# Quantum Tomography and Schwinger’s Picture of Quantum Mechanics

F. M. Ciaglia 1,6 , F. Di Cosmo1,2,7 , A. Ibort1,2,8 ,
G. Marmo3,4,9
1Departamento de Matemáticas, Univ. Carlos III de Madrid, Leganés, Madrid,
Spain
2 ICMAT, Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM)
3 INFN-Sezione di Napoli, Naples, Italy
4 Dipartimento di Fisica “E. Pancini”, Università di Napoli Federico II,
Naples, Italy
6 e-mail: fciaglia[at]math.uc3m.es
7 e-mail: fcosmo[at]math.uc3m.es
8 e-mail: albertoi[at]math.uc3m.es
9 e-mail: gmarmo[at]unina.it

###### Abstract

In this paper the problem of tomographic reconstruction of states is
investigated within the so-called Schwinger’s picture of Quantum Mechanics in
which a groupoid is associated with every quantum system. The attention is
focused on spin tomography: In this context the groupoid of interest is the
groupoid of pairs over a finite set. In a nutshell, this groupoid is made up
of transitions between all possible pairs of outcomes belonging to a finite
set. In addition, these transitions possess a partial composition rule,
generalizing the notion of groups. The main goal of the paper consists in
providing a reconstruction formula for states on the groupoid-algebra
associated with the observables of the system. Using the group of bisections
of this groupoid, which are special subsets in one-to-one correspondence with
the outcomes, a frame is defined and it is used to prove the validity of the
tomographic reconstruction. The special case of the set of outcomes being the
set of integers modulo n, with n odd prime, is considered in detail. In this
case the subgroup of discrete affine linear transformations, whose graphs are
linear subspaces of the groupoid, provides a quorum in close analogy with the
continuos case.

##  1 Introduction

The relation between classical and quantum mechanics is a long-standing
question in Quantum Theory. Since the seminal works by Bohr, Dirac and Weyl,
classical mechanics was meant to be classical Hamiltonian mechanics, due to
the direct analogy between Poisson brackets and commutators and between
Hamilton and Heisenberg equations. Many characteristic features of Quantum
Theories have been understood using methods from symplectic and Poisson
geometry and one fruit of this investigation was the geometric quantization
program by Kirillov, Konstant and Souriau [46, 41, 63]. However, quantization
is only one side of the correspondence between classical and quantum mechanics
and, if we insist on viewing quantum theories as being “more fundamental”,
quantization is not the most appropriate tool to obtain a deeper understanding
of quantum phenomena. Indeed a fundamental observation is the fact that
classical behaviours of physical systems are obtained from quantum ones by
taking a suitable limit. Consequently, there must always be a dequantization
procedure describing such a limit and classicality should appear in this way.
This was the rationale behind the generalization of Weyl-Wigner
correspondence[66, 67] to the Quantizer-Dequantizer formalism proposed by some
of the authors [51, 9]. Koopman’s paper [45] can be considered a seminal work
in this direction, since for the first time classical mechanics was described
using the same formalism of Schrödinger’s formulation of Quantum Mechanics.
Once a unified description for quantum and classical systems exists the
problem of the emergence of classicality can be approached more directly.
Despite such an unified approach, quantum and classical systems have very
different properties and being able to distinguish them is a crucial problem
which is to be faced especially for the development of Quantum Technologies
[8, 33, 65]. In particular, being able to discriminate between quantum and
classical states is one of the fundamental issues in current quantum computing
[58, 16, 23].

The modern notion of state has its roots back to von Neumann’s “Gesamtheiten”
[64]. As stated by Leonhardt, “knowing the state means knowing the maximally
available statistical information about all physical quantities of a physical
object” [47]. This corresponds to the knowledge of the probability
distributions associated with all the possible observables of a physical
system [8]. While classically a state is a single probability distribution on
the phase space of the system, in the quantum case there are infinitely many
different probability distributions associated with every quantum state and it
is not possible to obtain information about all of them simultaneously.
However, a big amount of this information is redundant, and, in many
instances, a quantum state can be actually reconstructed from the knowledge of
a finite number of probability distributions (certainly for a system with a
finite number of degrees of freedom). This reconstruction problem is the
fundamental question of Quantum Tomography. The problem of Quantum Tomography
originates from a question raised by Pauli [57] and answered in the negative
by Bargmann [60]: Is it possible to reconstruct any wave function from the
probability distributions associated with position and momentum operators?
Since then, Quantum Tomography has rapidly evolved and it has become a
fundamental chapter in Quantum Information Theory: in particular, here one can
distinguish exact tomography from real tomography (see for instance the books
[55, 56]). In exact tomography no source of error is considered and the exact
reconstruction of the original state can be achieved in a finite amount of
time. On the other hand, real tomography deals with physical situation where
the occurrence of errors needs to be taken into account in order to get
meaningful results (see for instance the papers [59, 39] where real
tomographic procedures for single and multiple qubits are described). In this
case, an effective estimation of the state of the system can be obtained only
after a post-measurement process, and adaptive procedures in this phase are
under investigation (see for instance the recent algorithm [29]).
Additionally, the features of the involved state may suggest more efficient
procedures which can be adopted, like tomography via compressed sensing
illustrated in [31, 26] which is suited for the reconstruction of low rank
states. In this paper we will focus only on exact tomography, whereas the
problem of state estimation and real tomography will be dealt in forthcoming
works.

There exist many different procedures for the exact tomographic reconstruction
of a state. From one point of view, it is possible to perform different
measures on copies of the state obtained from the same ensemble: each
measurement will provide the experimenter with a probability distribution and
the knowledge of a sufficient number of probabilities allows for the
reconstruction of the state. From this point of view, if we have a quantum
system with a finite number of degrees of freedom, say n, the maximum amount
of information about the ensemble can be got if one performs n+1 measurements
which are mutually unbiased [68, 38], i.e., the associated orthonormal bases
$\left\\{\mid e^{(r)}_{j}\rangle\right\\}$, with $r=1,2,\cdots,n+1$ and
$j=1,2,\cdots,n$ satisfy the property $|\langle
e^{(r)}_{j}|e^{(s)}_{k}\rangle|^{2}=1/n$ whenever $r\neq s$.

