# Quantum formalism on the plane: POVM-Toeplitz quantization, Naimark theorem
and linear polarisation of the light

Roberto Beneducia, Emmanuel Frionb, Jean-Pierre Gazeauc and Amedeo Perrid A
Dipartimento di Fisica, Università della Calabria
and Istituto Nationale di Fisica Nucleare, Gruppo c. Cosenza, 87036 Arcavacata
di Rende (Cs), Italy B Helsinki Institute of Physics, P. O. Box 64,
FIN-00014 University of Helsinki, Finland C Université Paris Cité,
CNRS, Astroparticule et Cosmologie 75013 Paris, France
D Independent Scholar,
Via Stazione 4, Spezzano della Sila 87058 Cosenza, Italy
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We investigate two aspects of the elementary example of POVMs on the Euclidean
plane, namely their status as quantum observables and their role as quantizers
in the integral quantization procedure. The compatibility of POVMs in the
ensuing quantum formalism is discussed, and a Naimark dilation is found for
the quantum operators. The relation with Toeplitz quantization is explained. A
physical situation is discussed, where we describe the linear polarization of
the light with the use of Stokes parameters. In particular, the case of
sequential measurements in a real bidimensional Hilbert space is addressed. An
interpretation of the Stokes parameters in the framework of unsharp or fuzzy
observables is given. Finally, a necessary condition for the compatibility of
two dichotomic fuzzy observables which provides a condition for the
approximate joint measurement of two incompatible sharp observables is found.

###### Contents

  1. 1 Introduction
  2. 2 General setting
    1. 2.1 Definition of a POVM
    2. 2.2 Naimark theorem and compatibility of POVMs
  3. 3 An outline of integral quantization and semi-classical portraits
    1. 3.1 Quantization map
    2. 3.2 Semi-classical portraits
    3. 3.3 POVMs and quantum measurement
  4. 4 Integral quantization, Toeplitz quantization and Naimark dilation
  5. 5 Integral POVM quantization of fonctions on the circle
    1. 5.1 Euclidean plane with Dirac notations
    2. 5.2 Integral POVM quantization
    3. 5.3 Quantization and symbol calculus
    4. 5.4 Integral POVM quantization on the circle and Toeplitz-Naimark formalism
  6. 6 An example of quantum measurement of orientations: polarization of light and Stokes parameters
  7. 7 Compatibility of POVMs
    1. 7.1 POVMs arising from the measurement of polarization
    2. 7.2 Dichotomic POVMs
  8. 8 Conclusions
  9. A Quantum orientations in $\mathbb{R}^{n}$
    1. A.1 Pure and mixed states
    2. A.2 SO$(n)$ covariant integral quantisation

##  1\. Introduction

Positive operator-valued measures (POVMs) play a central role in the Hilbert
space formulation of quantum mechanics. Indeed, observables are represented by
positive operator-valued measures, states are represented by positive trace-
class operators with trace one and observables and states are in statistical
duality (see next section). One of the main features of quantum physics is the
existence of incompatible observables that cannot be measured simultaneously
[1]. In other words, measurement of two observables in quantum physics can
show some incompatibility. This reflects in purely quantum phenomena, such as
the violation of Bell inequalities [2, 3] or the Kochen-Specker contextuality
[4], with implications in quantum information processing [5, 6, 7, 8, 9, 10,
11, 12] and in explaining the power of quantum computation [13, 14, 15, 16,
17, 18, 19, 20].

Spectral measures, which are in one-to-one correspondence with self-adjoint
operators, are particular cases of POVMs. The compatibility of two observables
represented by spectral measures is equivalent to their commutativity.
However, that is not true for general (unsharp) observables described by POVMs
for which compatibility does not imply commutativity. This opens new
perspectives and raises the question of compatibility between POVMs. General
measurements can realize arbitrary joint measurability structure [21, 22], and
studying incompatible sets can prove relevant for quantum information
protocols [23, 24, 25].

Several works have been devoted to find conditions for the compatibility of
POVMs. See [1] for a review, see also [26], where the necessary and sufficient
condition for compatibility of two dichotomic observables is reformulated as
semidefinite program, and [27] where it is shown that two POVMs are jointly
measurable if and only if they can be dilated to two commuting orthogonal
POVMs, though this result cannot be extended to the case of more than two
POVMs [28]. In the present paper, we analyze the simple case of two dichotomic
POVMs in a two-dimensional real Hilbert space, i.e. the Euclidean plane, and
show that a condition for the compatibility can be derived starting from its
definition. We provide an interpretation of fuzzy dichotomic observables in
terms of the Stokes parameters for the polarization of an electromagnetic
wave; this suggests an interpretation of the approximate joint measurement of
incompatible observable as well. From the mathematical viewpoint, this is a
particular case of the results found in Ref. [1], chapter 14, in which the
complex two-dimensional case is analyzed; it is the real Hilbert space case
obtained when the coefficient multiplying the Pauli matrix $\sigma_{2}$ in the
definition of the self-adjoint operators is equal to zero, i.e., we have
$A=a_{0}\mathbbm{1}+\mathbf{n}\cdot\mathbf{\sigma}$ with
$\mathbf{n}=(n_{1},0,n_{3})$,
$\mathbf{\sigma}=(\sigma_{1},\sigma_{2},\sigma_{3})$ and $\mathbbm{1}$ the
identity operator. In the framework of electromagnetic wave polarization, that
means we are focusing on the linear polarization and are not taking into
consideration the circular polarization.

Besides their role as quantum observables, POVMs also provide a quantization
procedure based on integral calculus. The properties of this integral
quantization procedure were studied in [29], where it was shown that a choice
of POVM gives a well-defined quantum model for a system, and offers the
possibility of a regularization effect as well as a full probabilistic
content. These features have been leveraged in quantum cosmological models,
where the Big Bang singularity is replaced by a bounce [30, 31, 32, 33, 34,
35, 36, 37, 38, 39, 40, 41]. POVMs play a key role in the phase-space
formulation of quantum mechanics as well [42, 43, 44].

Finally, we recall that an interesting class of measurements represented by
POVMs concerns weak measurements. They are measurements which disturb (in the
average) a quantum system very little, while extracting a little bit of
information (see e.g. [45, 46]). Weak measurements can be used experimentally
to show the breaking of Bell’s inequalities [47], which can be used in order
to set up experiments preserving entanglement even when performing many
sequential measurements [48, 49]. In the same vein, weak measurements are also
used to test the noise-disturbance relation stemming from the Heisenberg
uncertainty principle [50] in various experiments [51, 52, 53].

The content of the present article shows various non-trivial aspects connected
with the role of POVMs as both observables and quantization maps. One of its
objective is to describe a sequential measurement with POVMs acting on the
Euclidean plane, certainly one of the simplest examples of Hilbert space. We
particularly focus on the question of incompatibility. In this line, we begin
by reviewing the definition of a POVM and compatibility between POVMs in
Section 2. In Section 3, we give the definition and main properties of
integral quantization and its concomitant semi-classical portraits by
insisting on the genuine role of POVMs in implementing the procedure. In
Section 4 we establish an original result, namely the relationship between our
approach and the so-called Toeplitz quantization, which allows us to find the
Naimark dilation of our quantum operators. In Section 5, we apply our
formalism and Euclidean plane to the circle. We show that the integral POVM
quantisation of a real function defined on the circle results in a $2\times 2$
matrix acting on the Euclidean plane. We then describe the corresponding semi-
classical portrait or symbol calculus. We end the section by illustrating the
Naimark-Toeplitz correspondence within this $2d$ Euclidean framework. A nice
physical application of our approach is developed in Section 6, where we
establish the link with Stokes parameters for the linear polarization of
electromagnetic waves and describe the interaction of a linearly polarized
light with a polarizer as an example of quantum measurement. We then discuss
the approximate measurement of two incompatible sharp polarization observables
$E$ and $E^{\prime}$ realized by the joint measurement of their fuzzy versions
$F$ and $F^{\prime}$. This requires a characterization of the compatibility of
two POVMs. We begin Section 7 with an illustration of the incompatibility
problem, then we derive the necessary conditions for the compatibility of
dichotomic observables. As a last result, we show there is a direct relation
between the fuzziness of a measurement and the degree of linear polarization
of a physical observable. We generalize some technical aspects of our results
to $\mathbb{R}^{n}$ in Appendix A.

##  2\. General setting

What is an observable? In standard quantum mechanics an observable is
represented by a self-adjoint operator but it has been argued [54, 55, 56, 57,
58, 1, 59] that the choice of self-adjoint operators as representative of
observables is too restrictive. It has been suggested that an observable is
better represented by a POVM of which the spectral measures are particular
examples. Before we give motivation for such a generalization of the concept
of observable, let us recall the main definitions and properties of POVMs
[27]. In the following, the set of positive trace-class self-adjoint operators
with trace one is denoted by $S(\mathcal{H})$.

###  2.1. Definition of a POVM

######  Definition 2.1.

A normalized Positive-Operator Valued measure (POVM) is a map
$F:\mathcal{B}(\Omega)\to\mathcal{L}^{+}_{s}(\mathcal{H})$ from the Borel
$\sigma$-algebra of a topological space $\Omega$ to the space of linear
positive self-adjoint operators such that:

(2.1) $\displaystyle F\left(\bigcup_{n=1}^{\infty}\Delta_{n}\right)$
$\displaystyle=\sum_{n=1}^{\infty}F(\Delta_{n})\;,$ (2.2) $\displaystyle
F(\Omega)$ $\displaystyle=\mathbbm{1}\;,$

where $\\{\Delta_{n}\\}$ is a countable family of disjoint sets in
$\mathcal{B}(\Omega)$ and the series converges in the weak operator topology.
The POVM is said to be real if $\Omega=\mathbb{R}$. A projection-valued
measure (PVM) is a POVM such that $F(\Delta)$ is a projection operator for
every $\Delta\in\mathcal{B}(\Omega)$.

Spectral measures are real PVMs. By the spectral theorem, spectral measures
are in a one-to-one correspondence with self-adjoint operators. Therefore, if
we use POVMs to represent observables, the set of observables represented by
self-adjoint operators is just a subset of the set of observables. The
motivation for such an extension is rooted in the statistical nature of the
measurement process. It is well known that by repeating the measurement $M$ of
a system $S$ in the state $\rho\in S(\mathcal{H})$ we, in general, obtain
different results (that we suppose contained in a topological space $\Omega$).
On the other hand, by repeating the measurement $M$ several times, we observe
a statistical regularity, which is encoded in the statistical distribution
$f^{M}_{\rho}$ of the measurement outcomes, and depends only on the state of
the system. In other words, we have a map $\rho\mapsto f^{M}_{\rho}$ ensuring
the existence of regularities of a statistical nature. Note that this is a
necessary condition for the existence of the scientific enterprise. In its
turn, the statistical distribution $f_{\rho}^{M}$ is represented by a
probability measure $\mu^{M}_{\rho}:\mathcal{B}(\Omega)\to[0,1]$. At this
point of our argument, it is better to specify that the state of the system is
defined by the procedure we followed to prepare the system before the
measurement, and to recall that the space $S(\mathcal{H})$ of the states of a
given system is a convex space (this corresponds to the existence of mixed
states). For example, a filtering operation ensures that if a beam of
electrons passes through the filter, the electrons in the beam must
immediately afterward have positions within a restricted range of values. The
previous analysis can be summarized by saying that the measurement process $M$
defines a map from “states” to “probability measures,”
$\rho\mapsto\mu^{M}_{\rho}$. Moreover, if the state is given by a convex
combination of states, $\rho=t\rho_{1}+(1-t)\rho_{2}$, the measurement
outcomes are given by a convex combination of probability measures,
$t\mu^{M}_{1}+(1-t)\mu^{M}_{2}$. In mathematical terms, the map
$\rho\mapsto\mu^{M}_{\rho}$ is affine. Note that the previous argument is of a
general nature. It applies both to classical and quantum formalism (in
mechanics, optics, …). In the classical case, for example, the states are
represented by probability measures on the phase space $\Gamma$, and the
measurements are affine maps that transform probability measures on $\Gamma$
into probability measures on the space of possible outcomes of the measurement
process, $\Omega$ [56]. Actually, the scheme we just described applies to any
physical model where the states of the system define a convex set (see [1]
chapter 23). In order to introduce the concept of observable in this scheme,
it is sufficient to note that different measurements can correspond to the
same affine map. In this case, we say that the measurements are equivalent and
divide the set of measurement into equivalence classes. An observable is an
equivalent class of measurements. The same can happen with states: two states
$\rho_{1}$ and $\rho_{2}$ are said to be equivalent if they cannot be
separated by the set of measurements, i.e., if $\mu_{1}^{M}=\mu_{2}^{M}$ for
every measurement $M$. Now, let us focus on the particular case of the quantum
framework where, to each system, a complex and separable Hilbert space
$\mathcal{H}$ is associated and the states are represented by density
operators, i.e. positive, bounded self-adjoint operators with trace $1$. We
have the following characterization of the measurement process:

######  Theorem 2.1 (Holevo, [56]).

There is a one-to-one correspondence between POVMs
$F:\mathcal{B}(\Omega)\to\mathcal{L_{s}^{+}(H)}$ and affine maps
$S(\mathcal{H})\mapsto\mathcal{M}_{+}(\Omega)$ from states to probability
measures which is given by $\mu(\Delta)=Tr(\rho F(\Delta))$,
$\Delta\in\mathcal{B}(\Omega)$, $\rho\in S(\mathcal{H})$,
$\mu\in\mathcal{M}_{+}(\Omega)$.

Note that equivalent measurements are represented by the same POVM and this
establishes a one-to-one map between observables and POVMs. Moreover,
$\mu_{\rho}^{M}=Tr[F(\Delta)\rho]$ establishes a statistical duality between
observables (POVMs) and states (density operators) [1].

###  2.2. Naimark theorem and compatibility of POVMs

In the previous section, we have seen that observables are represented by
POVMs and that observables represented by spectral measures (self-adjoint
operators) define a subset of the space of observables. Now, we recall
Naimark’s dilation theorem which ensures that every POVM is the projection of
a projector-valued measure defined in an extended Hilbert space.

######  Theorem 2.2 (Naimark, [60]).

Let $F:\mathcal{B}(\Omega)\to\mathcal{L}^{+}_{s}(\mathcal{H})$ be a POVM in a
Hilbert space $\mathcal{H}$. Then, there is an extended Hilbert space
$\mathcal{H}^{+}$ and a projector-valued (PV) measure
$E^{+}:\mathcal{B}(\Omega)\to\mathcal{L}_{s}(\mathcal{H}^{+})$ such that

$PE^{+}(\Delta)\psi=F(\Delta)\psi,\quad\psi\in\mathcal{H},\quad\Delta\in\mathcal{B}(\Omega)\,,$

where $P$ is the projection operator onto $\mathcal{H}$.

Naimark’s theorem provides a necessary and sufficient condition for the
compatibility of two POVMs.

######  Definition 2.2.

Two POVMs $F_{1}:\mathcal{B}(\Omega_{1})\to\mathcal{L}^{+}_{s}(\mathcal{H})$
and $F_{2}:\mathcal{B}(\Omega_{2})\to\mathcal{L}^{+}_{s}(\mathcal{H})$ are
compatible if there is a third POVM
$F:\mathcal{B}(\Omega_{1}\times\Omega_{2})\to\mathcal{L}^{+}_{s}(\mathcal{H})$
of which $F_{1}$ and $F_{2}$ are the marginals, i.e.,

$F_{1}(\Delta_{1})=F(\Delta_{1}\times\Omega_{2}),\quad
F_{2}(\Delta_{2})=F(\Omega_{1}\times\Delta_{2}).$

We recall that the symbol $\mathcal{B}(\Omega_{1}\times\Omega_{2})$ denotes
the product $\sigma$-algebra generated by the family of sets
$\\{\Delta_{1}\times\Delta_{2}\,\,:\,\,\Delta_{1}\in\mathcal{B}(\Omega_{1}),\,\Delta_{2}\in\mathcal{B}(\Omega_{2})\\}$.

######  Theorem 2.3 ([27]).

Two POVMs $F_{1}:\mathcal{B}(\Omega_{1})\to\mathcal{L}^{+}_{s}(\mathcal{H})$
and $F_{2}:\mathcal{B}(\Omega_{2})\to\mathcal{L}^{+}_{s}(\mathcal{H})$ are
compatible if and only if there are two Naimark extensions
$E^{+}_{1}:\mathcal{B}(\Omega_{1})\to\mathcal{L}_{s}(\mathcal{H}^{+})$ and
$E^{+}_{2}:\mathcal{B}(\Omega_{2})\to\mathcal{L}_{s}(\mathcal{H}^{+})$ such
that $[E^{+}_{1},E^{+}_{2}]=0$.

