5)

Equation (5) is the Born-von Karman boundary condition. Because the number of
”site” states are $4$-even, mathematically this do not represent a prime
number which allows for finite field analyses. We seek out, from the general
discrete Fourier transform relation, reduced spaces that are represented by
prime number of sites. Obviously, the number $4$ is a product of two prime
numbers $2$. So we seek $2$-$D$ spaces connected by Hadamard transform, from
the general discrete Fourier transform relations. We have the Fourier
transformed states,

$\left|B_{k}\right\rangle=\frac{1}{\sqrt{N}}\sum_{R_{i},\ i=0...3}e^{ik\cdot
R_{i}}\left|R_{i}\right\rangle$ (6)

and the inverse transformation

$\left|R_{i}\right\rangle=\frac{1}{\sqrt{N}}\sum_{k_{j},\
i=0...3}e^{-ik_{j}\cdot R_{i}}\left|B_{k_{j}}\right\rangle$ (7)

where

$\displaystyle k_{n}$ $\displaystyle=$ $\displaystyle\frac{2\pi}{4}n\text{, \
}n=0,1,...3$ (8) $\displaystyle k_{0}$ $\displaystyle=$ $\displaystyle 0,\
k_{1}=\frac{\pi}{2},\ k_{2}=\pi,\ k_{3}=\frac{3\pi}{2}$ (9) $e^{ik\cdot
R}=\left(\begin{array}[]{cccc}1&1&1&1\\\ 1&i&-1&-i\\\ 1&-1&1&-1\\\
1&-i&-1&i\end{array}\right)$ (10)

From Eq. (6), we have

$\left|B_{k}\right\rangle=\frac{1}{\sqrt{N}}\left(\begin{array}[]{cccc}1&1&1&1\\\
1&i&-1&-i\\\ 1&-1&1&-1\\\
1&-i&-1&i\end{array}\right)\left(\begin{array}[]{c}\left|0\right\rangle\left|0\right\rangle\\\
\left|0\right\rangle\left|1\right\rangle\\\
\left|1\right\rangle\left|0\right\rangle\\\
\left|1\right\rangle\left|1\right\rangle\end{array}\right)$ (11)

For the inverse transformationfunction, we have,

$\displaystyle e^{-ik\cdot R}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}e^{i0\cdot 0}&e^{i0\cdot
1}&e^{i0\cdot 2}&e^{i0\cdot 3}\\\ e^{i0\cdot 1}&e^{-i\frac{\pi}{2}\cdot
1}&e^{-i\pi\cdot 1}&e^{-i\frac{3\pi}{2}\cdot 1}\\\ e^{i0\cdot
2}&e^{-i\frac{\pi}{2}\cdot 2}&e^{-i\pi\cdot 2}&e^{-i\frac{3\pi}{2}\cdot 2}\\\
e^{i0\cdot 3}&e^{-i\frac{\pi}{2}\cdot 3}&e^{-i\pi\cdot
3}&e^{-i\frac{3\pi}{2}\cdot 3}\end{array}\right)$ (16) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}1&1&1&1\\\ 1&-i&-1&i\\\ 1&-1&1&-1\\\
1&i&-1&-i\end{array}\right)$ (21) $\left(e^{ik\cdot R}\right)\left(e^{-ik\cdot
R}\right)=\frac{1}{4}\left(\begin{array}[]{cccc}1&1&1&1\\\ 1&i&-1&-i\\\
1&-1&1&-1\\\ 1&-i&-1&i\end{array}\right)\left(\begin{array}[]{cccc}1&1&1&1\\\
1&-i&-1&i\\\ 1&-1&1&-1\\\ 1&i&-1&-i\end{array}\right)$ (22) $\left(e^{ik\cdot
R}\right)\left(e^{-ik\cdot R}\right)=\left(\begin{array}[]{cccc}1&0&0&0\\\
0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&1\end{array}\right)=I_{4}$ (23)