On the other hand, instead of considering a set of unbiased measurements, one
could consider a more efficient tomographic reconstruction via Symmetric
Informationally Complete-POVMs (SIC-POVMs), i.e., a positive operator valued
measure whose statistics allows to reconstruct the state[4, 61]. A SIC-POVM
for a n-dimensional quantum system consists of $n^{2}$ positive operators,
each one being a multiple of a projector
$P_{j}=|\psi_{j}\rangle\langle\psi_{j}|$, such that their sum is the identity
operator and the adjective symmetric refers to the property that
$|\langle\psi_{j}|\psi_{k}\rangle|^{2}=1/(n+1)$, for any $j,k$. This number,
however, can be reduced in the particular instance of pure states tomography,
and the minimal number of vectors in an Informationally Complete POVM is 2N
[27].

The existence of mutually unbiased orthonormal bases (MUBs) and SIC-POVMs in
every dimension is still an open question and only partial results are known.
For instance MUBs have been proven to exist in every dimension power of a
prime number [68, 38], whereas, using group theory (S)IC-POVMs have been found
in many dimensions (see for instance [61, 20] for some first examples of these
covariant POVMs) but an existence theorem for every $n$ is still lacking.

In this paper, we exploit the so-called Schwingers picture of Quantum
Mechanics [13, 10] based on groupoids, in order to get reconstruction formulae
for a state of a quantum system with a finite number of degrees of freedom. In
this context, to every quantum system there is associated a groupoid
$\mathcal{G}(\Omega)$ over a finite set $\Omega$ and a von Neumann algebra
$\mathcal{V}(\mathcal{G})$ which is generated by suitable functions on
$\mathcal{G}(\Omega)$ endowed with a convolution product. Moreover,
$\mathcal{V}(\mathcal{G})$ is represented on the Hilbert space
$\mathcal{H}(\mathcal{G})$ of square-integrable functions on
$\mathcal{G}(\Omega)$. Using the notion of bisection [49, 15] we show that it
is possible to build a frame for $\mathcal{H}(\mathcal{G})$ when
$\mathcal{G}(\Omega)$ is the pair groupoid of a set $\Omega$ with finite
cardinality, say $n$. Then, using the decomposition of the identity associated
with this frame we are able to provide a reconstruction formula for any state.

A fundamental ingredient of this approach is that characteristic functions
supported on bisections determine a unitary representation of the permutation
group $\mathbb{S}_{n}$, for generic values of $n$. However, contrarily to the
situation investigated in many previous works [18, 19, 6], we do not use an
irreducible representation, but the fundamental one. Moreover, in the case of
odd prime dimension, a subgroup of the whole symmetric group $\mathbb{S}_{n}$
is sufficient to produce a frame, i.e., the group of discrete affine linear
transformations (see Sect.4.3). It is interesting to notice that the
associated graph is the discrete analogue of a line and one can interpret the
reconstruction formula as the discrete analogue of the generalized Radon
transform using the set of all Lagrangian subspace of a symplectic vector
space [54]. Despite of this nice geometric analogy, the frame which is
constructed in this paper is not tight and consequently it cannot be directly
used to construct a (S)IC-POVM. Nevertheless, generalizations to address this
problem can be conceived.

Another relevant aspect consists in the fact that using a different approach
directly inspired by classical Radon transform, we derive a tomographic
approach which is similar to the one based on the so called (generalized)
Pauli group in the discrete phase space [30, 43, 1], which has been shown to
have a deep connection with the notion of MUBs and IC-POVMs [44, 62]. Indeed,
in these works displacement operators are constructed satisfying the relations
$XY=\omega YX$ ($\omega=\exp(i2\pi/n)$) and using their powers a sufficient
number of mutually unbiased orthonormal bases is obtained. These operators can
be grouped into representations of the group of integers modulo $n$, each one
providing a basis of the Hilbert space $\mathbb{C}^{n}$. In the rest of this
paper, instead of using a representation of the Pauli group and the associated
displacement operators, we will prove that the discrete analogue of a
Lagrangian submanifold can be obtained for the pair groupoid
$\mathcal{G}(\Omega)=\Omega\times\Omega$, with $\Omega$ the finite field of
order $n$ and $n$ odd prime. In this case, a bisection defines an immersion
map of $\Omega$ within $\mathcal{G}(\Omega)$ and the bisections associated
with the graphs of linear maps form the analogue of linear subspaces over
which we integrate the function associated with a state. In this sense the
procedure outlined in this paper is closely related to the classical idea of
tomographic reconstruction, even if the setting is purely quantum. In
particular, we will also find a family of $n+1$ orthonormal bases, but they
are not mutually unbiased. A way to recover the previously mentioned mutually
unbiased bases will be presented in Sec.4.3.

Apart from introductory and conclusive sections, the paper contains three
sections. In Sec.2 the ingredients of a mathematical description of a physical
system are provided, to set the stage for the unified approach to classical
and quantum mechanics previously mentioned. Using such frame, we briefly
recall the main ingredients of the different formulations of Quantum
Mechanics. In Sec.3 we present the Weyl-Wigner correspondence and its
connection to Quantum Tomography. Finally, in Sec.4 we illustrate a
tomographic reconstruction procedure adapted to the groupoid picture of
Quantum Mechanics. The two cases of generic $n$ and $n$ odd prime, are
presented and the peculiarities of the second case are highlighted.

##  2 Mathematical description of physical systems

Every mathematical description of a physical system requires the introduction
of some basic ingredients:

  * •

a set $\mathcal{O}$ whose elements are the observables of the theory;

  * •

a set $\mathfrak{S}$ whose elements are the states of the theory (usually, it
is a convex set);

  * •

a statistical interpretation provided by a map,
$\mu\,\colon\mathfrak{S}\times\mathcal{O}\,\rightarrow\,\mathscr{M}\left(\mathbb{R}\right)$,
where $\mathscr{M}\left(\mathbb{R}\right)$ denotes the space of probability
measures on the real line;

  * •

a dynamical evolution law provided by a family of consistent evolution
operators,

$U_{t,s}\,\colon\,\mathfrak{S}\,\rightarrow\,\mathfrak{S}\,,\quad
t,s\in\mathbb{R}_{+}\,,\quad U_{t_{2},t_{1}}=U_{t_{2},s}\circ U_{s,t_{1}}\,;$
(1)

  * •

an algorithmic rule to compose physical systems.

Concerning the evolution map, with additional requirements about its
diferentiability, it can be determined by a differential equation. The
probability map $\mu$, instead, is interpreted as follows: the number
$\mu(\rho,A)(E)$, with $\rho\in\mathfrak{S}$, $A\in\mathcal{O}$, and $E$ a
measurable subset of $\mathbb{R}$, is the probability that a measurement of
the observable $A$ gives a result in the subset $E$.