##  3\. An outline of integral quantization and semi-classical portraits

In this section, we recall the fundamental role played by POVMs as quantizers
of classical models of physical systems, i.e. in the building of self-adjoint
operators and their subsequent PV measures [29, 61].

###  3.1. Quantization map

To define a quantization procedure for functions defined on a set $X$ (e.g. a
phase space), the map from elements in a “classical” vector space to operators
in some Hilbert space must fulfill basic requirements. A minimal program
should meet the four following criteria [62, 63]:

  1. (1)

Linearity. Quantization is a linear map $f\mapsto A_{f}$:

(3.1)
$\displaystyle\mathfrak{Q}:\mathcal{C}(X)\mapsto\mathcal{A}({\mathcal{H}})\,,\qquad\mathfrak{Q}(f)=A_{f}\,,$

where

     * •

$\mathcal{C}(X)$ is a vector space of complex or real-valued functions $f(x)$
on a set $X$, i.e. a “classical” mathematical model,

     * •

$\mathcal{A}({\mathcal{H}})$ is a vector space of linear operators in some
real or complex Hilbert space ${\mathcal{H}}$, i.e., a “quantum” mathematical
model, notwithstanding the question of common domains in the case of unbounded
operators.

  2. (2)

Unity. The map (3.1) is such that the function $f=1$ is mapped to the identity
operator $\mathbbm{1}$ on ${\mathcal{H}}$.

  3. (3)

Reality. A real function $f$ is mapped to a self-adjoint operator $A_{f}$ in
${\mathcal{H}}$ or, at least, a symmetric operator (in the infinite-
dimensional case).

  4. (4)

Covariance. Defining the action of a symmetry group G on X by
$\left(g,x\right)\in G\times X$ such as $\left(g,x\right)\mapsto g\cdot x\in
X$, there is a unitary representation $U$ of $G$ such that
$A_{T(g)f}=U(g)A_{f}U(g^{-1})$, with $(T(g)f)(x)=f\left(g^{-1}\cdot x\right)$.

In the remainder of this paper, we suppose ${\mathcal{H}}$ be a finite Hilbert
space. Therefore, the quantized objects $A_{f}$ are matrices [64]. Note that
the previous requirements are minimal. In this work, we make use of POVMs to
define the quantization procedure, and it is therefore necessary to consider
additional minimal structures, namely a measure $\nu$ on $X$, the
$\sigma$-algebra defined in (2.1), and local compactness. POVMs can be used in
conjunction with integral quantization to fulfill the first three above
requirements. However, covariance needs some additional structure [65, 61]. In
particular, Euclidean POVMs provide covariance, as we will see in section 5.2.
The integral quantization of a function $f(x)$ on the measure space $(X,\nu)$
is defined by the linear map

(3.2) $f\mapsto A_{f}=\int_{X}\,{\sf M}(x)\,f(x)\,\mathrm{d}\nu(x)\,,$

where the family of matrices ${\sf M}(x)$ solves the identity as

(3.3) $X\ni x\mapsto\mathsf{M}(x)\,,\quad\int_{X}\,{\sf
M}(x)\,\mathrm{d}\nu(x)=\mathbbm{1}\,.$

If the matrices ${\sf M}(x)$ are non-negative, the quantum operator related to
the characteristic function on $\Delta$, $A(\chi_{\Delta})$, defines a POVM .
Indeed, the restriction to $\Delta$ of the quantization map (3.2)

(3.4) $\displaystyle F(\Delta):=A(\chi_{\Delta})=\int_{X}\,{\sf
M}(x)\,\chi_{\Delta}(x)\,\mathrm{d}\nu(x)=\int_{\Delta}\,{\sf
M}(x)\,\mathrm{d}\nu(x)\;,$

solves the identity (2.2) by definition. Therefore, we have shown two key
roles of POVMs; they are the mathematical representatives of observables and
they provide a quantization procedure.

###  3.2. Semi-classical portraits

A quantum model for a system must have a classical or semi-classical
counterpart. We say there exists a dequantization map acting on an operator
$A_{f}$ to give back the original $f(x)$, in general with some corrections.
This procedure gives a semi-classical portrait of the original function $f$,
often denoted $\check{f}(x)$ (Husimi and Wigner functions [66], lower (Lieb)
[67] or covariant (Berezin) [68] symbols). Given two families ${\sf
M}_{\mathrm{a}}(x)$ (for “analysis”) and ${\sf M}_{\mathrm{r}}(x)$ (for
“reconstruction") resolving the identity in the sense of (3.3), we here
generalize the definition of these lower symbols in the following way:

(3.5)
$A_{f}\mapsto\check{f}(x):=\mathrm{tr}(\mathsf{M}_{\mathrm{r}}(x)\,A_{f})=\int_{X}\,\mathrm{tr}(\mathsf{M}_{\mathrm{r}}(x){\sf
M}_{\mathrm{a}}(x^{\prime}))\,f(x^{\prime})\,\mathrm{d}\nu(x^{\prime})\,,$

Choosing $\mathsf{M}_{\mathrm{a}}(x)=\rho_{\mathrm{a}}(x)$ and
$\mathsf{M}_{\mathrm{r}}(x)=\rho_{\mathrm{r}}(x)$ to be density matrices, the
map
$x^{\prime}\mapsto\mathrm{tr}(\rho_{\mathrm{r}}(x)\rho_{\mathrm{a}}(x^{\prime}))$
defines a probability distribution
$\mathrm{tr}(\rho_{\mathrm{r}}(x)\rho_{\mathrm{a}}(x^{\prime}))$ on the
measure space $(X,\mathrm{d}\nu(x^{\prime}))$, as it is easily proved by
multiplying the resolution of the unity (3.3) with $\rho_{\mathrm{r}}(x)$ and
tracing the result. Thus, the expectation value of the operator $A_{f}$,
$\check{f}(x)$, is built from the analysis POVMs based on the
$\mathsf{M}_{\mathrm{a}}(x)$’s and given by:

(3.6)
$f(x)\mapsto\check{f}(x)=\int_{X}f(x^{\prime})\,\mathrm{tr}(\rho_{\mathrm{r}}(x)\rho_{\mathrm{a}}(x^{\prime}))\,\mathrm{d}\nu(x^{\prime})\,.$

The map (3.6) represents in general a regularization of the original, possibly
extremely singular, $f$. Standard cases are for
$\rho_{\mathrm{a}}=\rho_{\mathrm{r}}$, in particular for rank-one density
matrices (coherent states).

###  3.3. POVMs and quantum measurement

We base the content of this section on [61], in which the analogy of the
present formalism with quantum measurement was discussed. In quantum
mechanics, a physically relevant $A_{f}$ is a self-adjoint operator whose
expectation value is the “unsharp” representation [56, 1]

(3.7)
$\displaystyle\mathrm{tr}\left(\rho_{m}A_{f}\right)=\int_{X}f(x)\,\mathrm{tr}(\rho_{m}\rho(x))\,\mathrm{d}\nu(x)\,.$

In compliance with the presentation given in Section 2.1, the operator

$\rho_{m}=\sum_{i}q_{i}|\kern 1.29167pt\phi_{i}\kern
1.29167pt\rangle\langle\kern 1.29167pt\phi_{i}\kern 1.29167pt|$

is here the density matrix describing the mixing of the pure states $|\kern
1.29167pt\phi_{i}\kern 1.29167pt\rangle$, each associated with the outcome
probability $q_{i}$. On the other hand, the expectation value (3.7) of any
quantum observable stems from the probability distribution
$\mathrm{tr}\left(\rho_{m}\rho(x)\right)$. We note that, unlike in (3.6),
$\rho_{m}$ does not depend on any variable. Since the family of density
operators used to construct our quantization procedure can provide the choice
$\rho_{m}=\rho_{r}(x_{0})$ for some $x_{0}$, we obtain a quantum measurement
based on the pair of POVMs $\left(\rho_{m},\rho(x)\right)$, which is
equivalent to the pair $\left(\rho_{r},\rho_{a}\right)$ introduced in (3.6).
We also note that the unsharp representation can be associated with the
“sharp” representation

(3.8)
$\mathrm{tr}\left(\rho_{m}A_{f}\right)=\int_{\mathbb{R}}\lambda\,\mathrm{tr}(\rho_{m}\,\mathrm{d}E_{f}(\lambda))\;,$

coming from the self-adjoint operator $A_{f}$ spectral decomposition

(3.9) $A_{f}=\int_{\mathbb{R}}\lambda\,\mathrm{d}E_{f}(\lambda)\;,$

with spectral measure $\mathrm{d}E_{f}(\lambda)$. While the sharp
representation is directly associated to the PV spectral measure $E_{f}$, a
family $\rho(x)$ resolving the identity is needed in the unsharp
representation. However, classical and quantum spectra can be considered
identical, regardless of the differences between their PV measure and POVMs.
Finally, we observe that both the quantum operator $A_{f}$ and its semi-
classical counterpart (3.6) involve probability distributions with respect to
some POVMs. Therefore, quantifying an object and measuring it are done through
a sole procedure.

##  4\. Integral quantization, Toeplitz quantization and Naimark dilation

The so-called Berezin-Klauder-Toeplitz quantization (see [69] for a clear
presentation of it 111We stress that the quantization procedure presented in
[69] is based on group action while our presentation does not.), and in
particular coherent state (CS) quantization, is one kind of integral
quantization. In this section, we describe their relationship, which will
allow us to find the Naimark dilation of our quantum operators. The
construction of family of operators solving the identity in (3.3) in the CS
case is implemented along the following scheme. Let us equip the set $X$ with
another measure $\mu$ and consider the Hilbert space
$L^{2}(X,\mathrm{d}\mu(x))$ of complex-valued functions on $X$ which are
Lebesgue square integrable with respect to $\mu$. One then chooses an
orthonormal set $\mathcal{O}$ of functions $\phi_{i}(x)$ (set aside the
question of evaluation map in their respective equivalence classes),
satisfying the finiteness and positiveness conditions

(4.1)
$0<\mathcal{N}(x)=\sum_{i}|\phi_{i}(x)|^{2}<\infty\quad(\mathrm{a.e.})\,,$

and a “companion” Hilbert space $\mathcal{H}$ (the space of quantum states)
with orthonormal basis $\\{|e_{i}\rangle\\}$ in one-to-one correspondence
$\\{|e_{i}\rangle\leftrightarrow\phi_{i}\\}$ with the elements of
$\mathcal{O}$. There results a family $\mathcal{C}$ of unit vectors
$|x\rangle$ in $\mathcal{H}$, named, in a generalised sense, coherent states
(CS). These vectors are labelled by elements of $X$ and resolve the identity
operator in $\mathcal{H}$:

(4.2) $X\ni
x\mapsto|x\rangle=\frac{1}{\sqrt{\mathcal{N}(x)}}\sum_{i}\overline{\phi_{i}(x)}|e_{i}\rangle\in\mathcal{H}\,.$
(4.3) $\langle x|x\rangle=1\,,\quad\int_{X}\,|x\rangle\langle
x|\,\mathrm{d}\nu(x)=\mathbbm{1}_{\mathcal{H}}\,,\quad\mathrm{d}\nu(x)=\mathcal{N}(x)\,\mathrm{d}\mu(x)\,.$

We now explain the meaning of the CS quantization within this framework. Let
us denote by $\mathbb{P}_{\mathfrak{o}}$ the orthogonal projector mapping
$L^{2}(X,\mathrm{d}\mu(x))$ onto the sub-Hilbert space
$\mathcal{H}_{\mathfrak{o}}$, which is the closure of the linear span of the
elements of $\mathcal{O}$. Clearly, $\mathbb{P}_{\mathfrak{o}}$ is the
identity operator on $\mathcal{H}_{\mathfrak{o}}$. Now, we identify
$\mathcal{H}_{\mathfrak{o}}\equiv\mathcal{H}$ through a unitary map
$U_{\mathfrak{o}}$ such that we have

(4.4) $U_{\mathfrak{o}}|e_{i}\rangle=\phi_{i}\,.$

Let $f(x)$ be a function defining the multiplication operator in
$L^{2}(X,\mathrm{d}\mu(x))$:

(4.5) $M_{f}:\phi\mapsto f\,\phi\,.$

######  Proposition 4.1.

(Toeplitz quantization) We have the identity

(4.6)
$\mathbb{P}_{\mathfrak{o}}M_{f}\mathbb{P}_{\mathfrak{o}}\phi_{i}=U_{\mathfrak{o}}A_{f}U^{-1}_{\mathfrak{o}}\phi_{i}\;,\quad\forall
i\in\mathbb{N}\;,$

where $A_{f}$ is the CS quantisation of $f$ as

(4.7) $A_{f}=\int_{X}\,f(x)\,|x\rangle\langle x|\,\mathrm{d}\nu(x)\,.$

###### Proof.

We will compare the respective matrix elements of the left- and right-hand
sides of (4.6) with respect to the basis $\\{|e_{i}\rangle\\}$ of
$\mathcal{H}$. Let us start with the action of
$U_{\mathfrak{o}}A_{f}U^{-1}_{\mathfrak{o}}$ on $\phi_{i}$:

$\displaystyle U_{\mathfrak{o}}A_{f}U^{-1}_{\mathfrak{o}}\phi_{i}$
$\displaystyle=U_{\mathfrak{o}}A_{f}|e_{i}\rangle=U_{\mathfrak{o}}\int_{X}\,f(x)\,|x\rangle\langle
x|e_{i}\rangle\,\mathrm{d}\nu(x)$
$\displaystyle=U_{\mathfrak{o}}\sum_{jk}\int_{X}\,f(x)\,\overline{\phi_{j}}(x)\,\phi_{k}(x)\,\mathrm{d}\mu(x)\,|e_{j}\rangle\langle
e_{k}|e_{i}\rangle$
$\displaystyle=U_{\mathfrak{o}}\sum_{j}\int_{X}\,f(x)\,\overline{\phi_{j}}(x)\,\phi_{i}(x)\,\mathrm{d}\mu(x)\,|e_{j}\rangle$
$\displaystyle=\sum_{j}\left(A_{f}\right)_{ji}\,U_{\mathfrak{o}}|e_{j}\rangle=\sum_{j}\left(A_{f}\right)_{ji}\,\phi_{j}.$

On the other hand,

$\displaystyle\mathbb{P}_{\mathfrak{o}}M_{f}\mathbb{P}_{\mathfrak{o}}\phi_{i}=\mathbb{P}_{\mathfrak{o}}M_{f}\phi_{i}$
$\displaystyle=\sum_{j}\left(\phi_{j},f\phi_{i}\right)_{L^{2}}\,\phi_{j}$
$\displaystyle=\sum_{j}\phi_{j}\,\int_{X}\overline{\phi_{j}}(x)\,f(x)\,\phi_{i}(x)\,\mathrm{d}\mu(x)$
$\displaystyle=\sum_{j}\left(A_{f}\right)_{ji}\,\phi_{j}\,.$

$\square$

Note that $M_{\Delta}:=M_{\chi_{\Delta}}$ defines a projection-valued measure
such that $M_{f}=\int f(x)\,\mathrm{d}M_{x}$. Indeed, let $g\in
L^{2}(X,\mathrm{d}\mu(x))$ and
$\mu_{g}(\cdot):=\int_{(\cdot)}|g(x)|^{2}\,\mathrm{d}\mu(x)=\left(g|M_{\chi_{(\cdot)}}g\right)_{L^{2}}$
be the corresponding measure of which $|g|^{2}$ is the Radon-Nikodym
derivative ($\mathrm{d}\mu_{g}(x)=|g(x)|^{2}\,\mathrm{d}\mu(x)$). Then, for
every $g\in L^{2}(X,\mathrm{d}\mu(x))$,

$\displaystyle\int f(y)\,\mathrm{d}\left(g|M_{\chi_{(\cdot)}}g\right)_{L^{2}}$
$\displaystyle=\int f(y)\,\mathrm{d}\mu_{g}(y)=\int
f(y)|g(y)|^{2}\,\mathrm{d}\mu(y)$
$\displaystyle=\left(g|fg\right)_{L^{2}}=\left(g|M_{f}g\right)_{L^{2}}\,.$

where $\mathrm{d}\left(g|M_{\chi_{(\cdot)}}g\right)_{L^{2}}$ denotes
integration with respect to the measure $\left(g|M_{(\cdot)}g\right)_{L^{2}}$.
Moreover, $M_{(\cdot)}:=$ $M_{\chi_{(\cdot)}}$ is a Naimark dilation of the
POVM $U_{\mathfrak{o}}A_{\Delta}U^{-1}_{\mathfrak{o}}:=$
$U_{\mathfrak{o}}A_{\chi_{\Delta}}U^{-1}_{\mathfrak{o}}=\int_{\Delta}U_{\mathfrak{o}}|x\rangle\langle
x|U^{-1}_{\mathfrak{o}}\,\mathrm{d}\nu(x)$ since, by proposition 4.1,
$\mathbb{P}_{\mathfrak{o}}M_{\chi_{\Delta}}\mathbb{P}_{\mathfrak{o}}\phi_{i}=U_{\mathfrak{o}}A_{\chi_{\Delta}}U^{-1}_{\mathfrak{o}}\phi_{i}$.
In other words, and with slight abuse of language, every operator $A_{f}=\int
f(x)\,\mathrm{d}A_{x}$ is the projection of the operator $M_{f}=\int
f(x)\,\mathrm{d}M_{x}$ where $\mathrm{d}A_{x}$ denotes integration with
respect to the POVM $A_{\chi_{\Delta}}$ and $\mathrm{d}M_{x}$ denotes
integration with respect to the PV measure $M_{\chi_{\Delta}}$. In its turn,
$M_{\chi_{\Delta}}$ is the Naimark’s dilation of $A_{\chi_{\Delta}}$. An
alternative construction of families of $X$-labelled coherent states rests
upon unitary transports of one unit-norm state $|\psi\rangle$ picked in the
Hilbert space $\mathcal{H}$. Let $\\{|e_{i}\rangle\\}$ be an orthonormal basis
of $\mathcal{H}$ and let $U(x)$ be a family of $X$-labelled unitary operators
on $\mathcal{H}$ which obey square-integrability and orthonormality on
$(X,\nu)$ in the sense given in the following.