From Eq.(7), we have,

$\left(\begin{array}[]{c}\left|0\right\rangle\left|0\right\rangle\\\
\left|0\right\rangle\left|1\right\rangle\\\
\left|1\right\rangle\left|0\right\rangle\\\
\left|1\right\rangle\left|1\right\rangle\end{array}\right)=\left(\begin{array}[]{cccc}1&1&1&1\\\
1&-i&-1&i\\\ 1&-1&1&-1\\\
1&i&-1&-i\end{array}\right)\left(\begin{array}[]{c}\left|B_{0}\right\rangle\\\
\left|B_{\frac{\pi}{2}}\right\rangle\\\ \left|B_{\pi}\right\rangle\\\
\left|B_{\frac{3\pi}{2}}\right\rangle\end{array}\right)$ (24)

where, from Eq.(6),

$\left(\begin{array}[]{c}\left|B_{0}\right\rangle\\\
\left|B_{\frac{\pi}{2}}\right\rangle\\\ \left|B_{\pi}\right\rangle\\\
\left|B_{\frac{3\pi}{2}}\right\rangle\end{array}\right)=\frac{1}{2}\left(\begin{array}[]{cccc}1&1&1&1\\\
1&i&-1&-i\\\ 1&-1&1&-1\\\
1&-i&-1&i\end{array}\right)\left(\begin{array}[]{c}\left|0\right\rangle_{1}\left|0\right\rangle_{2}\\\
\left|0\right\rangle_{1}\left|1\right\rangle_{2}\\\
\left|1\right\rangle_{1}\left|0\right\rangle_{2}\\\
\left|1\right\rangle_{1}\left|1\right\rangle_{2}\end{array}\right)$ (25)

However, as depicted in our ICL model, entanglement can only occurs for two
states, namely, either

$\left(\begin{array}[]{c}\left|0\right\rangle_{1}\left|0\right\rangle_{2}\\\
0\\\ 0\\\
\left|1\right\rangle_{1}\left|1\right\rangle_{2}\end{array}\right)\text{ or
}\left(\begin{array}[]{c}0\\\
\left|0\right\rangle_{1}\left|1\right\rangle_{2}\\\
\left|1\right\rangle_{1}\left|0\right\rangle_{2}\\\
0\end{array}\right)\text{.}$ (26)

These are faithfully represented by our ICL model, Fig. 1.

Figure 1: Physical diagrammatic model of “triplet” (left) and singlet (right)
entanglement. By construction, each diagram is viewed as a two-state system,
respectively. The actual physical implementation of the chain of inverters may
need frictionles male/female sliding tube coupling for large-angle swing, but
this is beside the point. We assume a rigid coupling model for simultaneity of
events at both ends.

So, we are only interested in the Bloch functions, coming from
$\left|B_{0}\right\rangle$ and $\left|B_{2}\right\rangle$ having Hadamard
Fourier transformation from two site states. We have the remaining matrix
entries,

$\left(\begin{array}[]{c}\left|B_{0}\right\rangle=\Phi^{+}\\\ 0\\\
\left|B_{2}\right\rangle=\Phi^{-}\\\
0\end{array}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cccc}1&0&0&1\\\
0&0&0&0\\\ 1&0&0&-1\\\
0&0&0&0\end{array}\right)\left(\begin{array}[]{c}\left|0\right\rangle_{1}\left|0\right\rangle_{2}\\\
\left|0\right\rangle_{1}\left|1\right\rangle_{2}\\\
\left|1\right\rangle_{1}\left|0\right\rangle_{2}\\\
\left|1\right\rangle_{1}\left|1\right\rangle_{2}\end{array}\right)$ (27)