According to this scheme, we can illustrate the different formalisms of both
Quantum and Classical Mechanics. Indeed, in Classical Mechanics a physical
system is described by a pair $(\mathcal{M},\omega)$, where $\mathcal{M}$ is a
suitable manifold and $\omega$ is a symplectic 2-form. The set of observables
is given by a set of measurable real-valued functions
$\mathscr{F}(\mathcal{M})=:\mathcal{O}$, which also forms an associative
commutative algebra. The set of states is given by the family of probability
measures $\mathfrak{S}:=\mathscr{P}(\mathcal{M})$ on the manifold
$\mathcal{M}$. Given a state $\nu\in\mathfrak{S}$ and an observable
$f\in\mathcal{O}$, the probability map $\mu$ is provided by the measure,
$\mu(\nu,f)(E)=\nu(f^{-1}(E))$, according to which the observables are
interpreted as random variables on a probability space. The evolution map is a
one-parameter group of symplectic transformations,
$\varphi_{t}\,\colon\,\mathcal{M}\,\rightarrow\,\mathcal{M}$ possessing
additional regularity properties. In the case of differentiable one-parameter
group of symplectic transformations we recover an infinitesimal description
given by the well-known Hamilton equations. Finally, if we have two physical
system $(\mathcal{M}_{1},\omega_{1})$ and $(\mathcal{M}_{2},\omega_{2})$,
respectively, we define their composition as the system
$(\mathcal{M}_{1}\times\mathcal{M}_{2},\omega_{1}\oplus\omega_{2})$.

On the other hand, we have different formulations, also called “pictures”, for
Quantum Mechanics. In the standard picture, the departure point is the
association of a Hilbert space $\mathcal{H}$ with a quantum system. Then, the
observables are represented as self-adjoint operators on $\mathcal{H}$; the
states of the system are identified with non-negative linear operators on
$\mathcal{H}$ with unit trace (density operators), and the extreme points
(pure states) of the space of states $\mathfrak{S}$ are rank-one projectors
which are in 1-1 correspondence with rays in $\mathcal{H}$ (elements of the
complex projective space $\mathbb{P}(\mathcal{H})$). The evolution map
$U_{t}\,\colon\,\mathcal{H}\,\rightarrow\,\mathcal{H}$ is a one-parameter
group of unitary transformations of the Hilbert space $\mathcal{H}$. At the
infinitesimal level the evolution is determined by the Schrödinger equation.
As for the probabilistic interpretation, given a self-adjoint operator
$A\in\mathscr{B}(\mathcal{H})$ we have its spectral decomposition
$A=\int_{\sigma(A)}\lambda\,\mathrm{d}\Pi_{\lambda}$, where $\Pi_{\lambda}$ is
a projection-valued measure on the spectrum $\sigma\subset\mathbb{R}$ of the
operator $A$. Then, given a pure state $\rho_{\psi}$ and an observable $A$ we
have the probability map

$\mu(\rho_{\psi},A)(E)=\int_{E}\frac{\,\left\langle\psi\,|\,\mathrm{d}\Pi_{\lambda}\psi\right\rangle}{\left\langle\psi\,|\,\psi\right\rangle}\,.$
(2)

The quantity
$e_{\psi}(A):=\frac{\left\langle\psi\,|\,A\psi\right\rangle}{\left\langle\psi\,|\,\psi\right\rangle}$
is interpreted as the expectation value of the observable $A$ on the state
associated to the vector $|\left.\psi\right\rangle\in\mathcal{H}$, whereas
$e_{\psi}(A^{2})-(e_{\psi}(A))^{2}$ is the corresponding dispersion. The
probabilistic interpretation is straightforwardly extended to mixed normal
states using the trace on $\mathscr{B}(\mathcal{H})$. Regarding the
composition, quantum systems are composed according to von Neumanns
composition rule, that is, the Hilbert space of the total system is the tensor
product of the composing systems.

It was noticed almost immediately that the Hilbert space formalism was not
suited only to Quantum Mechanics but it could be adapted to describe also
Classical Mechanics. This fundamental observation was elaborated in 1931 by
Koopman in the paper “Hamiltonian systems and transformations in Hilbert
space” [45] and it was the departing point of an extremely active field of
research on the relation between Classical and Quantum Mechanics. Then,
according to the previous scheme, given a classical Hamiltonian system
$(\mathcal{M},\,\omega)$, with $\mathrm{dim}(\mathcal{M})=2n$, we can
associate the Hilbert space $\mathcal{H}=\mathcal{L}^{2}(\mathcal{M},\nu)$
where $\nu$ is the measure determined by the volume form
$\mathrm{vol}_{\mathcal{M}}=\omega^{n}$. The set of observables,
$\mathcal{O}$, is formed by multiplication operators on $\mathcal{H}$, and the
states are probability measures absolutely continuous with respect to the
measure $\nu$. The evolution map is given by the one-parameter group of
unitary transformations, $(U_{t}\psi)(m)=\psi(\varphi_{t}(m))$, where
$m\in\mathcal{M}$ and $\varphi_{t}$ is a one-parameter group of
symplectomorphisms of $\mathcal{M}$. The Hilbert space associated with a
composite system is the tensor product of the subsystems. Therefore, the same
formalism originated in Quantum Mechanics can be used also for the description
of classical systems. The need for a formalism capable of accommodating the
description of both classical and quantum systems is now considered as an
important feature of a theoretical description of physical systems.