######  Proposition 4.2.

Let $\psi$ be arbitrarily picked in $\mathcal{H}$. The family of $X$-labelled
unit-norm states in $\mathcal{H}$ defined by

(4.8) $X\ni x\mapsto|x\rangle_{\psi}:=U(x)|\psi\rangle\,,$

solves the identity in $\mathcal{H}$ with respect to the measure $\nu$ on $X$,

(4.9) $\int_{X}|x\rangle_{\psi}{}_{\psi}\langle
x|\,\mathrm{d}\nu(x)=\mathbbm{1}\,,$

if and only if the $X$-labelled family of unitary operators $U(x)$ obeys
square-integrability and orthonormality on $(X,\nu)$ in the following sense

(4.10) $\displaystyle\delta_{ik}$
$\displaystyle=\int_{X}U_{ji}(x)\,\overline{U_{jk}}(x)\,\mathrm{d}\nu(x)\quad\forall\,j\,,$
(4.11) $\displaystyle 0$
$\displaystyle=\int_{X}\left[U_{ji}(x)\,\overline{U_{lk}}(x)+U_{li}(x)\,\overline{U_{jk}}(x)\right]\,\mathrm{d}\nu(x)\quad\forall\,j,l\,,\
j\neq l\,,$

where $U_{ij}(x):=\langle e_{i}|U(x)|e_{j}\rangle$.

###### Proof.

By expanding $\psi=\sum_{i}a_{i}\,|e_{i}\rangle$, we have,

$|x\rangle_{\psi}=U(x)|\psi\rangle=\sum_{i}a_{i}\,U(x)\,|e_{i}\rangle=\sum_{ij}a_{i}\,U_{ji}(x)\,|e_{j}\rangle\,,$

and so

$\displaystyle|x\rangle_{\psi}{}_{\psi}\langle x|$
$\displaystyle=\sum_{ijkl}a_{i}\bar{a}_{k}\,U_{ji}(x)\,\overline{U_{lk}}(x)\,|e_{j}\rangle\langle
e_{l}|$
$\displaystyle=\sum_{ik}a_{i}\bar{a}_{k}\,\sum_{j}U_{ji}(x)\,\overline{U_{jk}}(x)\,|e_{j}\rangle\langle
e_{j}|+\sum_{ik}a_{i}\bar{a}_{k}\,\sum_{j\neq
l}U_{ji}(x)\,\overline{U_{lk}}(x)\,|e_{j}\rangle\langle e_{l}|\,.$

We now integrate the previous expression and impose (4.9). From the
completeness of the basis, $\sum_{j}|e_{j}\rangle\langle e_{j}|=\mathbbm{1}$,
and linear independence of the operators $|e_{j}\rangle\langle e_{l}|$, we
infer

(4.12) $\displaystyle 1$
$\displaystyle=\sum_{ik}a_{i}\bar{a}_{k}\,\int_{X}U_{ji}(x)\,\overline{U_{jk}}(x)\,\mathrm{d}\nu(x)\quad\forall\,j\,,$
(4.13) $\displaystyle 0$
$\displaystyle=\sum_{ik}a_{i}\bar{a}_{k}\,\int_{X}U_{ji}(x)\,\overline{U_{lk}}(x)\,\mathrm{d}\nu(x)\quad\forall\,j,l,\
j\neq l\,,$

for any unit-norm complex sequence $(a_{i})\in\ell^{2}$. Let us consider the
two operators $\mathrm{M}^{j}$ and $\mathrm{M}^{j,l}$, $j\neq l$, with matrix
elements with respect to the basis $\\{e_{i}\\}$,

(4.14)
$\mathrm{M}^{j}_{ki}=\int_{X}U_{ji}(x)\,\overline{U_{jk}}(x)\,\mathrm{d}\nu(x)\,,\quad\mathrm{M}^{j,l}_{ki}=\int_{X}U_{ji}(x)\,\overline{U_{lk}}(x)\,\mathrm{d}\nu(x)\,.$

The operator $\mathrm{M}^{j}$ is Hermitian and the condition (4.12) means that
its mean value obeys $\langle\psi|\mathrm{M}^{j}|\psi\rangle=1$ for any
normalised $|\psi\rangle\in\mathcal{H}$. There results that
$\mathrm{M}^{j}=\mathbbm{1}$, and so

$\int_{X}U_{ji}(x)\,\overline{U_{jk}}(x)\,\mathrm{d}\nu(x)=\delta_{ik}\quad\forall\,j\,.$

The operator $\mathrm{M}^{j,l}$ obeys
$\left(\mathrm{M}^{j,l}\right)^{{\dagger}}=\overline{\mathrm{M}}^{l,j}$, and
so the operator $\mathrm{M}^{j,l}+\mathrm{M}^{l,j}$ is Hermitian. The
condition (4.13) means that its mean value obeys
$\langle\psi|\mathrm{M}^{j,l}+\mathrm{M}^{l,j}|\psi\rangle=0$ for any
normalised $|\psi\rangle\in\mathcal{H}$. There results that
$\mathrm{M}^{j,l}+\mathrm{M}^{l,j}=\mathbb{0}$, and so

$\int_{X}\left[U_{ji}(x)\,\overline{U_{lk}}(x)+U_{li}(x)\,\overline{U_{jk}}(x)\right]\,\mathrm{d}\nu(x)=0\quad\forall\,j,l\,,\
j\neq l\,.$

$\square$

The above result extends to density operators.

######  Proposition 4.3.

Let $\rho$ be a density operator on $\mathcal{H}$, and
$\rho(x):=U(x)\,\rho\,U^{{\dagger}}(x)$ its unitary transport $U(x)$ which has
square-integrability and orthonormality properties (4.10) and (4.11). We then
have the resolution of the identity:

(4.15) $\int_{X}\rho(x)\,\mathrm{d}\nu(x)=\mathbbm{1}\,.$

###### Proof.

From the decomposition of $\rho$ in a sum of rank-one projectors,

$\rho=\sum_{i}p_{i}\,|\psi_{i}\rangle\langle\psi_{i}|\,,\quad\|\psi_{i}\|^{2}=1\,\quad\sum_{i}p_{i}=1\,,\quad
0\leq p_{i}\leq 1\,,$

we have

$\rho(x)=\sum_{i}p_{i}\,|x\rangle_{\psi_{i}}{}_{\psi_{i}}\langle x|\,.$

By applying (4.9) to each CS projectors in this sum, we obtain

$\int_{X}\rho(x)\,\mathrm{d}\nu(x)=\sum_{i}p_{i}\,\mathbbm{1}=\mathbbm{1}\,.$

$\square$

Let us now understand the Toeplitz quantization within this alternative
approach. Implementing the construction (4.8) with the choice $\psi=e_{j}$ and
$U^{{\dagger}}(x)$, we obtain

(4.16) $|x\rangle_{e_{j}}=\sum_{i}\overline{U_{ji}}(x)\,|e_{i}\rangle\,.$

We denote by $\mathcal{O}_{j}=\\{U_{ji}\\}$ the orthonormal system in
$L^{2}(X,\mathrm{d}\nu(x))$ formed by the functions $U_{ji}(x)$ at fixed $j$
according to the condition (4.10), and by $\mathbb{P}_{\mathfrak{o}_{j}}$ the
corresponding orthogonal projector mapping $L^{2}(X,\mathrm{d}\nu(x))$ onto
the sub-Hilbert space $\mathcal{H}_{\mathfrak{o}_{j}}$ which is the closure of
the linear span of the elements of $\mathcal{O}_{j}$. With the identification
(4.4), i.e. $U_{\mathfrak{o}_{j}}|e_{i}\rangle=U_{ji}$, relation (4.6) holds
under the form,

(4.17)
$\mathbb{P}_{\mathfrak{o}_{j}}M_{f}\mathbb{P}_{\mathfrak{o}_{j}}U_{ji}=U_{\mathfrak{o}_{j}}A^{(j)}_{f}U^{-1}_{\mathfrak{o}_{j}}U_{ji}\,,\quad
A^{(j)}_{f}=\int_{X}f(x)\,|x\rangle_{e_{j}}{}_{e_{j}}\langle
x|\,\mathrm{d}\nu(x)\,.$

We now extend this result to the integral quantization with a density operator
$\rho$ on $\mathcal{H}$. We choose the orthonormal basis $\\{|e_{i}\rangle\\}$
of the latter as made of the eigenstates of $\rho$ (with possible null
eigenvalue),

(4.18) $\rho=\sum_{i}p_{i}\,|e_{i}\rangle\langle
e_{i}|\,,\quad\sum_{i}p_{i}=1\,,\quad 0\leq p_{i}\leq 1\,.$

Clearly we have,

(4.19)
$A^{(\rho)}_{f}=\int_{X}f(x)\,\rho(x)\,\mathrm{d}\nu(x)=\sum_{j}p_{j}\int_{X}f(x)\,|x\rangle_{e_{j}}{}_{e_{j}}\langle
x|\,\mathrm{d}\nu(x)=\sum_{j}p_{j}A^{(j)}_{f}\,.$

By linearity, from (4.17) we infer

(4.20)
$\sum_{j}p_{j}\mathbb{P}_{\mathfrak{o}_{j}}M_{f}\mathbb{P}_{\mathfrak{o}_{j}}=\sum_{j}p_{j}U_{\mathfrak{o}_{j}}A^{(j)}_{f}U^{-1}_{\mathfrak{o}_{j}}\,.$

Let us complete (4.10) by imposing the full orthonormality constraint on the
matrix elements $U_{ij}(x)$:

(4.21) $\displaystyle\delta_{jl}\delta_{ik}$
$\displaystyle=\int_{X}U_{ji}(x)\,\overline{U_{lk}}(x)\,\mathrm{d}\nu(x)\quad\forall\,i,j,k,l\,.$

It follows that $\mathbb{P}_{\mathfrak{o}_{j}}\mathbb{P}_{\mathfrak{o}_{l}}=0$
$\forall j\neq l$. Moreover, (4.19) becomes the direct sum

(4.22) $A^{(\rho)}_{f}=\bigoplus_{j}p_{j}A^{(j)}_{f}\,.$

Defining the direct sum of unitary operators
$U_{\mathfrak{o}}=\bigoplus_{j}U_{\mathfrak{o}_{j}}$ we write (4.20) as

(4.23)
$\bigoplus_{j}p_{j}\mathbb{P}_{\mathfrak{o}_{j}}M_{f}\mathbb{P}_{\mathfrak{o}_{j}}=\bigoplus_{j}p_{j}U_{\mathfrak{o}_{j}}A^{(j)}_{f}U_{\mathfrak{o}_{j}}^{-1}=U_{\mathfrak{o}}A^{(\rho)}_{f}U^{-1}_{\mathfrak{o}}$

where we recover an analogous result about Naimark dilation in the case of
density matrices.

##  5\. Integral POVM quantization of fonctions on the circle

Let us now describe the integral quantization for building observables acting
on the two-dimensional real Hilbert space, i.e., the Euclidean plane. In this
section, we give an overview of the material developed in [65] on the integral
quantization of functions on the unit circle. We describe rays in the
Euclidean plane, and real POVMs built from density matrices in dimension 2. We
then discuss the integral quantization procedure based on the group of
rotations SO$(2)$.

###  5.1. Euclidean plane with Dirac notations

First, a quantum state can be decomposed into a sum of pure states, each
identified with a unit vector. In the specific case of $\mathbb{R}^{2}$, the
polar angle $\phi\in[0,2\pi)$ is associated with the unit vector
$\widehat{\mathbf{u}}_{\phi}$. This unit vector corresponds to the pure state
$|\phi\rangle:=\left|\widehat{\mathbf{u}}_{\phi}\right\rangle$. In particular,
the two axes of the Euclidean plane can be described as a “horizontal
orientation" state $\widehat{\boldsymbol{\imath}}=|0\rangle$ and a “vertical
orientation" state
$\widehat{\boldsymbol{\jmath}}=\left|\dfrac{\pi}{2}\right\rangle$. The plane
is depicted in figure 1.

$\widehat{\boldsymbol{\imath}}=|0\rangle\equiv\begin{pmatrix}1\\\
0\end{pmatrix}$$|\phi\rangle=\begin{pmatrix}\cos\phi\\\
\sin\phi\end{pmatrix}$(quantum) state in “orientation
representation”$O$$\langle
0|0\rangle=1=\left\langle\frac{\pi}{2}\right|\left.\frac{\pi}{2}\right\rangle\,,\quad\langle
0\left|\frac{\pi}{2}\right\rangle=0$$\mathbbm{1}=|0\rangle\langle
0|+\left|\frac{\pi}{2}\right\rangle\left\langle\frac{\pi}{2}\right|\
\Leftrightarrow\ \left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right)=\left(\begin{array}[]{cc}1&0\\\
0&0\end{array}\right)+\left(\begin{array}[]{cc}0&0\\\
0&1\end{array}\right)$$\widehat{\boldsymbol{\jmath}}=|\pi/2\rangle=\begin{pmatrix}0\\\
1\end{pmatrix}$$1$$\phi$ Figure 1. The Euclidean plane and its unit vectors
viewed as pure quantum states in Dirac ket notations.

Therefore, a pure state in the horizontal-vertical representation can be
decomposed as

(5.1)
$|\phi\rangle=\cos{\phi}\,|0\rangle+\sin{\phi}\,\left|\frac{\pi}{2}\right\rangle\,,\quad\langle
0|\phi\rangle=\cos\phi\,,\qquad\left\langle\frac{\pi}{2}\right|\phi\rangle=\sin\phi\,,$

with the associated probability distributions $\phi\mapsto\cos^{2}\phi$ and
$\phi\mapsto\sin^{2}\phi$ on the unit circle equipped with the measure
$\mathrm{d}\phi/\pi$. One also have the unsharp unit circle representation of
a pure state given by the trigonometric function

(5.2) $\langle\eta|\phi\rangle=\cos(\phi-\eta)\,.$

with associated probability distribution families

(5.3)
$\eta\mapsto\mathcal{P}_{\phi}(\eta)=|\langle\eta|\phi\rangle|^{2}=\cos^{2}(\phi-\eta)\,,\quad\int_{0}^{2\pi}\mathcal{P}_{\phi}(\eta)\,\frac{\mathrm{d}\eta}{\pi}=1\,.$

To $|\phi\rangle$ corresponds the orthogonal projector
$E_{\phi}=|\phi\rangle\langle\phi|$ (also called pure state),

(5.4) $\displaystyle E_{\phi}=\begin{pmatrix}\cos{\phi}\\\
\sin{\phi}\end{pmatrix}\begin{pmatrix}\cos{\phi}&\sin{\phi}\end{pmatrix}=\begin{pmatrix}\cos^{2}{\phi}&\cos{\phi}\sin{\phi}\\\
\cos{\phi}\sin{\phi}&\sin^{2}{\phi}\end{pmatrix}=\mathcal{R}(\phi)|0\rangle\langle
0|\mathcal{R}(-\phi)\,,$

where $\mathcal{R}(\phi)=\begin{pmatrix}\cos\phi&-\sin\phi\\\
\sin\phi&\cos\phi\end{pmatrix}$. In contrast to pure states the mixed states
are defined by two-dimensional density matrices of the form

(5.5) $\rho:=\mathsf{M}(a,b)=\begin{pmatrix}a&b\\\ b&1-a\end{pmatrix}\,,\quad
0\leq a\leq 1\,,\quad\Delta:=\det\rho=a(1-a)-b^{2}\geq 0\,.$

From the spectral decomposition of $\rho$

(5.6) $\rho=\lambda E_{\phi}+(1-\lambda)E_{\phi+\pi/2}\,,$

we only have to consider the upper half-plane, i.e., $0\leq\phi\leq\pi$. The
highest eigenvalue of $\rho$ is $\lambda=\frac{1}{2}(1+\sqrt{1-4\Delta})$,
such that $1/2\leq\lambda\leq 1$. Defining $r:=2\lambda-1$, $0\leq r\leq 1$, a
real density matrix takes the form

(5.7)
$\displaystyle\rho=\left(\frac{1+r}{2}\right)E_{\phi}+\left(\frac{1-r}{2}\right)E_{\phi+\pi/2}\;,$

and, in polar coordinates $(r,\phi)$, we obtain

(5.8)
$\rho\equiv\rho_{r,\phi}=\frac{1}{2}\mathbbm{1}+\frac{r}{2}\mathcal{R}(\phi)\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\mathcal{R}(-\phi)=\begin{pmatrix}\frac{1}{2}+\frac{r}{2}\cos
2\phi&\frac{r}{2}\sin 2\phi\\\ \frac{r}{2}\sin
2\phi&\frac{1}{2}-\frac{r}{2}\cos 2\phi\end{pmatrix}\,.$

In particular, we retrieve the projector on the unit vector,
$\rho_{1,\phi}=E_{\phi}$. Note that the parameter $r$ encodes the distance of
$\rho$ to the pure state $E_{\phi}$ while $1-r$ measures the degree of
“mixing" [65]. Its statistical interpretation is given by the von Neumann
entropy defined as

(5.9)
$S_{\rho}:=-\mathrm{Tr}(\rho\,\ln\rho)=-\lambda\ln\lambda-(1-\lambda)\ln(1-\lambda)=-\frac{1+r}{2}\ln\frac{1+r}{2}-\frac{1-r}{2}\ln\frac{1-r}{2}\,.$

As a function of $\lambda\in[1/2,1]$, $S_{\rho}$ is non-negative, concave and
symmetric with respect to its maximum value $\log 2$ at $\lambda=1/2$, which
corresponds to $r=0$, i.e., $\rho_{0}\equiv\mathbbm{1}/2$, which describes the
state of completely random orientations.