where the $2$-$D$ transformation matrix is derived from the general relation,

$\left(\begin{array}[]{c}\left|B_{0}\right\rangle\\\
\left|B_{1}\right\rangle\\\ \left|B_{2}\right\rangle\\\
\left|B_{3}\right\rangle\end{array}\right)=\frac{1}{2}\left(\begin{array}[]{cccc}\delta&1&1&\delta\\\
1&i&-1&-i\\\ \delta&-1&1&-\delta\\\
1&-i&-1&i\end{array}\right)\left(\begin{array}[]{c}\left|0\right\rangle_{1}\left|0\right\rangle_{2}\\\
\left|0\right\rangle_{1}\left|1\right\rangle_{2}\\\
\left|1\right\rangle_{1}\left|0\right\rangle_{2}\\\
\left|1\right\rangle_{1}\left|1\right\rangle_{2}\end{array}\right)$ (28)

where all matrix entries are cross out except the $\delta$ entries in Eq.
(28). Therefore by contracting to prime number space, we have, using only the
$\delta$ entries of the transformation matrix,

$\left(\begin{array}[]{c}\left|\Phi^{+}\right\rangle\\\
\left|\Phi^{-}\right\rangle\end{array}\right)=\left(\begin{array}[]{c}\left|B_{0}\right\rangle\\\
\left|B_{2}\right\rangle\end{array}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}1&1\\\
1&-1\end{array}\right)\left(\begin{array}[]{c}\left|0\right\rangle_{1}\left|0\right\rangle_{2}\\\
\left|1\right\rangle_{1}\left|1\right\rangle_{2}\end{array}\right)$ (29)

or with other contraction of $\left|B_{0}\right\rangle$ and
$\left|B_{2}\right\rangle$ retaining the transformation,

$\left(\begin{array}[]{c}\left|B_{0}\right\rangle=\Psi^{+}\\\ 0\\\
\left|B_{2}\right\rangle=-\Psi^{-}\\\
0\end{array}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cccc}0&1&1&0\\\
0&0&0&0\\\ 0&-1&1&0\\\
0&0&0&0\end{array}\right)\left(\begin{array}[]{c}\left|0\right\rangle_{1}\left|0\right\rangle_{2}\\\
\left|0\right\rangle_{1}\left|1\right\rangle_{2}\\\
\left|1\right\rangle_{1}\left|0\right\rangle_{2}\\\
\left|1\right\rangle_{1}\left|1\right\rangle_{2}\end{array}\right)$ (30)

yielding $2$-$D$ transformation matrix

$\left(\begin{array}[]{c}\Psi^{+}\\\
-\Psi^{-}\end{array}\right)=\left(\begin{array}[]{c}\left|B_{0}\right\rangle\\\
\left|B_{2}\right\rangle\end{array}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}1&1\\\
-1&1\end{array}\right)\left(\begin{array}[]{c}\left|0\right\rangle_{1}\left|1\right\rangle_{2}\\\
\left|1\right\rangle_{1}\left|0\right\rangle_{2}\end{array}\right)$ (31)
$\left(\begin{array}[]{c}\Psi^{+}\\\
-\Psi^{-}\end{array}\right)\Longrightarrow\left(\begin{array}[]{c}\Psi^{+}\\\
\Psi^{-}\end{array}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}1&1\\\
1&-1\end{array}\right)\left(\begin{array}[]{c}\left|0\right\rangle_{1}\left|1\right\rangle_{2}\\\
\left|1\right\rangle_{1}\left|0\right\rangle_{2}\end{array}\right)$ (32)

where $\frac{1}{\sqrt{2}}$ becomes the proper normalization factor for the
reduced $2-D$ space.