Another way of looking at this feature comes from the Heisenberg picture of
Quantum Mechanics, a description which is dual to the Schrödinger’s one, in
the sense that the focus is on the space of observables, instead of the set of
states. In this case every physical system is associated with a
$C^{*}$-algebra [3], say $\mathcal{A}$, whose real elements, i.e. invariant
under the involution $\ast\,\colon\,\mathcal{A}\,\rightarrow\,\mathcal{A}$ of
the algebra, are interpreted as the observables of the system. The set
$\mathcal{O}$ of observables is not a subalgebra, since the product of two
real elements does not need to be real. However, it can be endowed with a
Jordan algebra structure, whose structure is connected with fundamental
questions of Quantum Information Theory which are currently under
investigation (see [14, 24]). The set $\mathfrak{S}$ of states is the set of
positive linear functionals of norm one, where the norm is the norm of the
dual Banach space $\mathcal{A}^{*}$. The statistical interpretation is
recovered by noticing that the functional calculus for $C^{*}$-algebras [3]
extends the functional calculus on the algebra $\mathscr{B}(\mathcal{H})$ and
the probability map is given by

$\mu(\rho,A)(E)=\rho(\int_{E}\,\mathrm{d}\Pi_{\lambda})\,,$ (3)

where $\Pi_{\lambda}$ is the projection-valued measure on the spectrum of a
real element of a $C^{*}$-algebra. Therefore, we get the expectation value map
$e_{\rho}(A)=\rho(A)$. The evolution map, instead, is given by a one-parameter
group $\varphi_{t}$ of automorphisms of the algebra $\mathcal{A}$ which at the
infinitesimal level is generated by a derivation of the algebra. In the
particular instance that $\varphi_{t}$ consists of inner automorphisms of the
algebra we recover at the infinitesimal level the Heisenberg equation.
Finally, given two physical systems described by the algebras
$\mathcal{A}_{1}$ and $\mathcal{A}_{2}$, respectively, the composite system is
associated with the tensor product algebra, where a suitable completion has to
be chosen for the tensor product in order to get a $C^{*}$-algebra. In this
context, classical systems arise when the $C^{*}$-algebra is Abelian while the
standard Hilbert space formalism of Quantum Mechanics is essentially recovered
when the $C^{*}$-algebra is $\mathcal{B}(\mathcal{H})$.

##  3 Weyl-Wigner correspondence

In order to contextualize our approach to tomography let us recall briefly the
Weyl-Wigner correspondence and the description of Quantum Mechanics on phase
space. Let us start from considering an Abelian vector group $\mathcal{M}$
with a symplectic structure $\omega$ invariant with respect to the action of
the group on itself. A Weyl system is a map
$W\,\colon\,(\mathcal{M},\,\omega)\,\rightarrow\,\mathcal{U}(\mathcal{H})$
satisfying

$W(v_{1})W(v_{2})W^{\dagger}(v_{1})W^{\dagger}(v_{2})=\mathrm{e}^{i\omega(v_{1},\,v_{2})}\mathbf{1}\,,\quad
v_{1},\,v_{2}\in\mathcal{M}\,,$ (4)

where $\mathcal{U}(\mathcal{H})$ is the group of unitary transformations of a
Hilbert space $\mathcal{H}$. In other words, a Weyl map defines a projective
representation on the Hilbert space $\mathcal{H}$ of the Abelian group
$\mathcal{M}$.

A standard way to construct such a representation is to consider a Lagrangian
subspace $L\subset\mathcal{M}$, so that $\mathcal{M}\cong\mathbf{T}^{*}L$.
Then we introduce the Hilbert space $\mathcal{H}_{L}=\mathcal{L}^{2}(L,\mu)$,
where $\mu$ is a measure invariant under the action of the group itself.
Therefore, $\mathcal{M}$ can be described by the coordinate functions $(x,f)$,
where $x$ belongs to $L$ and $f$ denotes a linear functional in the dual
Abelian group $L^{*}$. Then, we can define a Weyl map as follows

$\begin{split}U\,\colon\,L\,\rightarrow\,\mathcal{U}(\mathcal{H}_{L})\quad(U(x)\psi)(y)=(\mathrm{e}^{ix\hat{P}}\psi)(y)=\psi(y+x)\\\
V\,\colon\,L^{*}\,\rightarrow\,\mathcal{U}(\mathcal{H}_{L})\quad(V(f)\psi)(y)=(\mathrm{e}^{if\hat{Q}}\psi)(y)=\mathrm{e}^{if(y)}\psi(y)\end{split}$
(5)

so that the commutation relations in Eq.(4) hold. Eventually, by using the
symplectic Fourier transform $\tilde{A}$ of any function $A$ on $\mathcal{M}$

$A(q,p)=\int_{\mathcal{M}}\tilde{A}(x,f)\mathrm{e}^{i(p(x)-f(q))}\,\mathrm{d}x\,\mathrm{d}f\,,$
(6)

we can build an operator on the Hilbert space $\mathcal{H}_{L}$ as follows

$A(\hat{Q},\hat{P})=\int_{\mathcal{M}}\tilde{A}(x,f)\mathrm{e}^{i(x\hat{P}-f\hat{Q})}\,\mathrm{d}x\,\mathrm{d}f\,.$
(7)

It is also possible to invert the previous relation, so that given an operator
on $\mathcal{H}_{L}$ we can associate a function on the group $\mathcal{M}$.
In particular, if $F(x,x^{\prime})$ is the position representation of an
operator on $\mathcal{H}_{L}$ we have the following transformation

$\mathcal{W}_{F}(q,p)=\int_{L}F\left(q+\frac{x}{2},\,q-\frac{x}{2}\right)\mathrm{e}^{-ixp}\,\mathrm{d}x\,,$
(8)

which is called Wigner transform.

It is possible to look at the previous construction from a different
perspective. To this aim let us first make a short digression. For the sake of
simplicity let us consider a finite set $\mathcal{S}$ of elements, say
$s_{1},\,s_{2},\,\cdots\,,\,s_{n}$ and take the formal linear combinations

$A_{f}=\sum_{j=1}^{n}f(s_{j})s_{j}\,,$ (9)

where $f$ is a complex-valued function on $\mathcal{S}$ (let us note here that
the function $f$ does not need to be complex-valued but may be real,
quaternionic or even p-adic). The set of all these formal combinations forms a
vector space, say $\mathcal{V}(\mathcal{S})$. Then, if the set $\mathcal{S}$
is endowed with a binary composition rule

$\beta\,\colon\,\mathcal{S}\times\mathcal{S}\,\rightarrow\,\mathcal{S}\,,$
(10)

the vector space $\mathcal{V}(\mathcal{S})$ becomes an algebra with respect to
the product

$A_{f}\cdot
A_{g}=\left(\sum_{j=1}^{n}f(s_{j})s_{j}\right)\cdot\left(\sum_{k=1}^{n}g(s_{k})s_{k}\right)=\sum_{j=1}^{n}\sum_{k=1}^{n}f(s_{j})g(s_{k})\beta(s_{j},s_{k})\,.$
(11)

If the internal composition $\beta$ determines a group structure on the set
$\mathcal{S}$ the algebra $\mathcal{V}(\mathcal{S})$ is an associative algebra
called the group-algebra of the group $\mathcal{S}$. The passage to the
continuous case can be performed by introducing a measure invariant with
respect to the action of the group on itself and replacing the sum with an
integral over the group [28]. Then, if the group has a representation
$U\,\colon\,\mathcal{S}\,\rightarrow\,\mathcal{B}(\mathcal{H})$ on some
Hilbert space $\mathcal{H}$, the elements of the group-algebra are represented
as operators on $\mathcal{H}$ according to
$\hat{A}_{f}=\sum_{j=1}^{n}f(s_{j})U(s_{j})$.