###  5.2. Integral POVM quantization

The measure space $(X,\mathrm{d}\nu(x))$ for the Euclidean plane is the unit
circle with its uniform (Lebesgue) measure:

(5.10)
$X=\mathbb{S}^{1}\,,\quad\mathrm{d}\nu(x)=\frac{\mathrm{d}\phi}{\pi}\,,\quad\phi\in[0,2\pi)\,.$

$\mathbb{S}^{1}$ is here viewed as the abelian group SO$(2)$ of rotations in
the plane. As a consequence, the density operator (5.8) is manifestly
covariant under rotation

(5.11)
$\rho_{r,\phi_{0}}(\phi)=\mathcal{R}\left(\phi\right)\rho_{r,\phi_{0}}\mathcal{R}\left(-\phi\right)=\rho_{r,\phi_{0}+\phi}\,,\quad
0\leq\phi<2\pi\,.$

Additionally, one easily proves by elementary integration on matrix elements
that this family solves the identity

(5.12)
$\int_{0}^{2\pi}\rho_{r,\phi+\phi_{0}}\,\frac{\mathrm{d}\phi}{\pi}=\mathbbm{1}\,.$

This property allows to apply the procedure (3.2) to quantize a function (or
distribution) $f(\phi)$ on the circle. Its quantum counterpart is the
2$\times$2 matrix operator

(5.13) $\begin{split}f\mapsto
A_{f}&=\int_{0}^{2\pi}f(\phi)\rho_{r,\phi+\phi_{0}}\,\frac{\mathrm{d}\phi}{\pi}=\begin{pmatrix}\langle
f\rangle+\frac{r}{2}C_{c}\left(R_{\phi_{0}}f\right)&\frac{r}{2}C_{s}\left(R_{\phi_{0}}f\right)\\\
\frac{r}{2}C_{s}\left(R_{\phi_{0}}f\right)&\langle
f\rangle-\frac{r}{2}C_{c}\left(R_{\phi_{0}}f\right)\end{pmatrix}\\\ &=\langle
f\rangle\,I+\frac{r}{2}\left[C_{c}\left(R_{\phi_{0}}f\right)\,\sigma_{3}+C_{s}\left(R_{\phi_{0}}f\right)\,\sigma_{1}\right]\,,\end{split}$

where $\langle f\rangle:=\frac{1}{2\pi}\int_{0}^{2\pi}f(\phi)\,\mathrm{d}\phi$
is the average of $f$ on the unit circle and
$R_{\phi_{0}}(f)(\phi):=f(\phi-\phi_{0})$. The symbols $C_{c}$ and $C_{s}$ are
for the cosine and sine doubled angle Fourier coefficients of $f$,

(5.14) $C_{c}(f)=\int_{0}^{2\pi}f(\phi)\cos
2\phi\,\frac{\mathrm{d}\phi}{\pi}\,,\quad C_{s}(f)=\int_{0}^{2\pi}f(\phi)\sin
2\phi\,\frac{\mathrm{d}\phi}{\pi}\,.$

We observe that the quantisation of the components
$(\cos(\phi-\phi_{0}),\sin(\phi-\phi_{0}))$ of the density matrix eigenvectors
give $0$. Let us denote by $N_{\phi_{0}}$ the space of functions or
distributions (in a certain sense) $T(\phi)$ on the circle which obey

(5.15)
$\int_{0}^{2\pi}T(\phi)\rho_{r,\phi+\phi_{0}}\,\frac{\mathrm{d}\phi}{\pi}=0\,.$

Such a space includes the orthogonal in
$L_{\mathbb{R}}^{2}(\mathbb{S}^{1},\mathrm{d}\phi/\pi)$ to the $3$-dimensional
subspace spanned by the functions $1$, $\cos 2\phi$, and $\sin 2\phi$. Thus,
the operator $A_{f}$ is the $\rho_{r,\phi+\phi_{0}}$-quantization of $f$
modulo any element of $N_{\phi_{0}}$. Actually, by identifying the Euclidean
space $\mathbb{R}^{3}$ with the real span $V_{3}$ of the orthonormal set in
$L_{\mathbb{R}}^{2}(\mathbb{S}^{1},\mathrm{d}\phi/\pi)$,

(5.16)
$V_{3}=\mathrm{Span}\left\\{e_{0}(\phi):=\frac{1}{\sqrt{2}},e_{1}(\phi):=\cos
2\phi,e_{2}(\phi):=\sin 2\phi\right\\}\,,$

the linear map (5.13) yields a non-commutative version of $\mathbb{R}^{3}$. In
particular, we have

(5.17) $\displaystyle A_{e_{0}}$
$\displaystyle=\frac{\mathbbm{1}}{\sqrt{2}}\,,$ (5.18) $\displaystyle
A_{e_{1}}$ $\displaystyle=\frac{r}{2}[\cos 2\phi_{0}\,\sigma_{3}+\sin
2\phi_{0}\,\sigma_{1}]\equiv\frac{r}{2}\sigma_{2\phi_{0}}\,,$ (5.19)
$\displaystyle A_{e_{2}}$ $\displaystyle=\frac{r}{2}[-\sin
2\phi_{0}\,\sigma_{3}+\cos
2\phi_{0}\,\sigma_{1}]\equiv\frac{r}{2}\sigma_{2\phi_{0}+\pi/2}\,,$

with the non-zero commutation rule

(5.20)
$\left[A_{e_{1}},A_{e_{2}}\right]=-\frac{r^{2}}{2}\tau_{2}\,,\quad\tau_{2}:=\begin{pmatrix}0&-1\\\
1&0\end{pmatrix}\,.$

In the above we have associated with the unit vector
$\widehat{\mathbf{u}}_{\phi}$ (5.1) the symmetric matrix,

(5.21)
$\sigma_{\phi}:=\cos\phi\,\sigma_{3}+\sin\phi\,\sigma_{1}\equiv\overrightarrow{\boldsymbol{\sigma}}\cdot\widehat{\mathbf{u}}_{\phi}=\begin{pmatrix}\cos\phi&\sin\phi\\\
\sin\phi&-\cos\phi\end{pmatrix}=\mathcal{R}(\phi)\,\sigma_{3}\,,$

where we have introduced the vector notation
$\overrightarrow{\boldsymbol{\sigma}}=\sigma_{3}\widehat{\boldsymbol{\imath}}+\sigma_{1}\widehat{\boldsymbol{\jmath}}$.
Note the commutation rule

(5.22)
$\left[\sigma_{\phi},\sigma_{\phi^{\prime}}\right]=2\sin(\phi-\phi^{\prime})\tau_{2}\,.$

Also note that the $3$ matrices $\sigma_{1}$, $\tau_{2}$, $\sigma_{3}$ span
the Lie algebra $\mathfrak{sl}$(2,$\mathbb{R}$), while $\tau_{2}$ is precisely
the generator of rotations in the plane:

(5.23) $\mathcal{R}(\phi)=e^{\tau_{2}\phi}\,.$

The spectral decomposition of the quantum observable $\sigma_{2\phi}$
precisely involves its eigen-orientations $\left|\phi\right\rangle$
(eigenvalue $1$) and $\left|\phi+\frac{\pi}{2}\right\rangle$ (eigenvalue
$-1$),

(5.24)
$\sigma_{2\phi}=\left|\phi\right\rangle\left\langle\phi\right|-\left|\phi+\frac{\pi}{2}\right\rangle\left\langle\phi+\frac{\pi}{2}\right|=E_{\phi}-E_{\phi+\pi/2}\,.$

In agreement with (5.6), the density operator (5.8) is expressed in terms of
$\sigma_{2\phi}$ as

(5.25) $\rho_{r,\phi}=\frac{1}{2}\left(\mathbbm{1}+r\sigma_{2\phi}\right)\,.$

Finally, note that the covariance requirement emerges from the rotational
invariance of the density operator (5.11)

(5.26)
$\mathcal{R}\left(\theta\right)\,A_{f}\,\mathcal{R}\left(-\theta\right)=A_{R_{\theta}(f)}\,.$

An interesting question concerns the existence of functions $f(\phi)$ in
$V_{3}$ yielding density matrices through the quantization map (5.13). Solving
this problem amounts to find $f(\phi)=a_{0}+a_{2}\cos 2\phi+b_{2}\sin 2\phi$
such that $A_{f}=\rho_{s,\theta}$. Solutions are given from the relations

(5.27) $a_{0}=\frac{1}{2}\,,\quad\left\\{\begin{array}[]{c}r\left(a_{2}\cos
2\phi_{0}-b_{2}\sin 2\phi_{0}\right)=s\cos 2\theta\\\ r\left(a_{2}\sin
2\phi_{0}+b_{2}\cos 2\phi_{0}\right)=s\sin 2\theta\end{array}\right.\,,\quad
0\leq r\,,\,s\leq 1\,.$

Putting $a_{2}=l\cos 2\theta_{0}$ and $b_{2}=l\sin 2\theta_{0}$ with $l>0$, we
obtain from above,

(5.28) $l=\frac{s}{r}\,,\quad\theta_{0}=\theta-\phi_{0}\,,$

which yields the solution

(5.29) $f(\phi)=\frac{1}{2}+\frac{s}{r}\,\cos 2(\phi+\phi_{0}-\theta)\,.$

The representation of a given mixed state as a continuous superposition of
mixed states as

(5.30) $\rho_{s,\theta}=\int_{0}^{2\pi}\left[\frac{1}{2}+\frac{s}{r}\,\cos
2\phi\right]\,\rho_{r,\phi+\theta}\,\frac{\mathrm{d}\phi}{\pi}\,,$

which is convex for $r\geq 2s$, provides one more illustration of a typical
property of quantum-mechanical ensembles in comparison with their classical
counterparts. Note that if $r\geq 2s$ holds,
$\phi\mapsto\left[\frac{1}{2}+\frac{s}{r}\,\cos 2\phi\right]$ is a probability
distribution and (5.30) can also be viewed as the average of these mixed
states.

###  5.3. Quantization and symbol calculus

Here, we go forward in the algebraic and geometrical interpretation of the
quantisation map (5.13), yielded by the POVM built from the family of density
matrices

(5.31)
$\left\\{\rho_{r,\phi+\phi_{0}}\,,\,0\leq\phi<2\pi\,\mathrm{mod}\,2\pi\right\\}\,,$

with parameters $0\leq r\leq 1$ and $0\leq\phi_{0}<2\pi\,\mathrm{mod}\,2\pi$,
and restricted to the $3$-d Fourier realisation $V_{3}$ of the Euclidean space
$\mathbb{R}^{3}$. Given a vector
$\vec{f}=f_{1}\widehat{\boldsymbol{\imath}}+f_{2}\widehat{\boldsymbol{\jmath}}+f_{0}\widehat{\boldsymbol{k}}\in\mathbb{R}^{3}$,
identified with $\vec{f}=f_{1}e_{1}+f_{2}e_{2}+f_{0}e_{0}\in V_{3}$, its
quantization is the quantum observable

(5.32)
$A_{f}=f_{1}A_{e_{1}}+f_{2}A_{e_{2}}+f_{0}A_{e_{0}}=f_{1}\frac{r}{2}\sigma_{2\phi_{0}}+f_{2}\frac{r}{2}\sigma_{2\phi_{0}+\pi/2}+f_{0}\frac{\mathbbm{1}}{\sqrt{2}}\,.$

Three basic matrices, namely the identity $\mathbbm{1}$ and the two real Pauli
matrices $\sigma_{1}$ and $\sigma_{3}$, generate the Jordan algebra
$\mathcal{J}_{3}$ 222A Jordan algebra is an (not necessarily associative)
algebra over a field whose multiplication is commutative and satisfies the
_Jordan identity_ : $(xy)x^{2}=x(yx^{2})$ . of all real symmetric $2\times 2$
matrices. Any element $A$ of the algebra decomposes as

(5.33) $A\equiv\left(\begin{array}[]{cc}a&b\\\
b&d\end{array}\right)=\frac{a+d}{2}I+\frac{a-d}{2}\sigma_{3}+b\sigma_{1}\equiv\alpha
I+\delta\sigma_{3}+\beta\sigma_{1}\,.$

The product in this algebra is defined by

(5.34) $\mathcal{O^{\prime\prime}}=A\odot
A^{\prime}=\frac{1}{2}\left(AA^{\prime}+A^{\prime}A\right)\,,$

which entails on the level of components $\alpha,\delta,\beta$, the relations
:

(5.35)
$\alpha^{\prime\prime}=\alpha\alpha^{\prime}+\delta\delta^{\prime}+\beta\beta^{\prime},\
\delta^{\prime\prime}=\alpha\delta^{\prime}+\alpha^{\prime}\delta,\
\beta^{\prime\prime}=\alpha\beta^{\prime}+\alpha^{\prime}\beta\,.$

In particular we have

(5.36)
$\sigma_{1}\odot\sigma_{3}=\sigma_{3}\odot\sigma_{1}=0\,,\quad\mbox{while}\quad\sigma_{1}\sigma_{3}=-\sigma_{3}\sigma_{1}=\tau_{2}=\begin{pmatrix}0&-1\\\
1&0\end{pmatrix}\,.$

Equipped with the scalar product

(5.37) $\langle A|B\rangle_{\mathcal{J}_{3}}=\mathrm{Tr}(AB)$

the algebra $\mathcal{J}_{3}$ is also a $3$-d real Hilbert space. To any
$\phi\in[0,2\pi)\,\mathrm{mod}\,2\pi$ corresponds the orthonormal basis

(5.38)
$\left\\{\frac{\sigma_{\phi}}{\sqrt{2}}\,,\quad\frac{\sigma_{\phi+\pi/2}}{\sqrt{2}}\,,\quad\frac{\mathbbm{1}}{\sqrt{2}}\right\\}\,.$

It results from (5.32) that, for $0<r\leq 1$, we have

(5.39) $\|\vec{f}\|^{2}_{V_{3}}=f_{1}^{2}+f_{2}^{2}+f_{0}^{2}=\langle
A_{f}|\Delta_{r}A_{f}\rangle_{\mathcal{J}_{3}}\,,\quad\Delta_{r}:=\mathrm{diag}\left(\frac{2}{r^{2}},\frac{2}{r^{2}},1\right)\,,$

where $A_{f}$ is viewed as the vector

(5.40) $A_{f}=\begin{pmatrix}\frac{f_{1}r}{\sqrt{2}}\\\
\frac{f_{2}r}{\sqrt{2}}\\\ f_{0}\end{pmatrix}$

in the $3$-d real Hilbert space with basis (5.38), as shown in (5.32). From
the positiveness of $\Delta_{r}$, we infer that the map
$\mathbb{R}^{3}\ni\vec{f}\mapsto A_{f}\in\mathcal{J}_{3}$ is one-to-one but
not unitary. For this reason, any symmetric matrix $A$ is, for $r\neq 0$, the
$\rho_{r,\phi+\phi_{0}}$-quantization of a unique function
$f_{A}\in\mathbb{R}^{3}$ as