Of course, besides the above entangled Bell basis states, there are other
interesting Bloch functions with Hadamard transformations, by selectively
crossing out of the matrix entries of Eq. (28), such as,

$\displaystyle\left(\begin{array}[]{c}\left|H_{0}\right\rangle\\\
\left|H_{1}\right\rangle\end{array}\right)\text{{\tiny derived
from}}\left(\begin{array}[]{c}\left|B_{0}\right\rangle\\\
\left|B_{1}\right\rangle\end{array}\right)$ (37) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}1&1\\\
1&-1\end{array}\right)\left(\begin{array}[]{c}\left|0\right\rangle_{1}\left|0\right\rangle_{2}\\\
\left|1\right\rangle_{1}\left|0\right\rangle_{2}\end{array}\right)$ (42)
$\displaystyle\left(\begin{array}[]{c}\left|H_{2}\right\rangle\\\
\left|H_{3}\right\rangle\end{array}\right)\text{{\tiny derived
from}}\left(\begin{array}[]{c}\left|B_{0}\right\rangle\\\
\left|B_{2}\right\rangle\end{array}\right)$ (47) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}1&1\\\
1&-1\end{array}\right)\left(\begin{array}[]{c}\left|0\right\rangle_{1}\left|0\right\rangle_{2}\\\
\left|0\right\rangle_{1}\left|1\right\rangle_{2}\end{array}\right)$ (52)
$\displaystyle\left(\begin{array}[]{c}\left|H_{0}\right\rangle\\\
\left|H_{1}\right\rangle\end{array}\right)\text{{\tiny derived
from}}\left(\begin{array}[]{c}\left|B_{0}\right\rangle\\\
\left|B_{3}\right\rangle\end{array}\right)$ (57) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}1&1\\\
1&-1\end{array}\right)\left(\begin{array}[]{c}\left|0\right\rangle_{1}\left|0\right\rangle_{2}\\\
\left|1\right\rangle_{1}\left|0\right\rangle_{2}\end{array}\right)$ (62)
$\displaystyle\left(\begin{array}[]{c}\left|H_{0}\right\rangle\\\
\left|H_{1}\right\rangle\end{array}\right)\text{{\tiny derived from}
}\left(\begin{array}[]{c}\left|B_{2}\right\rangle\\\
\left|B_{3}\right\rangle\end{array}\right)$ (67) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}1&1\\\
1&-1\end{array}\right)\left(\begin{array}[]{c}\left|0\right\rangle_{1}\left|0\right\rangle_{2}\\\
\left|1\right\rangle_{1}\left|0\right\rangle_{2}\end{array}\right)$ (72)
$\displaystyle\left(\begin{array}[]{c}\left|H_{4}\right\rangle\\\
\left|H_{5}\right\rangle\end{array}\right)\text{{\tiny derived
from}}\left(\begin{array}[]{c}\left|B_{0}\right\rangle\\\
\left|B_{3}\right\rangle\end{array}\right)$ (77) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}1&1\\\
1&-1\end{array}\right)\left(\begin{array}[]{c}\left|1\right\rangle_{1}\left|0\right\rangle_{2}\\\
\left|1\right\rangle_{1}\left|1\right\rangle_{2}\end{array}\right)$ (82)

Equations (42)-(82) cannot be represented in terms of our ICL model, and
therefore do not represent entangled states. Thus, only entangled states are
faithfully representable in terms of our ICL model. Clearly, the entanglement
of two bare qubits is divided into two orthogonal spaces of triplet222The use
of the term “triplet” is actually a misnomer here since the entangled system
is not free to assume a singlet or zero spin state. Thus, this term is used
here only as a label and singlet entanglement states in Fig.1. These two
spaces are transformed by the transition-matrix, $\sigma_{x}$.

Mathematically, the inverter-chain link model of entanglement may be
formulated as a series of $\sigma_{x}$ operations, represented by physical
inverters or see-saw’s. Assume at first that there are two locally-entangled
qubits $A$ and $B$ in either singlet or triplet state, with singlet joined
with one see-saw ($\sigma_{x}$) or triplet joined by two see-saw’s
($\sigma_{x}\otimes\sigma_{x}$). Then using the unitary single-qubit
operation, $\sigma_{x}$, on one of the two qubits will result in an additional
extension of a inverter-chain-linked entangled two qubits, either $\Phi^{+}$
or $\Psi^{+}$, depending on the initial singlet or triplet state. Using a
series of $\sigma_{x}$ operations will then yield a physically longer
inverter-chain link between the two entangled qubits. A series of odd number
of $\sigma_{x}$ operations will result in eventual inversion of one of the
qubit, whereas an even number of $\sigma_{x}$ operations is equivalent to an
identify operation of one of the qubit, although the inverter-chain link is
always extended by one see-saw with each $\sigma_{x}$ operation.