Coming back to the Abelian group $\mathcal{M}$ and the maps $U$ and $V$, its
central extension by means of a 2-cocycle $\omega$ defines the Heisenberg-Weyl
group and the Weyl map is a representation of this group on the Hilbert space
$\mathcal{H}_{L}$ [42]. Therefore, the previous construction can be now
reinterpreted by saying that the Weyl map defines a representation of the
group-algebra of the Heisenberg-Weyl group on the Hilbert space
$\mathcal{H}_{L}$.

Summarizing the previous discussion we have seen that according to the Weyl-
Wigner correspondence it is possible to describe Quantum Mechanics using
functions on an auxiliary space $\mathcal{M}$ and then operators on a suitable
Hilbert space can be reconstructed from functions on $\mathcal{M}$. This
procedure provides an opposite point of view with respect to Koopman’s one: we
can analyse quantum systems also using tools coming from Classical Mechanics.
However, due to the difference between Quantum and Classical Mechanics, the
properties of the classical-like objects deriving from the quantum ones are
not as one would expect in the description of a classical system. For
instance, the function that one associates to the operator representing a
normal quantum state via the Wigner map does not need to be positive. This
undesired consequence can be avoided introducing the notion of tomogram (for
an introductory description see, for instance [36]).

In a sentence, the classical problem of tomography consists in the
reconstruction of a function defined over a certain space knowing its value on
specific subsets. For instance, in the case of tomography on $\mathbb{R}^{2}$
we have a function $f\,\colon\,\mathbb{R}^{2}\,\rightarrow\,\mathbb{R}$ and
one wants to reconstruct it just knowing its mean value on straight lines in
$\mathbb{R}^{2}$. More specifically, if we have a nonnegative normalized and
smooth function $f\,\colon\,\mathbb{R}^{2}\,\rightarrow\,\mathbb{R}$, i.e.,
$f(q,p)\geq 0$ $\forall(q,p)\in\mathbb{R}^{2}$ and
$\int_{\mathbb{R}^{2}}f(q,p)\,\mathrm{d}q\,\mathrm{d}p=1$, we define its
tomogram (also called its Radon transform) as

$\tilde{f}(\lambda\,;\,\mu,\,\nu)=\int_{\mathbb{R}^{2}}f(q,p)\delta(\lambda-\mu
q-\nu p)\,\mathrm{d}q\,\mathrm{d}p\,,$ (12)

which can be read as the expectation value of the family of observables
$\delta(\lambda-\mu q-\nu p)$ with respect to the probability density $f$.
There exists an inversion formula according to which it is possible to
reconstruct the function $f$ knowing a sufficient number of tomograms, called
a quorum. In $\mathbb{R}^{2}$ the formula is a Fourier-like transform [50]
expressed as follows

$f(q,p)=\frac{1}{(2\pi)^{2}}\int\tilde{f}(\lambda\,;\,\mu,\,\nu)\mathrm{e}^{i(\lambda-\mu
q-\nu p)}\,\mathrm{d}\lambda\,\mathrm{d}\mu\,\mathrm{d}\nu\,.$ (13)

Tomograms $\tilde{f}(\lambda\,;\,\mu,\,\nu)$ have the following properties:

  * •

nonnegative $\tilde{f}(\lambda\,;\,\mu,\,\nu)\geq 0$;

  * •

integrable $\int\tilde{f}(\lambda\,;\,\mu,\,\nu)\,\mathrm{d}\lambda<\infty$ ;

  * •

homogeneous
$\tilde{f}(c\lambda\,;\,\mu,\,\nu)=\frac{1}{c}\tilde{f}(\lambda;\,\mu,\,\nu)$,
$\forall c>0$.

The birth of Quantum Tomography, instead, is associated with a question raised
by Pauli [57], and then it evolved towards the modern problem of Quantum State
Tomography, as we have already mentioned in the introduction. From the point
of view of exact tomography, the Weyl-Wigner formalism allows to obtain a
reconstruction formula by introducing the operators

$\delta(\lambda\mathbf{1}-\mu\hat{Q}-\nu\hat{P})\,,\quad\mathrm{e}^{i(\mu\hat{Q}+\nu\hat{P})}\,.$
(14)

Then, given a state $\rho$, the corresponding tomograms are defined as

$\tilde{\rho}(\lambda\,;\,\mu\,,\,\nu)=\mathrm{Tr}(\rho\,\delta(\lambda\mathbf{1}-\mu\hat{Q}-\nu\hat{P}))$
(15)

and one can get the initial state $\rho$ via the following integral transform
[36]

$\rho=\int\tilde{\rho}(\lambda\,;\,\mu,\,\nu)\mathrm{e}^{i(\lambda\mathbf{1}-\mu\hat{Q}-\nu\hat{P})}\,\mathrm{d}\lambda\,\mathrm{d}\mu\,\mathrm{d}\nu\,.$
(16)

Let us notice here that the tomograms in the quantum settings can be read also
as the Radon transform of the Wigner function associated with a normal quantum
state. However, while in the quantum case Wigner functions need not be
nonnegative, the associated tomograms are still nonnegative functions and the
inversion formula Eq.(13) still holds.

From a purely mathematical point of view Eqs.(15,16) have to be understood
only at a formal level and suitable domains for the operators and the
distributions should be considered to get a rigorous formulation. To avoid
these analyitical difficulties which are not essential for the development of
the main idea of the paper, in the remaining sections we will focus on spin
tomography where the quantum system possesses only a finite number of levels.