(5.41) $\begin{split}A\equiv
A_{f_{A}}&=a_{1}\frac{1}{\sqrt{2}}\sigma_{2\phi_{0}}+a_{2}\frac{1}{\sqrt{2}}\sigma_{2\phi_{0}+\pi/2}+a_{0}\frac{\mathbbm{1}}{\sqrt{2}}=\int_{0}^{2\pi}\frac{\mathrm{d}\phi}{\pi}\,f_{A}(\phi)\,\rho_{r,\phi+\phi_{0}}\,,\\\
f_{A}(\phi)&=a_{1}\frac{\sqrt{2}}{r}\cos 2\phi+a_{2}\frac{\sqrt{2}}{r}\sin
2\phi+\frac{a_{0}}{\sqrt{2}}\,.\end{split}$

Adopting the terminology of Berezin-Lieb mentioned in Section 3.2, $f_{A}$ is
the upper (Lieb) or contravariant (Berezin) symbol $\check{A}$ of $A$.
Similarly, the lower (Lieb) or covariant (Berezin) symbol of a symmetric
matrix $A=\begin{pmatrix}a&b\\\ b&d\end{pmatrix}$ stands for the semi-
classical portrait of the latter. It is here defined coherently with (3.5):

(5.42)
$\check{A}(\phi)=\mathrm{Tr}\left(A\rho_{r,\phi_{0}}(\phi)\right)=\frac{a+d}{2}+r\left(\frac{a-d}{2}\cos
2(\phi+\phi_{0})+b\sin 2(\phi+\phi_{0})\right)\,.$

If $A\equiv A_{f}$ is the $\rho_{r,\phi_{0}}$-quantized of $f\in V_{3}$, we
obtain

(5.43) $\check{A}_{f}(\theta)\equiv\check{f}(\theta)=\langle
f\rangle+rs\int_{0}^{2\pi}\frac{\mathrm{d}\phi}{\pi}\,f(\phi)\,\cos
2\left(\phi+\phi_{0}-\theta-\theta_{0}\right)\,.$

Applying this to each component of $f$, we show the map $f\mapsto\check{f}$ in
$V_{3}$ corresponds to the matrix transformation in $\mathbb{R}^{3}$:

(5.44)
$\displaystyle\overrightarrow{\check{f}}=\begin{pmatrix}\check{f}_{1}\\\
\check{f}_{2}\\\ \check{f}_{0}\end{pmatrix}=\begin{pmatrix}rs\cos
2\left(\theta_{0}-\phi_{0}\right)&-rs\sin
2\left(\theta_{0}-\phi_{0}\right)&0\\\ rs\sin
2\left(\theta_{0}-\phi_{0}\right)&rs\cos
2\left(\theta_{0}-\phi_{0}\right)&0\\\
0&0&1\end{pmatrix}\begin{pmatrix}f_{1}\\\ f_{2}\\\ f_{0}\end{pmatrix}\,,$

describing a rotation of angle $2\left(\theta_{0}-\phi_{0}\right)$ around the
direction $f_{0}$ together with a possible contraction of the components
$f_{1}$ and $f_{2}$ by a factor $rs$ (remember $0\leq r,s\leq 1$). The upper
or contravariant symbol $\hat{A}(\phi)$ appearing in the operator-valued
integral

(5.45)
$A=\frac{1}{\pi}\int_{0}^{2\pi}d\phi\,\hat{A}(\phi)|\phi\rangle\langle\phi|\,.$

is highly non-unique, but is chosen as the simplest one.

###  5.4. Integral POVM quantization on the circle and Toeplitz-Naimark
formalism

Let us see how the material introduced in Section 4 fits the quantization map
(5.13). In the present case the large Hilbert space is the whole
$L_{\mathbb{R}}^{2}(\mathbb{S}^{1},\mathrm{d}\phi/\pi)$, i.e., the Hilbert
space of real-valued functions which are square-integrable on the circle
w.r.t. the measure $\mathrm{d}\phi/\pi$. Now consider the quantization of a
function $f(\phi)$ on the circle, as is described in Eq. (5.13). We start from
the linear operator in $L_{\mathbb{R}}^{2}(\mathbb{S}^{1},\mathrm{d}\phi/\pi)$
formally defined by

(5.46) $L_{\mathbb{R}}^{2}(\mathbb{S}^{1},\mathrm{d}\phi/\pi)\ni v\mapsto
M_{f}v=fv\,.$

and valid if $fv\in L_{\mathbb{R}}^{2}(\mathbb{S}^{1},\mathrm{d}\phi/\pi)$. We
now consider the family of rotation matrices $\mathcal{R}\left(\phi\right)$
acting on the plane $\mathcal{H}=\mathbb{R}^{2}$. The matrix elements

$\left\\{\mathcal{R}_{ij}\left(\phi\right)\right\\}=\left\\{\mathcal{R}_{11}\left(\phi\right)=\mathcal{R}_{22}\left(\phi\right)=\cos\phi,\mathcal{R}_{12}\left(\phi\right)=-\mathcal{R}_{21}\left(\phi\right)=-\sin\phi\right\\}$

satisfy the conditions of square-integrability and orthonormality (4.10) and
(4.11) on the circle.

(5.47) $\displaystyle\delta_{ik}$
$\displaystyle=\int_{0}^{2\pi}\mathcal{R}_{ji}(\phi)\,\mathcal{R}_{jk}(\phi)\,\frac{\mathrm{d}\phi}{\pi}\quad\forall\,j\,,$
(5.48) $\displaystyle 0$
$\displaystyle=\int_{0}^{2\pi}\left[\mathcal{R}_{ji}(\phi)\,\mathcal{R}_{lk}(\phi)+\mathcal{R}_{li}(\phi)\,\mathcal{R}_{jk}(\phi)\right]\,\frac{\mathrm{d}\phi}{\pi}\quad\forall\,j,l\,,\
j\neq l\,.$

Explicitly,

(5.49) $\displaystyle 1$
$\displaystyle=\int_{0}^{2\pi}\left(\mathcal{R}_{11}(\phi)\right)^{2}\,\frac{\mathrm{d}\phi}{\pi}=\int_{0}^{2\pi}\left(\mathcal{R}_{12}(\phi)\right)^{2}\,\frac{\mathrm{d}\phi}{\pi}$
(5.50) $\displaystyle 0$
$\displaystyle=\int_{0}^{2\pi}\mathcal{R}_{11}(\phi)\,\mathcal{R}_{12}(\phi)\,\frac{\mathrm{d}\phi}{\pi}$
(5.51) $\displaystyle 0$
$\displaystyle=\int_{0}^{2\pi}\left[\mathcal{R}_{11}(\phi)\,\mathcal{R}_{21}(\phi)+\mathcal{R}_{21}(\phi)\,\mathcal{R}_{11}(\phi)\right]\,\frac{\mathrm{d}\phi}{\pi},$
(5.52) $\displaystyle 0$
$\displaystyle=\int_{0}^{2\pi}\left[\mathcal{R}_{11}(\phi)\,\mathcal{R}_{22}(\phi)+\mathcal{R}_{21}(\phi)\,\mathcal{R}_{12}(\phi)\right]\,\frac{\mathrm{d}\phi}{\pi}\,,$

and so on. Within the framework established with Proposition 4.2, let us
identify the two orthonormal systems $\mathcal{O}_{j}\subset
L_{\mathbb{R}}^{2}(\mathbb{S}^{1},\mathrm{d}\phi/\pi)$, $j=1,2$. They are

(5.53)
$\mathcal{O}_{1}=\\{\cos\phi,\sin\phi\\}=\\{v_{1}(\phi),v_{2}(\phi)\\}\,\quad\mathcal{O}_{2}=\\{-\sin\phi,\cos\phi\\}=\\{-v_{2}(\phi),v_{1}(\phi)\\}\,.$

We denote by $\mathbb{P}_{\mathfrak{o}_{j}}$ the orthogonal projector mapping
$L_{\mathbb{R}}^{2}(\mathbb{S}^{1},\mathrm{d}\phi/\pi)$ onto the sub-Hilbert
space $\mathcal{H}_{\mathfrak{o}_{j}}$ which is the linear span of the
elements of $\mathcal{O}_{j}$. Then we identify
$\mathcal{H}_{\mathfrak{o}_{j}}\equiv\mathcal{H}$ through

(5.54) $U_{\mathfrak{o}_{j}}|e_{i}\rangle=\mathcal{R}_{ji}\,.$

With this identification the relations (4.17) hold under the form,

(5.55)
$\mathbb{P}_{\mathfrak{o}_{1}}M_{f}\mathbb{P}_{\mathfrak{o}_{1}}=U_{\mathfrak{o}_{1}}A^{(1)}_{f}U^{-1}_{\mathfrak{o}_{1}}\,,\quad
A^{(1)}_{f}=\int_{0}^{2\pi}f(\phi)\,|\phi\rangle_{e_{1}}{}_{e_{1}}\langle\phi|\,\frac{\mathrm{d}\phi}{\pi}\,.$
(5.56)
$\mathbb{P}_{\mathfrak{o}_{2}}M_{f}\mathbb{P}_{\mathfrak{o}_{2}}=U_{\mathfrak{o}_{2}}A^{(2)}_{f}U^{-1}_{\mathfrak{o}_{2}}\,,\quad
A^{(2)}_{f}=\int_{0}^{2\pi}f(\phi)\,|\phi\rangle_{e_{2}}{}_{e_{2}}\langle\phi|\,\frac{\mathrm{d}\phi}{\pi}\,.$

With the identification $|e_{1}\rangle=|\phi_{0}\rangle$,
$|e_{2}\rangle=|\phi_{0}+\pi/2\rangle$, we finally have

(5.57)
$A^{(\rho_{r,\phi_{0}})}_{f}=\frac{1+r}{2}A^{(\phi_{0})}_{f}+\frac{1-r}{2}A^{(\phi_{0}+\pi/2)}_{f}\,,$

and we retrieve the same form as the density matrix (5.7).

##  6\. An example of quantum measurement of orientations: polarization of
light and Stokes parameters

We now wish to describe the relationship between POVMs and light polarization
within the context of integral quantization, which was briefly presented in
[70]. Let us first introduce the notion of polarization, and explain how it
can be related to density matrices. As was pointed out in the preface of the
textbooks [71, 72], polarization is a fundamental characteristic of the
transverse wave that is light. There exists a well-known physical
interpretation of $2\times 2$ real or complex density matrices in terms of the
so-called Stokes parameters for the polarization of light, see for instance
[73] and references therein. The significance and measurements of such
parameters is clearly explained in [74]. We follow here the presentation given
by Landau and Lifshitz in Section 50 of [75], where is introduced the
polarization “tensor” for a quasi-monochromatic plane wave propagating along
the $z$-axis with the following notations involving the three Pauli matrices:

(6.1)
$\left(\rho_{\alpha\beta}\right)=\frac{1}{2}\begin{pmatrix}1+\xi_{3}&\xi_{1}-\mathsf{i}\xi_{2}\\\
\xi_{1}+\mathsf{i}\xi_{2}&1-\xi_{3}\end{pmatrix}=\frac{1}{2}\left(\mathbbm{1}+\sum_{i=1}^{3}\xi_{i}\sigma_{i}\right)\,,$

with the Stokes parameters

(6.2) $\xi_{1}=r\sin 2\phi\,,\quad\xi_{2}=A\,,\quad\xi_{3}=r\cos 2\phi\,.$

The labels $\alpha$, $\beta$ run over the coordinates $(x,y)$ in the plane
orthogonal to the $z$-axis. In (6.2) the parameter $0\leq r\leq 1$
characterizes the degree of maximal linear polarization whilst $-1\leq A\leq
1$ characterizes the circular polarization of the beam.

$\widehat{\boldsymbol{k}}$$\widehat{\boldsymbol{\jmath}}$$\widehat{\boldsymbol{\imath}}$$\bullet$$\mathfrak{Re}\left(\overrightarrow{\mathcal{E}}\right)$$y$$z$$x$
Figure 2. A quasi-monochromatic plane wave propagates to the right along the
$z$-axis. The electric field, as the real part of its complex description,
lies in the $x-y$ plane.

Let us explain the physical origin of the polarization tensor (6.1). In its
complex description, the electric field reads

(6.3)
$\overrightarrow{\mathcal{E}}(t)=\overrightarrow{\mathcal{E}_{0}}(t)\,e^{\mathsf{i}\omega
t}=\mathcal{E}_{x}\,\widehat{\boldsymbol{\imath}}+\mathcal{E}_{y}\,\widehat{\boldsymbol{\jmath}}=\left(\mathcal{E}_{\alpha}\right)\,,$

where $\overrightarrow{\mathcal{E}_{0}}(t)$ has slow temporal variation and
$\omega$ is the mean frequency. $\overrightarrow{\mathcal{E}_{0}}$ determines
the polarization of light. The latter is measured through Nicol prisms, or
other devices, by measuring the intensity of the light yielded by mean values
of quadratic expressions of field components, which are proportional to
$\mathcal{E}_{\alpha}\mathcal{E}_{\beta}$,
$\mathcal{E}_{\alpha}\mathcal{E}^{\ast}_{\beta}$ and their respective complex
conjugates. Now,
$\mathcal{E}_{\alpha}\mathcal{E}_{\beta}=\mathcal{E}_{0\alpha}\mathcal{E}_{0\beta}e^{2\mathsf{i}\omega
t}$ and
$\mathcal{E}^{\ast}_{\alpha}\mathcal{E}^{\ast}_{\beta}=\mathcal{E}^{\ast}_{0\alpha}\mathcal{E}^{\ast}_{0\beta}e^{-2\mathsf{i}\omega
t}$ have rapidly oscillating factors and so have null temporal average
$\langle\cdot\rangle_{t}$. It results that the properties of a partially
polarized light are completely described by the tensor

(6.4)
$J_{\alpha\beta}:=\left\langle\mathcal{E}_{0\alpha}\mathcal{E}^{\ast}_{0\beta}\right\rangle_{t}\,.$

The quantity
$J=\sum_{\alpha}J_{\alpha\alpha}=\left\langle|\mathcal{E}_{0x}|^{2}\right\rangle_{t}+\left\langle|\mathcal{E}_{0y}|^{2}\right\rangle_{t}$
determines the intensity of the wave, obtained from the measurement of the
energy flux transported by the wave. Since this quantity does not concern the
properties of polarization of the wave, one has eventually to deal with the
normalized tensor (6.1), i.e.,

(6.5) $\rho_{\alpha\beta}=\frac{J_{\alpha\beta}}{J}\,.$

The light is said completely polarized when the complex amplitude
$\overrightarrow{\mathcal{E}}_{0}$ is time-independent, and so is equal to its
time average. Then, the polarization tensor factorizes as

(6.6)
$\left(\rho_{\alpha\beta}\right)=\frac{1}{J}\begin{pmatrix}|\mathcal{E}_{0x}|^{2}&\mathcal{E}_{0x}\mathcal{E}^{\ast}_{0y}\\\
\mathcal{E}^{\ast}_{0x}\mathcal{E}_{0y}&|\mathcal{E}_{0y}|^{2}\end{pmatrix}=\begin{pmatrix}\mathcal{E}_{0x}/\sqrt{J}\\\
\mathcal{E}_{0y}/\sqrt{J}\end{pmatrix}\begin{pmatrix}\mathcal{E}^{\ast}_{0x}/\sqrt{J}&\mathcal{E}^{\ast}_{0y}/\sqrt{J}\\\
\end{pmatrix}\,,$

i.e., is the orthogonal projector along $\overrightarrow{\mathcal{E}}_{0}$,
and, in quantum terms, a pure state. This case should be put in regard to the
other extreme case, namely the non-polarized or natural light, for which all
directions in the $x-y$ plane are equivalent:

(6.7) $\left(\rho_{\alpha\beta}\right)=\frac{1}{2}\,\delta_{\alpha\beta}\,.$

In the general case and with $\xi_{i}$ coordinates,
$\det\left(\rho_{\alpha\beta}\right)=\frac{1}{4}\left(1-\xi_{1}^{2}-\xi_{2}^{2}-\xi_{3}^{2}\right)\equiv\frac{1}{4}\left(1-P^{2}\right)$,
where $0\leq P\leq 1$ is called the degree of polarization, from $P=0$ (random
polarization) to $P=1$ (total polarization). Another extreme case holds with
circular polarization. Then $\overrightarrow{\mathcal{E}_{0}}$ is constant and
$\mathcal{E}_{0y}=\pm\mathsf{i}\mathcal{E}_{0x}$, which gives
$\left(\rho_{\alpha\beta}\right)=(1/2)(\mathbbm{1}\pm\mathsf{i}\sigma_{2})$.
As a consequence, the parameter $-1\leq A\leq 1$ in (6.2) is interpreted as
the degree of circular polarization, with $A=1$ (resp. $A=-1$) for right
(resp. left) circular polarization, and $A=0$ for linear polarization. This
distinction between linear and circular polarizations corresponds to the
decomposition of (6.1) into its symmetric and antisymmetric parts:

(6.8)
$\left(\rho_{\alpha\beta}\right)=\rho_{r,\phi}+\frac{A}{2}\sigma_{2}=\frac{1+r}{2}E_{\phi}+\frac{1-r}{2}E_{\phi+\pi/2}+\frac{A}{2}\sigma_{2}\,,$

where we have introduced our own notations in (5.6) and (5.8). Hence, the
light can be viewed as the superposition of two waves which are incoherent and
elliptically polarized, with similar and orthogonal polarization ellipses.
Within the context of the present paper, circular polarization is not
considered. We then put $A=0$, ultimately reducing the polarization tensor to
our density matrix. Let us now describe the interaction polarizer-partially
linear polarized light as an example of simple quantum measurements. Here, we
follow and generalize the example presented by Peres in [76]. Two planes and
their tensor product are under consideration. The first one is the Hilbert
space on which act the states $\rho^{M}_{s,\theta}$ of the polarizer viewed as
an orientation pointer. We note that the action of the generator of rotations
$\tau_{2}=-\mathsf{i}\sigma_{2}$ on these states corresponds to a $\pi/2$
rotation :

(6.9)
$\tau_{2}\rho^{M}_{s,\theta}\tau_{2}^{-1}=-\tau_{2}\rho^{M}_{s,\theta}\tau_{2}=\rho^{M}_{s,\theta+\pi/2}\,.$

The second plane is the Hilbert space on which act the partially linearized
polarization states $\rho^{L}_{r,\phi}$ of the plane wave crossing the
polarizer. As was explained above, the spectral decomposition of the latter
corresponds to the incoherent superposition of two completely linearly
polarized waves

(6.10)
$\rho^{L}_{r,\phi}=\frac{1+r}{2}\,E_{\phi}+\frac{1-r}{2}\,E_{\phi+\pi/2}\,.$

The pointer is designed to detect the orientation in the plane determined by
the angle $\phi$. The interaction pointer-system generating a measurement
whose time duration is the interval $I_{M}=(t_{M}-\eta,t_{M}+\eta)$ centred at
$t_{M}$ is described by the (pseudo-) Hamiltonian operator

(6.11)
$\widetilde{H}_{\mathrm{int}}(t)=\,g^{\eta}_{M}(t)\tau_{2}\otimes\rho^{L}_{r,\phi}\,,$

where $g^{\eta}_{M}$ is a Dirac sequence with support in $I_{M}$, i.e.,

$\lim_{\eta\to
0}\int_{-\infty}^{+\infty}\mathrm{d}t\,f(t)\,g^{\eta}_{M}(t)=f(t_{M})\,.$

The operator (6.11) is the tensor product of an antisymmetric operator (i.e.,
or pseudo-Hamiltonian, see [65]) for the pointer with an operator for the
system which is symmetric (i.e. Hamiltonian). Nevertheless, the operator
$U(t,t_{0})$ defined for $t_{0}<t_{M}-\eta$ as

(6.12)
$U(t,t_{0})=\exp\left[\int_{t_{0}}^{t}\mathrm{d}t^{\prime}\,g^{\eta}_{M}(t^{\prime})\,\tau_{2}\otimes\rho^{L}_{r,\phi}\right]=\exp\left[G_{M}^{\eta}(t)\,\tau_{2}\otimes\rho^{L}_{r,\phi}\right]\,,$

with
$G_{M}^{\eta}(t)=\int_{t_{0}}^{t}\mathrm{d}t^{\prime}\,\,g^{\eta}_{M}(t^{\prime})$,
is an evolution operator. Hence, as soon as $t>t_{M}+\eta$,
$G_{M}^{\eta}(t)=1$, and from the general formula involving an orthogonal
projector $P$,

(6.13) $\exp(\theta\tau_{2}\otimes P)=\mathcal{R}(\theta)\otimes
P+\mathbbm{1}\otimes(\mathbbm{1}-P)\,,$

we obtain

(6.14)
$U(t,t_{0})=\mathcal{R}\left(G_{M}^{\eta}(t)\,\frac{1+r}{2}\right)\otimes
E_{\phi}+\mathcal{R}\left(G_{M}^{\eta}(t)\,\frac{1-r}{2}\right)\otimes
E_{\phi+\pi/2}\,.$

For $t_{0}<t_{M}-\eta$ and $t>t_{M}+\eta$, we finally obtain

(6.15) $U(t,t_{0})=\mathcal{R}\left(\frac{1+r}{2}\right)\otimes
E_{\phi}+\mathcal{R}\left(\frac{1-r}{2}\right)\otimes E_{\phi+\pi/2}\,.$

One easily checks that
$U(t,t_{0})U(t,t_{0})^{{\dagger}}=U(t,t_{0})^{{\dagger}}U(t,t_{0})=\mathbbm{1}\otimes\mathbbm{1}$.
After having prepared the polarizer in the state
$\rho^{M}_{s_{0},\theta_{0}}$, the action of (6.15) on the initial state
$\rho^{M}_{s_{0},\theta_{0}}\otimes\rho^{L}_{r_{0},\phi_{0}}$ reads for
$t>t_{M}+\eta$

$\displaystyle
U(t,t_{0})\,\rho^{M}_{s_{0},\theta_{0}}\otimes\rho^{L}_{r_{0},\phi_{0}}\,U(t,t_{0})^{{\dagger}}=$
$\displaystyle\rho^{M}_{s_{0},\theta_{0}+\frac{1+r}{2}}\otimes\frac{1+r_{0}\cos
2(\phi-\phi_{0})}{2}\,E_{\phi}$
$\displaystyle+\rho^{M}_{s_{0},\theta_{0}+\frac{1-r}{2}}\otimes\frac{1-r_{0}\cos
2(\phi-\phi_{0})}{2}\,E_{\phi+\pi/2}$
$\displaystyle+\frac{1}{4}\left(\mathcal{R}(r)+s_{0}\sigma_{2\theta_{0}+1}\right)\otimes
r_{0}\sin 2(\phi-\phi_{0})\,E_{\phi}\tau_{2}$ (6.16)
$\displaystyle-\frac{1}{4}\left(\mathcal{R}(-r)+s_{0}\sigma_{2\theta_{0}+1}\right)\otimes
r_{0}\sin 2(\phi-\phi_{0})\,\tau_{2}E_{\phi}\,.$

As expected from the standard theory of quantum measurement, this formula
indicates that the probability for the pointer to rotate by $\frac{1+r}{2}$,
corresponding to the polarization along the orientation $\phi$, is

(6.17)
$\mathrm{Tr}\left[\left(U(t,t_{0})\,\rho^{M}_{s_{0},\theta_{0}}\otimes\rho^{L}_{r_{0},\phi_{0}}\,U(t,t_{0})^{{\dagger}}\right)\left(\mathbbm{1}\otimes
E_{\phi}\right)\right]=\frac{1+r_{0}\cos 2(\phi-\phi_{0})}{2}\,,$

whereas it is

(6.18)
$\mathrm{Tr}\left[\left(U(t,t_{0})\,\rho^{M}_{s_{0},\theta_{0}}\otimes\rho^{L}_{r_{0},\phi_{0}}\,U(t,t_{0})^{{\dagger}}\right)\left(\mathbbm{1}\otimes
E_{\phi+\pi/2}\right)\right]=\frac{1-r_{0}\cos 2(\phi-\phi_{0})}{2}\,,$

for the perpendicular orientation $\phi+\pi/2$ and the pointer rotation by
$\frac{1-r}{2}$. For the completely linear polarization of the light, i.e.
$r_{0}=1$, we recover the familiar Malus laws, $\cos^{2}(\phi-\phi_{0})$ and
$\sin^{2}(\phi-\phi_{0})$ respectively.

##  7\. Compatibility of POVMs

We now argue that the physical situation where two measurements are made one
after another is described by a dichotomic POVM. We then recall the necessary
conditions for the compatibility of two dichotomic POVMs, and what form these
conditions take in the Euclidean plane.

###  7.1. POVMs arising from the measurement of polarization

It is well known [77] that the statistics of the sequential measurement of two
incompatible observables is described by a POVM. Let us analyze the case of
the sequential measurement of two incompatible polarization observables.
Suppose an incident light ray is polarized according to the state $\rho$ and
is sent toward a first vertical polarizer ($\pi/2$) and subsequently toward a
second oblique ($\pi/4$) polarizer. The light ray is then detected by a light
detector. Suppose the state of the detector is $D=1$ if it detects a light ray
and $D=0$ if it does not. What is the probability that the light ray triggers
the detector (in which case $D=1$ and the polarization of the transmitted
light ray is $\pi/4$)? We first note that the probability that the light ray
exits the first polarizer is $Tr\left(\rho E_{\frac{\pi}{2}}\right)$ where
$\rho$ is an initial state and
$E_{\frac{\pi}{2}}=|\frac{\pi}{2}\rangle\langle\frac{\pi}{2}|$. If the light
ray has passed the first polarizer, its state is

$\rho^{\prime}{}=\frac{E_{\frac{\pi}{2}}\rho
E_{\frac{\pi}{2}}}{Tr\left(E_{\frac{\pi}{2}}\rho\right)}\;.$

Then, the probability that it passes the second polarizer is
$Tr\left(\rho^{\prime}{}E_{\frac{\pi}{4}}\right)$. Therefore, the probability
that the state of the detector is $D=1$ at the end of the measurement process
(the light ray is transmitted) is

(7.1) $\displaystyle p\left(D=1\right)=Tr\left(\rho
E_{\frac{\pi}{2}}\right)Tr\left(\rho^{\prime}{}E_{\frac{\pi}{4}}\right)=Tr\left(\rho
E_{\frac{\pi}{2}}E_{\frac{\pi}{4}}E_{\frac{\pi}{2}}\right)=\frac{1}{2}Tr\left(\rho
E_{\frac{\pi}{2}}\right).$

Note that $E_{\frac{\pi}{2}}E_{\frac{\pi}{4}}E_{\frac{\pi}{2}}$ is an effect,
i.e. a symmetric operator $E$ such that $0\leq E\leq\mathbbm{1}$, but not a
projection operator (it is a multiple of a projection operator). Now, the
probability that the light ray passes the first polarizer but does not pass
the second is

(7.2) $\displaystyle Tr\left(\rho
E_{\frac{\pi}{2}}\right)Tr\left[\rho^{\prime}{}\left(\mathbbm{1}-E_{\frac{\pi}{4}}\right)\right]=Tr\left[\rho
E_{\frac{\pi}{2}}\left(\mathbbm{1}-E_{\frac{\pi}{4}}\right)E_{\frac{\pi}{2}}\right]=\frac{1}{2}Tr\left(\rho
E_{\frac{\pi}{2}}\right).$

The probability that, at the end of the measurement process, the detector is
in the state $D=0$ is

(7.3) $\displaystyle p(D=0)=Tr(\rho E_{0})+\frac{1}{2}Tr(\rho
E_{\frac{\pi}{2}})=Tr\left[\rho\left(E_{0}+\frac{1}{2}E_{\frac{\pi}{2}}\right)\right].$

Note that $p(D=1)+p\big{(}D=0\big{)}=1$. Again,
$E_{0}+\frac{1}{2}E_{\frac{\pi}{2}}$ is an effect but not a projection
operator. The measurement process is then described by the dichotomic POVM

(7.4) $\displaystyle
F=\left\\{\frac{1}{2}E_{\frac{\pi}{2}}\;,\;E_{0}+\frac{1}{2}E_{\frac{\pi}{2}}\right\\}\;,$

which is not the spectral measure corresponding to a self-adjoint operator. If
we analyze the sequential measurement where the light ray first passes the
oblique polarizer and then the vertical one, we obtain that the overall
measurement process is characterized by the POVM
$\left\\{\frac{1}{2}E_{\frac{\pi}{4}},\mathbbm{1}-\frac{1}{2}E_{\frac{\pi}{4}}\right\\}$
which is different from the previous one describing the measurement process
with the light ray passing first through the vertical polarizer. The two
measurement procedures give different results. That is explained by the
incompatibility of the measurement of the vertical and the oblique
polarization and is analogous to the Heisenberg microscope experiment where a
sequential measure of the position and momentum is realized.

###  7.2. Dichotomic POVMs

A dichotomic POVM $F$ is a pair $F=\\{A,\mathbbm{1}-A\\}$ with $A$ an effect.
We consider two dichotomic POVMs $F$ and $F^{\prime}$, and look for necessary
and sufficient conditions for their compatibility. We show that every
dichotomic observable $F$ is the fuzzy version of a sharp observable $E$; $F$
can also be interpreted as an approximation of $E$. Then, inspired by [1], we
answer the following question: given two incompatible sharp observables $E$
and $E^{\prime}$, is it possible to define two compatible fuzzy versions $F$
and $F^{\prime}$ of $E$ and $E^{\prime}$ respectively whose joint measurement
can be interpreted as the approximate joint measurement of $E$ and
$E^{\prime}$? First we recall the main definitions and theorems concerning the
compatibility of POVMs that we adapt to the case of a real, bidimensional
Hilbert space. Two dichotomic POVMs $F_{1}=\\{A_{1},\mathbbm{1}-A_{1}\\}$ and
$F_{2}=\\{A_{2},\mathbbm{1}-A_{2}\\}$ are compatible if there is a third POVM
$F=\\{G_{11},G_{00},G_{10},G_{01}\\}$ of which $F_{1}$ and $F_{2}$ are the
marginals, i.e.,

(7.5) $\displaystyle A_{1}=G_{11}+G_{10},\,\,A_{2}=G_{11}+G_{01},$
$\displaystyle\mathbbm{1}-A_{1}=G_{00}+G_{01},\,\,I-A_{2}=G_{00}+G_{10}\;.$

A necessary and sufficient condition for the compatibility [58] of $F_{1}$ and
$F_{2}$ is the existence of a positive operator $G_{11}$ such that

(7.6) $\begin{split}&G_{11}\leq A_{1},A_{2}\;,\\\
&A_{1}+A_{2}-G_{11}\leq\mathbbm{1}\;.\end{split}$

Indeed, if $G_{11}$ exists, we can define the effects

(7.7) $\displaystyle G_{00}$ $\displaystyle=\mathbbm{1}-A_{1}-A_{2}+G_{11}\geq
0\;,$ (7.8) $\displaystyle G_{10}$ $\displaystyle=A_{1}-G_{11}\geq 0\;,$ (7.9)
$\displaystyle G_{01}$ $\displaystyle=A_{2}-G_{11}\geq 0\;,$

which satisfy (7.5). Now we translate the above compatibility conditions in
the present context where an observable is represented by the dichotomic
observable corresponding to a symmetric operator of the kind defined below