This arbitrary extension of the $\sigma_{x}$ inverter-chain is the reason why
$A$ and $B$ can be separated at arbitrarily large distances while still being
entangled, providing the mysterious link raised by the EPR [1]. Thus, the
inverter-chain link arbitrary extension defines local and nonlocal
entanglements. It should be emphasized that our inverter-chain link model is a
physical representation of $\sigma_{x}$ operations. Indeed, a more fundamental
geometric representation of the mysterious link raised by the EPR [1] will
unify general relativity and quantum mechanics and has been challenging
theoretical physicists for decades.

Clearly, a single qubit $\sigma_{x}$ operation is equivalent to extending the
inverter- link chain link by one physical inverter, which basically serves as
a single-qubit unitary transformation from $\Phi^{+}$ to $\Psi^{+}$ and vice
versa. Note in all these entangled cases, the following unitary
transformations, namely, $\sigma_{x}$ and $\sigma_{z}$ on a qubit partner
generates the other entangled states and its phase $\sigma_{z}$-transformed
entangled partner, namely,

$\sigma_{x}\left|\Phi^{+}\right\rangle=\left|\Psi^{+}\right\rangle$ (83)
$\displaystyle\sigma_{z}\left|\Phi^{+}\right\rangle$ $\displaystyle=$
$\displaystyle\left|\Phi^{-}\right\rangle$ (84)
$\displaystyle\sigma_{z}\left|\Psi^{+}\right\rangle$ $\displaystyle=$
$\displaystyle\left|\Psi^{-}\right\rangle$ (85)

where in Eqs. (83) - (85), the Pauli operators operates only on single
qubit.The inverse Hadamard transformation yields the Wannier states, i.e.,

$\displaystyle\frac{1}{\sqrt{2}}\left(\left|\Phi^{+}\right\rangle+\left|\Phi^{-}\right\rangle\right)$
$\displaystyle=$
$\displaystyle\left|0\right\rangle_{1}\left|0\right\rangle_{2}$ (86)
$\displaystyle\frac{1}{\sqrt{2}}\left(\left|\Phi^{+}\right\rangle-\left|\Phi^{-}\right\rangle\right)$
$\displaystyle=$
$\displaystyle\left|1\right\rangle_{1}\left|1\right\rangle_{2}$ (87)
$\displaystyle\frac{1}{\sqrt{2}}\left(\left|\Psi^{+}\right\rangle+\left|\Psi^{-}\right\rangle\right)$
$\displaystyle=$
$\displaystyle\left|0\right\rangle_{1}\left|1\right\rangle_{2}$ (88)
$\displaystyle\frac{1}{\sqrt{2}}\left(\left|\Psi^{+}\right\rangle-\left|\Psi^{-}\right\rangle\right)$
$\displaystyle=$
$\displaystyle\left|1\right\rangle_{1}\left|0\right\rangle_{2}$ (89)

Equations (86) - (89) are crucial in our ICL diagrammatic analysis of
superdense coding.