##  4 Spin tomography and the groupoid picture of Quantum Mechanics

###  4.1 Finite dimensional quantum tomography: the q-bit

Spin tomography is concerned with finite dimensional quantum system. In this
case, the Hilbert space which we consider is $\mathcal{H}=\mathbb{C}^{n}$,
with the standard inner product, and the algebra of observables is given by
the self-adjoint elements of the algebra
$\mathcal{B}(\mathcal{H})=M_{n}(\mathbb{C})$, where $M_{n}(\mathbb{C})$
denotes the algebra of $n\times n$ matrices with complex entries. Given an
observable $A\in\mathcal{B}(\mathcal{H})$, i.e., a Hermitean matrix, there is
an orthonormal basis
$\left\\{\left.|e_{j}\right\rangle\right\\}_{j=1,\cdots\,,n}$ of the Hilbert
space $\mathcal{H}$ such that
$A=\sum_{j=1}^{n}\lambda_{j}\left.|e_{j}\right\rangle\left\langle
e_{j}|\right.$. Given this basis, it is possible to construct a matrix
$E=(E_{jk})$ of rank-one operators
$E_{jk}=\left.|e_{j}\right\rangle\left\langle e_{k}|\right.$, and for every
normalized vector $\left.|\psi\right\rangle\in\mathcal{H}$, there is a
corresponding pure state
$\rho_{\psi}=\left.|\psi\right\rangle\left\langle\psi|\right.$. Using these
two ingredients it is possible to define a new matrix with entries
$\Psi_{jk}=\left\langle\psi|e_{j}\right\rangle\left\langle
e_{k}|\psi\right\rangle$, whose diagonal elements form a probability vector
called the tomogram of $\rho_{\psi}$ with respect to $A$. The problem of exact
spin tomography [22, 52, 18] amounts to reconstruct the state $\rho_{\psi}$
using a sufficient number of tomograms associated with a suitable family of
orthonormal bases, or of observables $A$. As shown in [53], in general such a
procedure is successful if we use a family of projectors which form a total
set in the Hilbert space $\mathcal{I}_{2}\subset\mathcal{B}(\mathcal{H})$ of
Hilbert-Schmidt operators whose scalar product is $\left\langle
A|B\right\rangle=\mathrm{Tr}(A^{\dagger}B)$. In particular, when $\mathcal{H}$
is finite dimensional
$\mathcal{I}_{2}\cong\mathcal{B}(\mathcal{H})=M_{n}(\mathbb{C})$ a family of
projectors $\left\\{P_{\mu}\right\\}_{\mu\in\mathcal{M}}$ forms a quorum if it
determines a decomposition of the identity superoperator on $\mathcal{I}_{2}$.

Then, the reconstruction formula $\rho_{\psi}=\sum_{\mu\in
M}\mathrm{Tr}(P_{\mu}\rho_{\psi})P_{\mu}$ holds and the quantities
$\mathrm{Tr}(P_{\mu}\rho_{\psi})$ are the components of the tomograms for this
tomographic set of observables. It is obvious that the above reconstruction
procedure still holds if one replaces pure states with generic mixed states.

As an example we can deal with the q-bit case, which corresponds to a two-
dimensional quantum system. In such a case the Hilbert space is
$\mathcal{H}=\mathbb{C}^{2}$ and we can consider a family of observables given
by the Pauli matrices

$\sigma_{1}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right)\,,\quad\sigma_{2}=\left(\begin{array}[]{cc}0&-i\\\
i&0\end{array}\right)\,,\quad\sigma_{3}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)$ (17)

and the corresponding decomposing projectors contains the following
tomographic set

$\begin{split}P_{1}&=\left(\begin{array}[]{cc}1&0\\\
0&0\end{array}\right)\,,\quad P_{2}=\left(\begin{array}[]{cc}0&0\\\
0&1\end{array}\right)\,\\\ P_{3}&=\frac{1}{2}\left(\begin{array}[]{cc}1&1\\\
1&1\end{array}\right)\,,\quad
P_{4}=\frac{1}{2}\left(\begin{array}[]{cc}1&-i\\\
i&1\end{array}\right)\,.\end{split}$ (18)

Indeed, this set forms a non orthonormal basis of the space
$\mathcal{B}(\mathcal{H})=M_{2}(\mathbb{C})$ from which it is possible to get
the othonormal basis given by the four operators

$\begin{split}W_{1}=P_{1}&=\left(\begin{array}[]{cc}1&0\\\
0&0\end{array}\right)\,,\quad W_{2}=P_{2}=\left(\begin{array}[]{cc}0&0\\\
0&1\end{array}\right)\,\\\ W_{3}&=\left(\begin{array}[]{cc}0&1\\\
0&0\end{array}\right)\,,\quad W_{4}=\left(\begin{array}[]{cc}0&0\\\
1&0\end{array}\right)\,.\end{split}$ (19)

Then, if we call $(p_{1},1-p_{1})$, $(p_{2},1-p_{2})$, $(p_{3},1-p_{3})$ the
tomograms corresponding to the decompositions of
$\sigma_{1}\,,\,\sigma_{2}\,,\,\sigma_{3}$, a state $\rho$ can be expressed as
follows

$\rho=\frac{1}{2}\mathbf{1}+\left(p_{1}-\frac{1}{2}\right)\sigma_{1}+\left(p_{2}-\frac{1}{2}\right)\sigma_{2}+\left(p_{3}-\frac{1}{2}\right)\sigma_{3}\,.$
(20)

In the particular instance of a pure state, $\rho^{2}=\rho$, the following
identity holds

$\left(p_{1}-\frac{1}{2}\right)^{2}+\left(p_{2}-\frac{1}{2}\right)^{2}+\left(p_{3}-\frac{1}{2}\right)^{2}=\frac{1}{4}$
(21)

whereas, mixed states lie in the interior of the (Bloch) sphere. On the other
hand, uncorrelated probability vectors form a cube in the three-dimensional
space $(p_{1}\,,\,p_{2}\,,\,p_{3})$, so that quantum states are contained in a
sphere within the cube of the uncorrelated triplets
$(p_{1}\,,\,p_{2}\,,\,p_{3})$.