(7.10) $A(\alpha,\phi,r)=\begin{pmatrix}\frac{\alpha}{2}+\frac{r}{2}\cos
2\phi&\frac{r}{2}\sin 2\phi\\\ \frac{r}{2}\sin
2\phi&\frac{\alpha}{2}-\frac{r}{2}\cos
2\phi\end{pmatrix}=\alpha\rho_{\frac{r}{\alpha},\phi}\,.$

with eigenvalues $\frac{\alpha}{2}\pm\frac{r}{2}$. The condition $0\leq A\leq
I$ then becomes $r\leq\alpha\leq 2-r$ (so that $0\leq\alpha\leq 2$ and $0\leq
r\leq 1$) which ensures that $A\left(\alpha,\phi,r\right)$ is an effect. Note
that there are two meanings we can give to matrices of the form (7.10): if
$\alpha=r=1$, they are density operators which can be used to represent pure
states or sharp observables; in the case $\alpha=1$, $r\neq 1$, they can
represent mixed states or fuzzy (unsharp) observables; in the general case
$\alpha,r\neq 1$ they represent fuzzy unbiased observables (see below). We
recall that in the context of the linear polarisation of the light as
described by (6.5), the parameter $\alpha$ may be interpreted as the intensity
of the light while $r=\xi_{1}^{2}+\xi_{3}^{2}=P$ provides a measure of the
degree of polarization. That is relevant in order to give an interpretation of
the fuzzy observables as approximations of the sharp observables and of the
joint measurability of fuzzy observables as the approximate joint
measurability of incompatible sharp polarization observables (see below). The
dichotomic observable corresponding to $A(\alpha,\phi,r)$ is
$\big{\\{}F_{+}=A(\alpha,\phi,r),\,F_{-}=\mathbbm{1}-A(\alpha,\phi,r)\big{\\}}$
and can be interpreted as the randomization of the dichotomic PVM
$\big{\\{}E_{+}=A(1,\phi,1),\,E_{-}=\mathbbm{1}-A(1,\phi,1)\big{\\}}$ through
the Markov kernel $\mu(+,+)=\frac{\alpha+r}{2}$,
$\mu(+,-)=\frac{\alpha-r}{2}$, $\mu(-,+)=1-\mu(+,+)$, $\mu(-,-)=1-\mu(+,-)$.
Indeed, $F_{+}=\mu(+,+)E_{+}+\mu(+,-)E_{-}$ and
$F_{-}=\mu(-,+)E_{+}+\mu(-,-)E_{-}$. As an example, we can consider the POVMs
$\left\\{\frac{1}{2}E_{\frac{\pi}{2}},\mathbbm{1}-\frac{1}{2}E_{\frac{\pi}{2}}\right\\}$
and
$\left\\{\frac{1}{2}E_{\frac{\pi}{4}},\mathbbm{1}-\frac{1}{2}E_{\frac{\pi}{4}}\right\\}$
that we found in section 7.1 which represent sequential measurements. In
particular, we have that
$\frac{1}{2}E_{\frac{\pi}{2}}=A\left(\frac{1}{2},\frac{\pi}{2},\frac{1}{2}\right)$,
$\frac{1}{2}E_{\frac{\pi}{4}}=A\left(\frac{1}{2},\frac{\pi}{4},\frac{1}{2}\right)$.
They are not compatible as can be seen by showing that there is no symmetric
positive operator satisfying conditions (7.6). Note also that
$\left\\{\frac{1}{2}E_{\frac{\pi}{2}},\mathbbm{1}-\frac{1}{2}E_{\frac{\pi}{2}}\right\\}$
is the randomization of the PVM
$\left\\{E_{+}=A\left(1,\frac{\pi}{2},1\right),\,E_{-}=\mathbbm{1}-A\left(1,\frac{\pi}{2},1\right)\right\\}$
through the Markov kernels $\mu(+,+)=\frac{1}{2}$, $\mu(+,-)=0$,
$\mu(-,+)=\frac{1}{2}$, $\mu(-,-)=1$. The same holds for the POVM
$\left\\{\frac{1}{2}E_{\frac{\pi}{4}},\mathbbm{1}-\frac{1}{2}E_{\frac{\pi}{4}}\right\\}$
which is the randomization of the PVM
$\left\\{E_{+}=A\left(1,\frac{\pi}{4},1\right),\,E_{-}=\mathbbm{1}-A\left(1,\frac{\pi}{4},1\right)\right\\}$
through the same Markov kernel. Let

$F_{1}=\\{A(\alpha_{1},\phi_{1},r_{1}),\mathbbm{1}-A(\alpha_{1},\phi_{1},r_{1})\\}$

and

$F_{2}=\\{A(\alpha_{2},\phi_{2},r_{2}),\mathbbm{1}-A(\alpha_{2},\phi_{2},r_{2})\\}$

be two dichotomic POVMs. We look for conditions for the compatibility of
$F_{1}$ and $F_{2}$. An analogous problem has been dealt with in Ref. [58, 1].
The present case can be considered as the real version of the qubit case
analyzed in [1], chapter 14. If circular polarization were included, the
relevant Hilbert space would be $\mathbb{C}^{2}$ and the formalism would
coincide with the one in [1]. First, we note that to each symmetric operator
$A(\alpha,\phi,r)$ there corresponds a vector $v=(r,\phi)$ in the unit closed
disc. We then have

$\displaystyle A_{1}$ $\displaystyle=A(\alpha_{1},\phi_{1},r_{1})\mapsto
v_{1}=(r_{1},\phi_{1})\;,$ $\displaystyle A_{2}$
$\displaystyle=A(\alpha_{2},\phi_{2},r_{2})\mapsto v_{2}=(r_{2},\phi_{2})\;,$
$\displaystyle G_{11}$ $\displaystyle=A(\alpha,\phi,r)\mapsto v=(r,\phi)\;.$

Moreover, using (7.5)

$\displaystyle A_{1}-G_{11}$
$\displaystyle=A(\alpha_{1},\phi_{1},r_{1})-A(\alpha,\phi,r)=G_{10}=A(\alpha_{1}-\alpha,r_{10},\phi_{10})\;,$
$\displaystyle A_{2}-G_{11}$
$\displaystyle=A(\alpha_{2},\phi_{2},r_{2})-A(\alpha,\phi,r)=G_{01}=A(\alpha_{2}-\alpha,r_{01},\phi_{01})\;,$
$\displaystyle\mathbbm{1}-A_{1}-A_{2}+G_{11}$
$\displaystyle=\mathbbm{1}-A(\alpha_{1},\phi_{1},r_{1})-A(\alpha_{2},\phi_{2},r_{2})+A(\alpha,\phi,r)$
$\displaystyle=G_{00}=A(1-\alpha_{1}-\alpha_{2}+\alpha,r_{00},\phi_{00})\;,$

and consequently, we can define the vectors

$\displaystyle v_{10}$ $\displaystyle=(r_{10},\phi_{10})=v_{1}-v\;,$
$\displaystyle v_{01}$ $\displaystyle=(r_{01},\phi_{01})=v_{2}-v\;,$
$\displaystyle v_{00}$ $\displaystyle=(r_{00},\phi_{00})=v-v_{1}-v_{2}\;.$

This can be seen by a simple application of Carnot’s theorem for triangles.
Therefore, the respective eigenvalues imply that the compatibility between
POVMs is achieved if

(7.11) $\displaystyle\|v\|$ $\displaystyle=r\leq\alpha$
$\displaystyle(\text{positivity of}\,\,G_{11})\;,$ $\displaystyle\|v_{1}-v\|$
$\displaystyle=r_{10}\leq\alpha_{1}-\alpha$ $\displaystyle(\text{positivity
of}\,\,G_{10})\;,$ $\displaystyle\|v_{2}-v\|$
$\displaystyle=r_{01}\leq\alpha_{2}-\alpha$ $\displaystyle(\text{positivity
of}\,\,G_{01})\;,$ $\displaystyle\|v-v_{1}-v_{2}\|$ $\displaystyle=r_{00}\leq
2-\alpha_{1}-\alpha_{2}+\alpha$
$\displaystyle(A_{1}+A_{2}-G_{11}\leq\mathbbm{1})\;.$

We then have

(7.12)
$\displaystyle\|v_{1}+v_{2}\|+\|v_{1}-v_{2}\|\leq\alpha+(2-\alpha_{1}-\alpha_{2}+\alpha)+(\alpha_{1}-\alpha)+(\alpha_{2}-\alpha)=2$

which is a necessary condition for the compatibility. That is exactly the same
condition obtained in [58] for the case of a two-dimensional complex Hilbert
space of which the real case we are considering is a particular case. However,
the condition is not sufficient, as can be seen by noting that the two
dichotomic observables introduced in the previous subsection, i.e.,
$A\left(\frac{1}{2},\frac{\pi}{2},\frac{1}{2}\right)$ and
$A\left(\frac{1}{2},\frac{\pi}{4},\frac{1}{2}\right)$, satisfy the necessary
condition but are not compatible as we remarked above. Note also that in the
case of unbiased dichotomic observables $A(1,\phi_{1},r_{1})$,
$A(1,\phi_{2},r_{2})$, the condition is also sufficient. Indeed, one can
choose $v=\frac{v_{1}+v_{2}}{2}$ and $\alpha=1-\frac{|v_{1}-v_{2}|}{2}$ which,
thanks to condition (7.12), ensures that the inequalities (7.11) are
satisfied. In order to give an interpretation of both the fuzzy observable
$\\{A(\alpha,\phi,r),\mathbbm{1}-A(\alpha,\phi,r)\\}$ and the necessary
condition (7.12), we can resort again to the Stokes parameters. In section 6,
we have shown that the quantity $P=\xi_{1}^{2}+\xi_{3}^{2}=r$ provides a
measure of the degree of linear polarization; with $P$ varying between $1$
(total polarization) and $0$ (random polarization). In the present framework,
the same quantity is interpreted as a measure of the fuzziness of the
polarization measurement. For example, a measurement of polarization
represented by the PVM
$E=\big{\\{}E_{+}=A(1,\phi,1),\,E_{-}=\mathbbm{1}-A(1,\phi,1)\big{\\}}$ (sharp
observable) corresponds to a sharp measurement of polarization in the
direction defined by $\phi$ while a measurement represented by the POVM
$F=\big{\\{}F_{+}=A(1,\phi,r),\,F_{-}=\mathbbm{1}-A(1,\phi,r)\big{\\}}$, $r<1$
(unsharp observable) corresponds to a fuzzy measurement; the fuzziness
depending on $r$ which measures the degree with which $F$ approximates $E$.
The POVM
$\hat{F}=\big{\\{}F_{+}=A(\alpha,\phi,r),\,F_{-}=\mathbbm{1}-A(\alpha,\phi,r)\big{\\}}$
is a fuzzy version of $E$ as well and the degree of approximation depends both
on $\alpha$ and $r$. If two sharp observables
$E=\big{\\{}E_{+}=A(1,\phi,1),\,E_{-}=\mathbbm{1}-A(1,\phi,1)\big{\\}}$ and
$E^{\prime}=\big{\\{}E_{+}=A(1,\phi^{\prime},1),\,E_{-}=\mathbbm{1}-A(1,\phi^{\prime},1)\big{\\}}$
are not compatible, they could be approximated by two compatible fuzzy
versions
$F=\big{\\{}F_{+}=A(1,\phi,r),\,F_{-}=\mathbbm{1}-A(1,\phi,r)\big{\\}}$ and
$F^{\prime}=\big{\\{}F_{+}=A(1,\phi^{\prime},r^{\prime}),\,F_{-}=\mathbbm{1}-A(1,\phi^{\prime},r^{\prime})\big{\\}}$.
Equation (7.12) gives a necessary and sufficient condition for the joint
measurability of $F$ and $F^{\prime}$ to be connected to the degree of
fuzziness one has to add in the measurement process; the latter being measured
by $r$ and $r^{\prime}$. In the case we use fuzzy observables
$F=\big{\\{}F_{+}=A(\alpha,\phi,r),\,F_{-}=\mathbbm{1}-A(\alpha,\phi,r)\big{\\}}$
and
$F^{\prime}=\big{\\{}F_{+}=A(\alpha^{\prime},\phi^{\prime},r^{\prime}),\,F_{-}=\mathbbm{1}-A(\alpha^{\prime},\phi^{\prime},r^{\prime})\big{\\}}$,
with $\alpha\neq 1$ and/or $\alpha^{\prime}\neq 1$, to approximate $E$ and
$E^{\prime}$, condition (7.12) is only necessary.

##  8\. Conclusions

We considered in this work the role of real Positive Operator-Valued Measures
as quantizers of a physical system as well as their status as quantum
observables in finite dimension. We focused in particular in dimension 2,
which is the lowest one for which we can develop a non-trivial quantum
formalism. The starting point, after having recalled the definition of POVMs,
was to lay down some necessary and sufficient conditions for two POVMs to be
compatible. We then emphasized that POVMs, besides their role as observables,
can also provide a quantization scheme in the framework of integral
quantization. In particular, a first result emerged when considering Toeplitz
quantization as an integral quantization, where we showed that Naimark theorem
is valid in our case. We continued the study by restraining ourselves to
integral quantizations of functions on the unit circle and their actions on
real two-dimensional systems, and where we supposed non-negative POVMs, i.e.
density matrices. In this case, the quantization map gives a non-commutative
version of $\mathbb{R}^{3}$ that can be identified as a Fourier subspace, and
the density matrix (5.8) is particularly simple. Quantizing a function on this
subspace gives the simple $2$x$2$ matrix representation (5.13), which allows
for a simple non-commutative version of $\mathbb{R}^{3}$. The most general
functions living in this Fourier space were found in (5.29). They were
subsequently used to prove that a mixed state can be represented as a
continuous superposition of mixed state, which is a typical quantum mechanical
behavior. Applying Naimark theorem made possible the identification between
the quantum operator of a POVM and its density matrix. A physical illustration
of the previous results was made by using light polarization. Remembering the
construction of the polarization tensor (6.8), we showed that it reduces to
the general form of our two-dimensional density matrices when circular
polarization is neglected. Then, the interaction between a polarizer and the
system was described during a quantum measurement, where we showed that the
probability to obtain a given polarization is given by the evolution equation
(6). The case of sequential measurements was analysed at last. It was shown
that two measurements in which a light ray goes first through an oblique
polarizer before passing through a vertical polarizer is described by a
dichotomic POVM, while a measurement in the reverse order is described by
another dichotomic POVM, showing the incompatibility of the measurement
procedures. We finally searched for the necessary condition (7.12) for the
compatibility of two dichotomic POVMs in a real bidimensional Hilbert space.
In the last section, we related the density matrix’s parameters to the Stokes
parameters defining a polarization tensor for linearly polarized light. It
turns out that we can identify the degree of mixing of a density matrix to the
fuzziness of a quantum observable. We concluded that compatibility conditions
of two POVMs can be expressed in terms of Stokes parameters.

### Acknowledgments

JPG thanks the Brazilian Center for Research in Physics (CBPF) for its
hospitality. EF thanks both the CBPF and the Helsinki Institute of Physics
(HIP) for their hospitality. The present work has been realized in the
framework of the activities of the INDAM (Istituto Nazionale di Alta
Matematica).

##  Appendix A Quantum orientations in $\mathbb{R}^{n}$

###  A.1. Pure and mixed states

We now generalize to $\mathbb{R}^{n}$ a part of the material developed at
length in the previous sections. A vector in $\mathbb{R}^{n}$ is denoted by
$\boldsymbol{v}$ (Euclidean geometry) or $|\boldsymbol{v}\rangle$ (Dirac or
Hilbertian notation). Thus a pure state is represented by an element of
$\mathbb{S}^{n-1}/\mathbb{Z}_{2}$, and a mixed state by the general density
matrix

(A.1)
$\rho\equiv\rho_{\boldsymbol{\eta},\boldsymbol{\phi}}=\frac{1}{n}\mathbbm{1}+\mathcal{R}(\boldsymbol{\phi})\,\mathsf{D}(\boldsymbol{\eta})\,{}^{t}\mathcal{R}(\boldsymbol{\phi})\,,$

with notations similar to (5.8). The diagonal matrix
$\mathbbm{1}/n\equiv\rho_{\mathrm{rm}}$ which appears in this expression
describes the full random mixing. The symbol $\mathsf{D}$ stands for the
diagonal matrix

(A.2) $\mathsf{D}(\boldsymbol{\eta})=\begin{pmatrix}\eta_{1}&\dots&0\\\
\vdots&\ddots&\vdots\\\ 0&\dots&\eta_{n}\end{pmatrix}\,,$

where the vector $\boldsymbol{\eta}={}^{t}(\eta_{1},\eta_{2},\cdots,\eta_{n})$
lies in the simplex defined by the conditions

(A.3) $\sum_{i=1}^{n}\eta_{i}=0\,,\quad-\frac{1}{n}\leq\eta_{i}\leq
1-\frac{1}{n}\,,$

i.e., the intersection of the hyperplane normal to the first diagonal and
containing the origin with the unit hypercube shifted by the vector
${}^{t}(-1/n,-1/n,\dotsc,-1/n)$. The rotation matrix
$\mathcal{R}(\boldsymbol{\phi})\in\mathrm{SO}(n)$ and the symbol
$\boldsymbol{\phi}$ stands for the set of $n(n-1)/2$ (angular) parameters of
the group SO$(n)$. Thus, the manifold of density matrices (A.1) is real with
dimension equal to $n(n+2)/2$. The representation of the SO$(n)$ group
composition law is written here as

(A.4)
$\mathcal{R}(\boldsymbol{\phi})\mathcal{R}(\boldsymbol{\phi}^{\prime})=\mathcal{R}(\boldsymbol{\phi}\circ\boldsymbol{\phi}^{\prime})\;.$

We adopt the Euler parametrisation based on the descending sequence
$\mathrm{SO}(n)\supset\mathrm{SO}(n-1)\supset\cdots\supset\mathrm{SO}(2)$
[78]. First, for $1\leq k\leq n-1$, we denote $\mathcal{R}_{k}(\phi)$ the
rotation by the angle $\phi$ in the $\left(x_{k},x_{k+1}\right)$-plane. Next,
we introduce the successive products of such elementary rotations:

(A.5)
$\mathcal{R}^{(k)}\left(\boldsymbol{\phi}^{(k)}\right)=\mathcal{R}_{1}(\phi_{1}^{k})\,\mathcal{R}_{2}(\phi_{2}^{k})\,\cdots\mathcal{R}_{k}(\phi_{k}^{k})\,,\quad\boldsymbol{\phi}^{(k)}=\left(\phi_{1}^{k},\phi_{2}^{k},\dotsc,\phi_{k}^{k}\right)\,.$