For the unentangled superpositions or other Bloch function states, we also
have

$\displaystyle\frac{1}{\sqrt{2}}\left(\left|H_{0}\right\rangle+\left|H_{1}\right\rangle\right)$
$\displaystyle=$
$\displaystyle\left|0\right\rangle_{1}\left|0\right\rangle_{2}$ (90)
$\displaystyle\frac{1}{\sqrt{2}}\left(\left|H_{0}\right\rangle-\left|H_{1}\right\rangle\right)$
$\displaystyle=$
$\displaystyle\left|1\right\rangle_{1}\left|0\right\rangle_{2}$ (91)
$\displaystyle\frac{1}{\sqrt{2}}\left(\left|H_{2}\right\rangle+\left|H_{3}\right\rangle\right)$
$\displaystyle=$
$\displaystyle\left|0\right\rangle_{1}\left|0\right\rangle_{2}$ (92)
$\displaystyle\frac{1}{\sqrt{2}}\left(\left|H_{2}\right\rangle-\left|H_{3}\right\rangle\right)$
$\displaystyle=$
$\displaystyle\left|0\right\rangle_{1}\left|1\right\rangle_{2}$ (93)
$\displaystyle\frac{1}{\sqrt{2}}\left(\left|H_{4}\right\rangle+\left|H_{5}\right\rangle\right)$
$\displaystyle=$
$\displaystyle\left|1\right\rangle_{1}\left|0\right\rangle_{2}$ (94)
$\displaystyle\frac{1}{\sqrt{2}}\left(\left|H_{4}\right\rangle-\left|H_{5}\right\rangle\right)$
$\displaystyle=$
$\displaystyle\left|1\right\rangle_{1}\left|1\right\rangle_{2}$ (95)

Equations (90) - (95) has been useful in the conventional quantum circuit
diagrams, using controlled gates and Hadamard transforms, for superdense
coding. Again, in our ICL diagrams we find no need for these other unentangled
Bloch function states to elucidate the fundamental concept.

###  2.1 Unitary transformation of qubits

In matrix representation, a single qubit is represented by a matrix,

$\displaystyle\left|0\right\rangle$ $\displaystyle\equiv$
$\displaystyle\left(\begin{array}[]{c}\left|0\right\rangle\\\
0\end{array}\right)\Longrightarrow\left(\begin{array}[]{c}1\\\
0\end{array}\right)$ (100) $\displaystyle\left|1\right\rangle$
$\displaystyle\equiv$ $\displaystyle\left(\begin{array}[]{c}0\\\
\left|1\right\rangle\end{array}\right)\Longrightarrow\left(\begin{array}[]{c}0\\\
1\end{array}\right)$ (105)

The unitary transformations of a qubit are the $\sigma_{x},$ $\sigma_{z},$ and
their combinations, where the $\sigma_{x},$ and $\sigma_{z}$ are the Pauli
matrices. $\sigma_{x}$ acts as an inverter, whereas $\sigma_{z}$ acts as a
phase operator, when these operators are acting on the Wannier states. Thus,
we

$\displaystyle\sigma_{x}\left|0\right\rangle$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}0\\\
1\end{array}\right)=\left|1\right\rangle$ (108)
$\displaystyle\sigma_{x}\left|1\right\rangle$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}1\\\
0\end{array}\right)=\left|0\right\rangle$ (111)
$\displaystyle\sigma_{z}\left|0\right\rangle$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}1\\\
0\end{array}\right)=\left|0\right\rangle$ (114)
$\displaystyle\sigma_{z}\left|1\right\rangle$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}0\\\
-1\end{array}\right)=-\left|1\right\rangle$ (117)

##  3 ICL implementation of quantum teleportation

The diagrammatic perspective relies on a couple of principles, namely, (1)
that unitary transformations of a partner qubit generate all other possible
entanglements from a given entangled qubits, and (2) that quantum information
or quantum configurations (entropy) is conserved in quantum teleportation. The
first principle allows us to examine all possible scenarios of quantum
teleportation. Coupled with classical communication (CC) protocol, the second
principle conserved the configurations of the system.