From the perspective of Quantum State Tomography, on the other hand, one
imposes additional requirements on the operators that are used to reconstruct
any state. In particular, using unbiased measurements allows to maximize the
information one gains about the ensamble performing n+1 measurements. In the
case of q-bit tomography, a family of mutually unbiased measurements is made
up of the measurements associated with the three Pauli matrices:

$\left\\{\left(\begin{array}[]{c}1\\\
0\end{array}\right)\,,\,\left(\begin{array}[]{c}0\\\
1\end{array}\right)\right\\}\,,\;\left\\{\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}1\\\
1\end{array}\right)\,,\,\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}1\\\
-1\end{array}\right)\right\\}\,,\;\left\\{\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}1\\\
i\end{array}\right)\,,\,\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}1\\\
-i\end{array}\right)\right\\}\,.$

On the other hand, it is possible to get reconstruction formulae also using
SIC-POVMs, which are made up of $n^{2}$ positive operators multiple of rank-
one projectors $P_{j}=\mid\psi_{j}\rangle\langle\psi_{j}\mid$, where $n$ is
the dimension of the finite dimensional Hilbert space under investigation.
These operators form an informationally complete POVM when they are
independent and form a basis of the space of all operators on the
corresponding Hilbert space. The additional property of being symmetric means
that $\mid\langle\psi_{j}\mid\psi_{k}\rangle\mid^{2}=1/(n+1)$. Among the IC-
POVM, the symmetric ones satisfy an additional condition: They form minimal
spherical 2-designs (in the sense that have the minimum number of elements a
2-design can have) and minimize the second frame potential (see [61] for
definition and proof of this result). For the q-bit case a SIC-POVM is
obtained via the following operators [59] depending on three-dimensional
vectors $\overrightarrow{a}_{j}$, $j=1,\cdots,4$:

$P_{j}=\frac{1}{4}\left(\mathbf{1}+\overrightarrow{a}_{j}\cdot\overrightarrow{\sigma}\right)\,,\quad\overrightarrow{a}_{j}\cdot\overrightarrow{a}_{k}=\frac{4}{3}\delta_{jk}-\frac{1}{3}\,,\quad
j,k=1,\cdots,4\,,$ (22)

where we denoted $\overrightarrow{\sigma}=(\sigma_{1},\sigma_{2},\sigma_{3})$
the vector whose components are the three Pauli matrices.

###  4.2 Groupoid approach to Quantum Mechanics

In the rest of this section we are going to revisit exact quantum tomography
adopting a different formalism based on the use of groupoids and derived from
Schwinger’s picture of Quantum Mechanics [10]. Let us shortly recall the basic
ingredients of this description:

  * •

A physical system is described in terms of a pair
$\left(\mathcal{G}\,,\,[\mu]\right)$, where
$\mathcal{G}\rightrightarrows\Omega$ is a groupoid and $[\mu]$ is an
equivalence class of measures on the groupoid $\mathcal{G}$ (two measures are
equivalent if they have the same sets of zero measure).

  * •

The space of measurable functions is endowed with a convolution product
$\star_{\mu}$ and an involution operation $\dagger$, making it a
$\ast$-algebra. Choosing a suitable completion [32] it is possible to make
this algebra into a von-Neumann algebra $\mathcal{V}(\mathcal{G})$, called
von-Neumann groupoid-algebra, whose real elements form the Jordan-Lie Banach
algebra $\mathcal{O}$ of observables of the theory.

  * •

The states of the groupoid-algebra $\mathcal{V}(\mathcal{G})$, i.e., positive,
linear and normalized functionals define the states of the physical system.

  * •

Given two groupoids $\mathcal{G}_{1}\rightrightarrows\Omega_{1}$ and
$\mathcal{G}_{2}\rightrightarrows\Omega_{2}$ there are two possible
compositions in this setting: on one side there is the direct product of
groupoids
$\mathcal{G}_{1}\times\mathcal{G}_{2}\rightrightarrows\Omega_{1}\times\Omega_{2}$,
whereas on the other side we have the free product of groupoids
$\mathcal{G}_{1}\ast\mathcal{G}_{2}\rightrightarrows\Omega_{12}$, which
depends, however, on a pair of additional inclusion maps
$i_{j}\,\colon\,\Omega_{j}\,\rightarrow\,\Omega_{12}$, with $j=1,2$. These
composition rules describe different physical notions of “composition” of
systems, the first one being von Neumann’s composition, while the second one
reflects Lieb-Yngvason description of composition of thermodynamical systems
[48]. For more details we refer to [11].

  * •

Concerning the evolution map, Schwinger’s picture of Quantum Mechanics is
characterized by an Action Principle which determines the propagator of the
theory. It has been shown that such propagator can be determined from a
q-Lagrangian, which is an observable in $\mathcal{V}(\mathcal{G})$ associated
to a special family of states called DSF (Dirac-Schwinger-Feynman states)[12].
On the other hand, one can define an evolution map also as a one-parameter
group of automorphims of the groupoid-algebra $\mathcal{V}(\mathcal{G})$.

For the sake of self-consistence, let us add some more details about finite
connected groupoids which we will focus on for the rest of the paper.

A finite groupoid $\mathcal{G}\rightrightarrows\Omega$ consists of a set
$\Omega$ of objects, often representing observed outcomes of actual
measurements on the system, and a set of morphisms $\alpha:x\rightarrow y$
with $x,y\in\Omega$, abstracting the idea of transitions of the system.
Moreover, there is a pair of maps

$s\,\colon\,\mathcal{G}\,\rightarrow\,\Omega\,,\quad
t\,\colon\,\mathcal{G}\,\rightarrow\,\Omega\,,$ (23)

associating each morphism $\alpha\,\colon\,x\rightarrow\,y$ with its source,
$s(\alpha)=x$, and its target $t(\alpha)=y$. Transitions can be composed
according to a partial composition law $\beta\circ\alpha$, defined whenever
$t(\alpha)=s(\beta)$, which satisfies the following axioms:

  * •

Associativity: $(\gamma\circ\beta)\circ\alpha=\gamma\circ(\beta\circ\alpha)$;

  * •

Units: For every outcome $x\in\Omega$ there is a transition, $1_{x}$, called
the unit at $x$, which leaves invariant any other composable transition, i.e.,
$\alpha\circ 1_{x}=1_{y}\circ\alpha=\alpha\,$;

  * •

Inverses: For every transition $\alpha\,\colon\,x\rightarrow\,y$, there exists
an inverse transition $\alpha^{-1}\,\colon\,y\,\rightarrow\,x$, such that
$\alpha\circ\alpha^{-1}=1_{y}$ and $\alpha^{-1}\circ\alpha=1_{x}$ (this
property implements Feynman’s principle of “microscopic reversibility”[25]).