For each $1\leq k\leq n-1$, the Euler angles $\phi_{j}^{k}$, $1\leq j\leq k$
vary as follows

(A.6) $\left\\{\begin{array}[]{cc}0\leq\phi_{1}^{k}<2\pi&\\\
0\leq\phi_{j}^{k}<\pi&j\neq 1\end{array}\right.\,.$

Then any rotation $\mathcal{R}(\boldsymbol{\phi})$ of the group SO$(n)$ can be
written [78] as the product

(A.7)
$\mathcal{R}(\boldsymbol{\phi})=\prod_{k=1}^{n-1}\mathcal{R}^{(k)}\left(\boldsymbol{\phi}^{(k)}\right)\,,\quad\boldsymbol{\phi}=\left(\boldsymbol{\phi}^{(1)},\dotsc,\boldsymbol{\phi}^{(n-1)}\right)\,.$

The non-normalized invariant (Haar) measure on SO$(n)$ reads in terms of the
above $n(n-1)/2$ Euler angular variables:

(A.8)
$\mathrm{d}\boldsymbol{\phi}=\prod_{k=1}^{n-1}\prod_{j=1}^{k}\,\sin^{j-1}\phi_{j}^{k}\,\mathrm{d}\phi_{j}^{k}\,,$

from which we obtain the volume of SO$(n)$

(A.9)
$\mathrm{Vol}\left(\mathrm{SO}(n)\right)=\int_{\mathrm{SO}(n)}\mathrm{d}\boldsymbol{\phi}=\prod_{i=1}^{n}\mathrm{Area}\left(\mathbb{S}^{i-1}\right)\,,\quad\mathrm{Area}\left(\mathbb{S}^{i-1}\right)=2\frac{\pi^{i/2}}{\Gamma(i/2)}\,.$

Finally, note that the spectral decomposition of
$\rho_{\boldsymbol{\eta},\boldsymbol{\phi}}$ is easily deduced from (A.1) by
using the orthonormal set of column vectors of the rotation matrix
$\mathcal{R}(\boldsymbol{\phi})$:

(A.10)
$\rho_{\boldsymbol{\eta},\boldsymbol{\phi}}=\sum_{i=1}^{n}\left(\frac{1}{n}+\eta_{i}\right)\,|\boldsymbol{v}_{i}\rangle\langle\boldsymbol{v}_{i}|\,,\quad\mathcal{R}(\boldsymbol{\phi})=(\boldsymbol{v}_{1},\dotsc,\boldsymbol{v}_{n})\,.$

###  A.2. SO$(n)$ covariant integral quantisation

Let us choose a set of $(n+2)(n-1)/2$ parameters
$(\boldsymbol{\eta},\boldsymbol{\phi}_{0})$. The covariant integral
quantisation of functions (or distributions) on the SO$(n)$ manifold is based
on the resolution of the identity provided by the rotational transport of the
matrix density $\rho_{\boldsymbol{\eta},\boldsymbol{\phi}_{0}}$, similarly as
what we have seen in the two-dimensional case (5.12). The resolution of
identity reads

(A.11)
$\mathbbm{1}=\int_{\mathrm{SO}(n)}\frac{\mathrm{d}\boldsymbol{\phi}}{c_{n}}\,\mathcal{R}(\boldsymbol{\phi})\,\rho_{\boldsymbol{\eta},\boldsymbol{\phi}_{0}}\,{}^{t}\mathcal{R}(\boldsymbol{\phi})=\int_{\mathrm{SO}(n)}\frac{\mathrm{d}\boldsymbol{\phi}}{c_{n}}\,\rho_{\boldsymbol{\eta},\boldsymbol{\phi}\circ\boldsymbol{\phi}_{0}}\,,$

where $\mathrm{d}\boldsymbol{\phi}$ is the Haar measure on SO$(n)$ and
$\boldsymbol{\phi}\circ\boldsymbol{\phi}_{0}$ stands for the group composition
law in the parameter space. Equality (A.11) is validated by Schur’s Lemma.
Indeed, due to the invariance of the Haar measure, the right-hand side
commutes with the (fundamental) $n$-dimensional unitary irreducible
representation $\boldsymbol{\beta}\mapsto\mathcal{R}(\boldsymbol{\beta})$ of
SO$(n)$. The constant $c_{n}$ is calculated by taking the trace on both sides,

(A.12) $c_{n}=\frac{\mathrm{Vol}\left(\mathrm{SO}(n)\right)}{n}\,.$

Note that the resolution of the identity could be also proved by hand using
the orthonormality between matrix elements of the fundamental (symmetric) UIR
of SO$(n)$, $\boldsymbol{\phi}\mapsto\mathcal{R}(\boldsymbol{\phi})$,

(A.13)
$\int_{\mathrm{SO}(n)}\frac{\mathrm{d}\boldsymbol{\phi}}{c_{n}}\,\mathcal{R}_{ij}(\boldsymbol{\phi})\,\mathcal{R}_{i^{\prime}j^{\prime}}(\boldsymbol{\phi})=\delta_{ii^{\prime}}\,\delta_{jj^{\prime}}\,,$

which, with $\sum_{i=1}^{n}\eta_{i}=0$, entails

(A.14)
$\int_{\mathrm{SO}(n)}\frac{\mathrm{d}\boldsymbol{\phi}}{c_{n}}\,\mathcal{R}(\boldsymbol{\phi})\,\mathsf{D}(\boldsymbol{\eta})\,{}^{t}\mathcal{R}(\boldsymbol{\phi})=0_{n\times
n}\,.$

From (A.11), we can derive the quantisation of a real-valued function (or
distribution) $f(\boldsymbol{\phi})$ on SO$(n)$. It is precisely given by the
map

(A.15) $\begin{split}f(\boldsymbol{\phi})\mapsto
A_{f}&=\int_{\mathrm{SO}(n)}\frac{\mathrm{d}\boldsymbol{\phi}}{c_{n}}\,f(\boldsymbol{\phi})\,\rho_{\boldsymbol{\eta},\boldsymbol{\phi}\circ\boldsymbol{\phi}_{0}}=\int_{\mathrm{SO}(n)}\frac{\mathrm{d}\boldsymbol{\phi}}{c_{n}}\,f\left(\boldsymbol{\phi}\circ(\boldsymbol{\phi}_{0})^{-1}\right)\,\rho_{\boldsymbol{\eta},\boldsymbol{\phi}}\\\
&=\langle
f\rangle+\int_{\mathrm{SO}(n)}\frac{\mathrm{d}\boldsymbol{\phi}}{c_{n}}\,f\left(\boldsymbol{\phi}\circ(\boldsymbol{\phi}_{0})^{-1}\right)\,\mathcal{R}(\boldsymbol{\phi})\,\mathsf{D}(\boldsymbol{\eta})\,{}^{t}\mathcal{R}(\boldsymbol{\phi})\\\
&\equiv\langle f\rangle+\mathsf{N}_{f}\,.\end{split}$

In the above equation, $\langle f\rangle$ denotes the mean value of $f$ on
$\mathrm{SO}(n)$,

(A.16) $\langle
f\rangle=\int_{\mathrm{SO}(n)}\frac{\mathrm{d}\boldsymbol{\phi}}{\mathrm{Vol}\left(\mathrm{SO}(n)\right)}\,f(\boldsymbol{\phi})\,.$

We find the elements of the matrix $\mathsf{N}_{f}$ using the decomposition of
the tensor product

(A.17)
$\mathcal{R}\otimes\mathcal{R}=\oplus_{\boldsymbol{t}}\mathrm{T}^{\boldsymbol{t}}\;,$

as a (finite) direct sum of UIR’s $\mathrm{T}^{\boldsymbol{t}}$ of SO$(n)$
($\boldsymbol{t}$ is a multi-index). Actually, the range of this multi-index
is quite narrow in our case. From the Peter-Weyl theorem for continuous
functions on a compact group [79], this expansion reads on the level of matrix
elements of $\mathcal{R}$:

(A.18)
$\mathcal{R}_{il}(\boldsymbol{\phi})\,\mathcal{R}_{jl}(\boldsymbol{\phi})=\sum_{\boldsymbol{t},\boldsymbol{I},\boldsymbol{J}}C^{\boldsymbol{t}}_{il,jl,\boldsymbol{I},\boldsymbol{J}}\mathrm{T}^{\boldsymbol{t}}_{\boldsymbol{I},\boldsymbol{J}}(\boldsymbol{\phi})\,,$

where the $C^{\boldsymbol{t}}_{il,jl,\boldsymbol{I},\boldsymbol{J}}$ are
Clebsch-Gordan coefficients and $\boldsymbol{I}$, $\boldsymbol{J}$, are multi-
indices. The range of $\boldsymbol{\eta}$ is obviously quite limited. On the
other hand, with minimal assumptions on $f$, e.g. square integrability on
SO$(n)$, and from the completeness of the
$\mathrm{T}^{\boldsymbol{t}}_{\boldsymbol{I},\boldsymbol{J}}(\boldsymbol{\phi})$’s,
one can Fourier expand $f$ as

(A.19)
$f(\boldsymbol{\phi})=\sum_{\boldsymbol{t},\boldsymbol{I},\boldsymbol{J}}f^{\boldsymbol{t}}_{\boldsymbol{I},\boldsymbol{J}}\mathrm{T}^{\boldsymbol{t}}_{\boldsymbol{I},\boldsymbol{J}}(\boldsymbol{\phi})\,,$

and eventually,

(A.20)
$f\left(\boldsymbol{\phi}\circ(\boldsymbol{\phi}_{0})^{-1}\right)=\sum_{\boldsymbol{t},\boldsymbol{I},\boldsymbol{J},\boldsymbol{K}}f^{\boldsymbol{t}}_{\boldsymbol{I},\boldsymbol{J}}\mathrm{T}^{\boldsymbol{t}}_{\boldsymbol{I},\boldsymbol{K}}(\boldsymbol{\phi})\,\mathrm{T}^{\boldsymbol{t}}_{\boldsymbol{J},\boldsymbol{K}}\left(\boldsymbol{\phi}_{0}\right)\,,$

With (A.18) and (A.20) in hand, and from the orthogonality of the
$\mathrm{T}^{\boldsymbol{t}}_{\boldsymbol{I},\boldsymbol{J}}(\boldsymbol{\phi})$’s,
the matrix elements of $\mathsf{N}_{f}$ read as

(A.21)
$\left[\mathsf{N}_{f}\right]_{ij}=\sum_{l=1}^{n}\eta_{l}\sum_{\boldsymbol{t},\boldsymbol{I},\boldsymbol{J},\boldsymbol{K}}f^{\boldsymbol{t}}_{\boldsymbol{I},\boldsymbol{J}}\,C^{\boldsymbol{t}}_{il,jl,\boldsymbol{I},\boldsymbol{J}}\,\mathrm{T}^{\boldsymbol{t}}_{\boldsymbol{J},\boldsymbol{K}}\left(\boldsymbol{\phi}_{0}\right)\,.$

The covariance of the quantisation map (A.15) means

(A.22)
$\mathcal{R}(\boldsymbol{\beta})\,A_{f}\,{}^{t}\mathcal{R}(\boldsymbol{\beta})=A_{\mathrm{R}(\boldsymbol{\beta})f}\,,\quad(\mathrm{R}(\boldsymbol{\beta})f)(\boldsymbol{\phi}):=f\left((\boldsymbol{\beta})^{-1}\circ\boldsymbol{\phi}\right)\,.$

Let us determine the function $f=1/n+g$ such that $A_{f}$ is the density
matrix $\rho_{\boldsymbol{\zeta},\boldsymbol{\beta}}$, i.e.,

(A.23)
$\rho_{\boldsymbol{\zeta},\boldsymbol{\beta}}=\int_{\mathrm{SO}(n)}\frac{\mathrm{d}\boldsymbol{\phi}}{c_{n}}\,\left(\frac{1}{n}+g(\boldsymbol{\phi})\right)\,\rho_{\boldsymbol{\eta},\boldsymbol{\phi}\circ\boldsymbol{\phi}_{0}}\,.$

Taking the trace on both sides implies that

(A.24)
$\int_{\mathrm{SO}(n)}\mathrm{d}\boldsymbol{\phi}\,g(\boldsymbol{\phi})=0\,,\quad\mbox{{i.e.}},\quad\langle
g\rangle=0\,.$

Using (A.11), (A.14), and (A.24) makes (A.23) equivalent to

(A.25)
$\mathcal{R}(\boldsymbol{\beta})\,\mathsf{D}(\boldsymbol{\zeta})\,{}^{t}\mathcal{R}(\boldsymbol{\beta})=\int_{\mathrm{SO}(n)}\frac{\mathrm{d}\boldsymbol{\phi}}{c_{n}}\,g(\boldsymbol{\phi})\,\mathcal{R}(\boldsymbol{\phi}\circ\boldsymbol{\phi}_{0})\,\mathsf{D}(\boldsymbol{\eta})\,{}^{t}\mathcal{R}(\boldsymbol{\phi}\circ\boldsymbol{\phi}_{0})\,.$

Using the SO$(n)$-covariance, one gets for each $1\leq l\leq n$

(A.26)
$\frac{\zeta_{l}}{\eta_{l}}=\sum_{i,j}\int_{\mathrm{SO}(n)}\frac{\mathrm{d}\boldsymbol{\phi}}{c_{n}}\,g\left((\boldsymbol{\beta})^{-1}\circ\boldsymbol{\phi}\right)\,\mathcal{R}_{il}(\boldsymbol{\phi}\circ\boldsymbol{\phi}_{0})\,\mathcal{R}_{jl}(\boldsymbol{\phi}\circ\boldsymbol{\phi}_{0})\,.$

Using the reduction formula (A.18), expanding
$g(\boldsymbol{\phi})=\sum_{\boldsymbol{t},\boldsymbol{I},\boldsymbol{J}}g^{\boldsymbol{t}}_{\boldsymbol{I},\boldsymbol{J}}\mathrm{T}^{\boldsymbol{t}}_{\boldsymbol{I},\boldsymbol{J}}(\boldsymbol{\phi})$
as in (A.19), using the SO$(n)$ representation law and the orthogonality of
the corresponding matrix elements yields the linear system for the components
$g^{\boldsymbol{t}}_{\boldsymbol{I},\boldsymbol{J}}$:

(A.27)
$\frac{\zeta_{l}}{\eta_{l}}=\sum_{i,j,\boldsymbol{t},\boldsymbol{I},\boldsymbol{J},\boldsymbol{K},\boldsymbol{I}^{\prime}}g^{\boldsymbol{t}}_{\boldsymbol{I}^{\prime},\boldsymbol{J}}\,C^{\boldsymbol{t}}_{il,jl,\boldsymbol{I}\boldsymbol{J}}\mathrm{T}^{\boldsymbol{t}}_{\boldsymbol{I}^{\prime},\boldsymbol{J}}(\boldsymbol{\beta})\,\mathrm{T}^{\boldsymbol{t}}_{\boldsymbol{K},\boldsymbol{J}}(\boldsymbol{\phi}_{0})\,.$

Let us impose in our choice of $g$ the relations

(A.28)
$\sum_{i,j}g^{\boldsymbol{t}}_{\boldsymbol{I}^{\prime},\boldsymbol{J}}\,C^{\boldsymbol{t}}_{il,jl,\boldsymbol{I}\boldsymbol{J}}=h^{\boldsymbol{t}}_{l,\boldsymbol{I}}\delta_{\boldsymbol{I}^{\prime}\boldsymbol{J}}\,.$

We next choose $\boldsymbol{\phi}_{0}=\left(\boldsymbol{\beta}\right)^{-1}$.
We finally end with the following values of $\eta_{l}$:

(A.29)
$\eta_{l}=\frac{\zeta_{l}}{\sum_{\boldsymbol{I},\boldsymbol{t}}h^{\boldsymbol{t}}_{l,\boldsymbol{I}}}\,.$

Finally, let us consider the $n^{2}$-dimensional Lie algebra
$\mathfrak{gl}(n,\mathbb{R})$ and its Cartan decomposition into antisymmetric
and symmetric matrices,

(A.30) $\mathfrak{gl}(n,\mathbb{R})=\mathfrak{k}\oplus\mathfrak{p}\,,$

it is easy to prove that the quantisation (A.15) maps real-valued functions
into the $n(n+1)/2$-dimensional space $\mathfrak{p}$ of symmetric matrices,
i.e., the quantum observables, while $\mathfrak{k}\simeq\mathfrak{so}(n)$ is
the $n(n-1)/2$-dimensional Lie algebra of SO$(n)$. In particular, any element
$X$ in the sub-algebra $\mathfrak{k}$ may play the role of a
(pseudo)-Hamiltonian as a generator of the evolution (i.e. unitary) operator
$e^{tX}$ in $\mathbb{R}^{n}$. moreauthor morelabelname

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