The virtue of the diagrammatic perspective is that it simplifies the process
and gives a very intuitive understanding of quantum teleportation. Let Alice
and Bob share a maximally entangled state initially,

$\left|\Phi^{+}\right\rangle_{AB}=\frac{1}{\sqrt{2}}\left(\left|0\right\rangle_{A}\left|0\right\rangle_{B}+\left|1\right\rangle_{A}\left|1\right\rangle_{B}\right)$
(118)

So, Alice has two particles ($U$, the one she wants to teleport, and $A$, one
of the entangled pair), and Bob has the other entangled-pair particle, $B$.
Mathematically, we have,

$\displaystyle\left|\psi\right\rangle_{U}\otimes\left|\Phi^{+}\right\rangle_{AB}=C_{B}^{\left(+\right)}\left|\Phi^{+}\right\rangle_{UA}+C_{B}^{\left(-\right)}\left|\Phi^{-}\right\rangle_{UA}+D_{B}^{\left(+\right)}\left|\Psi^{+}\right\rangle_{UA}...$
(119) $\displaystyle=$
$\displaystyle\frac{1}{2}\left|\Phi^{+}\right\rangle_{UA}\
\otimes\psi_{{}_{B}}^{\left(0\right)}+\frac{1}{2}\left|\Phi^{-}\right\rangle_{UA}\
\otimes\psi_{{}_{B}}^{\left(1\right)}$
$\displaystyle+\frac{1}{2}\left|\Psi^{+}\right\rangle_{UA}\
\otimes\psi_{{}_{B}}^{\left(2\right)}+\frac{1}{2}\left|\Psi^{-}\right\rangle_{UA}\otimes\psi_{{}_{B}}^{\left(3\right)}$

where, $\psi_{{}_{B}}^{\left(i\right)}$, $\left(\ i=1,2,3\right)$ are the
coefficients of the expansion in terms of complete entangled basis states of
$A$ with $U$, namely, $\left|\Phi^{+}\right\rangle_{UA}$,
$\left|\Phi^{-}\right\rangle_{UA}$, $\left|\Psi^{+}\right\rangle_{UA}$,
$\left|\Psi^{-}\right\rangle_{UA}$. More explicitly, we have,

$\displaystyle\left(\begin{array}[]{c}\alpha\\\
\beta\end{array}\right)_{U}\otimes\left|\Phi^{+}\right\rangle_{AB}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left|\Phi^{+}\right\rangle_{UA}\left(\begin{array}[]{c}\alpha\\\
\beta\end{array}\right)$ (139)
$\displaystyle+\frac{1}{2}\left|\Phi^{-}\right\rangle_{UA}\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)\left(\begin{array}[]{c}\alpha\\\
\beta\end{array}\right)$
$\displaystyle+\frac{1}{2}\left|\Psi^{+}\right\rangle_{UA}\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right)\left(\begin{array}[]{c}\alpha\\\
\beta\end{array}\right)$
$\displaystyle+\frac{1}{2}\left|\Psi^{-}\right\rangle_{UA}\left(\begin{array}[]{cc}0&-1\\\
1&0\end{array}\right)\left(\begin{array}[]{c}\alpha\\\
\beta\end{array}\right).$

The last line of Eq. (139) differs only by a global sign factor with the term,

$+\frac{1}{2}\left|\Psi^{-}\right\rangle_{UA}\left(\begin{array}[]{cc}0&1\\\
-1&0\end{array}\right)\left(\begin{array}[]{c}\alpha\\\
\beta\end{array}\right)=+\frac{1}{2}\left|\Psi^{-}\right\rangle_{UA}\
\sigma_{z}\sigma_{x}\left(\begin{array}[]{c}\alpha\\\ \beta\end{array}\right)$

Thus, the situation looks exactly like the one shown in Fig.2.

Figure 2: Explicit superposition principle in quantum teleportation, depicted
in terms of our ICL model of quantum entanglement. The term ’cc’ means
classical communication.

##  4 ICL perspective of superdense coding

The use of entanglement also plays an important resource in superdense coding
With Alice and Bob sharing an entangled state, two classical bits per qubit
can be transmitted. Superdense refers to this information capacity in
communication channels. Our diagrammatic approach greatly simplifies the
concept than that of the quantum circuit, controlled gate and Hadamard
tranformation, approach.

Superdense coding is based on the observation that given some entangled Bell
basis state shared by Alice and Bob, there are local unitaries belonging to
either Alice or Bob which will map their shared Bell basis onto any of the
other Bell basis states apart from overall phase. Hadamard transformation,
$H$, with its phase-transformed ($\sigma_{z}$) partner Bloch states then
decode the resulting pair of Bloch functions to their computational basis or
Wannier function states. This is schemtically illustrated in Fig.3

Figure 3: ICL implementation of superdense coding. H is the Hadamard
transformation of a Bell basis and its phase-transformed partner.

##  5 Concluding Remarks

We have demonstrated that the discrete phase-space physics coupled with the
ICL implementation of entanglements have a much wider implications in
elucidating entanglement physics. For one thing, ICL has clearly shed more
light on the fundamental concept of concurrence, entropy of entanglement
formation, and entropic-distance measure as the fundamental characterization
of entanglement [2]; thus, as a natural measures of entanglement. The
derivation of entangled basis states through the ICL diagrams has also given a
straightforward generalization of Bell basis states for any number multi-
partite entanglements [3].

On the more fundamental question on the completeness of quantum physics, ICL
description of entanglement lends strong support on the significance of EPR
assertion that entanglements need a mysterious link, perhaps geometric, which
may call for extra dimensions. The apparently perfect scheme of the ICL model
in describing quantum entanglement seems to advocate the existence of
entangled spacetime-medium to take the place of the ICL in some unknown forms.
An interesting analogy is the bound/entangled pair of vortices and antivortex
on the surface of fluids which are really the ends of a vortex chain (tube)
forming a U-shape underneath the surface. For example, in $3$-$D$ the
superfluid quantized vortices form a metastable closed ring or open chain
ending at the surface. A vortex chain with both ends ending at the same
surface appears as a bound vortex pair at the surface.

Thus, it seems extra dimensions are needed in spacetime to have a fundamental
theory of entanglement. The work of Ooguri [6, 7] and collaborators shows that
this quantum entanglement generates the extra dimensions of the gravitational
theory. ”It appears that it seems possible to generate a geometric connection
between entangled qubits, even though there is no direct interaction between
the two systems [8]. Furthermore, the structure of spacetime is proposed to be
due to the ghostly features of entanglement. Could it be that besides the
geometrical spacetime aspects of gravity, there is a purely quantum mechanical
aspect of spacetime geometry with extra dimensions that give rise to
entanglement?

A hint along this idea is also given by Malcedona [8] when he stated that ”One
can consider, therefore, a pair of black holes where all the microstates are
“entangled.” Namely, if we observe one of the black holes in one particular
microstate, then the other has to be in exactly the same microstate. A pair of
black holes in this particular EPR entangled state would develop a wormhole,
or Einstein-Rosen bridge, connecting them through the inside. The geometry of
this wormhole is given by the fully extended Schwarzschild geometry. It is
interesting that both wormholes and entanglement naively appear to lead to a
propagation of signals faster than light.

It was known that quantum entanglement is related to deep issues in the
unification of general relativity and quantum mechanics, such as the black
hole information paradox and the firewall paradox,” says Hirosi Ooguri” [7].
Remarkably, all this can very simply be viewed as bound/entangled pair of
vortices and antivortex on the surface of fluids which are really the ends of
a vortex chain (vortex tube or ”Einstein-Rosen bridge”) forming a U-shape
chain underneath the surface or as a meeting two vertical funnels if surface
is theoretically bowed into U-shaped itself. This has led to the proposition,
ER=EPR conjecture in physics, stating that the EPR paradox should lead to the
unification of general relativity and quantum field theory. Indeed, the ER =
EPR conjecture [9, 10] is the bold statement that a large amount of
entanglement between two localized regions of space-time implies the existence
of a geometric connection between two regions of space-time.

## References

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