The set of transitions $\gamma\,\colon\,x\,\rightarrow\,x$ is the isotropy
group of the object $x$ and it is denoted by $Aut(x)$. Since the groupoid is
connected, i.e., for any $x,y\in\Omega$, there is
$\alpha\,\colon\,x\,\rightarrow\,y$, the isotropy subgroups are all isomorphic
to each other, thus we can identify all of them with an abstract group, say
$\Gamma$. The isotropy subgroupoid,
$\mathcal{G}_{0}=\cup_{x\in\Omega}Aut(x)\subset\mathcal{G}$, is a normal
subgroupoid such that the following exact sequence of morphisms of groupoids
holds

$\mathbf{1}_{\Omega}\rightarrow\mathcal{G}_{0}\stackrel{{\scriptstyle
i}}{{\to}}\mathcal{G}\stackrel{{\scriptstyle\Pi}}{{\rightarrow}}\mathcal{G}(\Omega)\rightarrow\mathbf{1}\,,$
(24)

where $\mathcal{G}(\Omega)=(t\times s)(\mathcal{G)}$ is a subgroupoid of the
pair groupoid $\Omega\times\Omega$ of the set $\Omega$; since $\mathcal{G}$ is
connected, $\mathcal{G}(\Omega)=\Omega\times\Omega$. The groupoid
$\mathbf{1}_{\Omega}$ is made up only of the units and $\mathbf{1}$ is the
trivial groupoid with only one object and one unit. The homomorphisms $i$ is
the canonical embedding of $\mathcal{G}_{0}$ as a subgroupoid of $\mathcal{G}$
and the canonical projection $\Pi$ is the groupoid homomorphism $t\times s$.
This sequence splits and there exists an isomorphism of groupoids such that
$\mathcal{G}\cong\Gamma\times\mathcal{G}(\Omega)$ [37].

A finite dimensional quantum system (e.g., a quantum system with a finite
number, say $n$, of energy levels) is described by the pair groupoid
$\mathcal{G}(\Omega)\rightrightarrows\Omega$ over the set of integers
$\Omega=\left\\{1,\,2,\,\cdots,\,n\right\\}$. A transition is given by the
pair $(k,j)\,\colon\,j\,\rightarrow\,k$ with $j,k=1,2,\cdots,n$. Two
transitions $(m,l)$ and $(k,j)$ can be composed if $k=l$ and the resulting
transition is $(m,k)\circ\,(k,j)=(m,j)$, which is the groupoidal expression of
the Ritz-Rydberg combination principle of frequencies in spectral lines
(something that was already noted by Connes in [17] chapter 1.1). Given this
finite set we can construct an algebra following the procedure in Sec.3.
Indeed, we have the set $\mathcal{V}(\mathcal{G})$ of all formal linear
combinations of elements of the set $\mathcal{G}(\Omega)$, say
$A_{f}=\sum_{j,k=1}^{n}f(k,j)(k,j)$, where $f$ is a complex-valued function on
$\mathcal{G}(\Omega)$. This set forms a vector space of dimension $n^{2}$
isomorphic to $M_{n}(\mathbb{C})$, which is endowed with the following
convolution product

$\begin{split}A_{f}\star
A_{g}&=\left(\sum_{m,l=1}^{n}f(m,l)(m,l)\right)\cdot\left(\sum_{j,k=1}^{n}g(k,j)(k,j)\right)=\\\
&=\sum_{j,k=1}^{n}\sum_{m,l=1}^{n}f(m,l)g(k,j)(m,l)\circ(k,j)=\\\
=\sum_{j,k=1}^{n}\sum_{m,l=1}^{n}&f(m,l)g(k,j)\delta_{lk}(m,j)=\sum_{m,j=1}^{n}\left(\sum_{k=1}^{n}f(m,k)g(k,j)\right)(m,j)\,.\end{split}$

Realizing that $A_{f}$ can also be thought of as a square matrix, a direct
check shows that $A_{f}\star A_{g}$ is nothing but the standard row-column
product among matrices. On the other hand, note that $A_{f}\star
A_{g}=A_{f\ast g}$, where $(f\ast g)(m,j)=\sum_{k=1}^{n}f(m,k)g(k,j)$. An
involution operation is given by the following rule
$(A_{f})^{\dagger}=\sum_{j,k=1}^{n}\overline{f(k,j)}(j,k)$, and the vector
space $\mathcal{V}(\mathcal{G})$ endowed with the convolution product $\star$
and the involution $\dagger$ form an associative algebra which is isomorphic
to $M_{n}(\mathbb{C})$. Any element of $\mathcal{V}(\mathcal{G})$ acts as an
operator on the Hilbert space $\mathcal{H}=\mathcal{L}^{2}(\mathcal{G})$ via
the left regular representation, i.e.,
$\pi\,\colon\,\mathcal{V}(\mathcal{G})\,\rightarrow\,\mathcal{B}(\mathcal{H})$
where $\pi(A_{f})\psi=f\ast\psi$. The weak-closure (which for a finite
groupoid is equivalent to the norm-closure) of $\pi(\mathcal{V}(\mathcal{G}))$
is the groupoid-algebra of the system and it is isomorphic to the Von-Neumann
algebra $M_{n}(\mathbb{C})$, which is a finite type-I factor. A state of the
system is a normalized positive linear functional, say $\omega$, on
$\mathcal{V}(\mathcal{G})$ and, in this discrete case, it can be identified
with the function $\varphi(k,j)=\omega(\delta_{(k,j)})$, where
$\delta_{(k,j)}$ is the function on $\mathcal{G}(\Omega)$ which is zero except
at the transition $(k,j)$. For such a factor there exists a tracial state [3]
$\tau$ whose action on an element of the algebra $\mathcal{V}(\mathcal{G})$
reads as

$\tau(A_{f})=\frac{1}{n}\sum_{k}f(k,k)\,.$ (25)

Using such tracial state any other state can be written as

$\omega(A_{f})=\tau(A_{\varphi}^{\dagger}\star A_{f})=\tau(A_{\varphi}\star
A_{f})=\frac{1}{n}\sum_{(j,k)\in\mathcal{G}}\overline{\varphi(j,k)}f(j,k)\,,$
(26)

where $\varphi$ is a positive semi-definite function on the groupoid, i.e.,
for any $N\in\mathbb{N}$, $\xi_{r}\in\mathbb{C}$ and
$\alpha_{r}\in\mathcal{G}$, $r,s=1,\cdots\,,N$, with
$t(\alpha_{s})=t(\alpha_{r})$, one has that:

$\sum_{r,s=1}^{N}\overline{\xi}_{r}\xi_{s}\varphi(\alpha_{r}^{-1}\circ\alpha_{s})\geq
0$ (27)

Under this identification of states and positive semi-definite functions on
$\mathcal{G}$, the evaluation of the state $\omega$ on the element of the
groupoid-algebra $A_{f}$ reads as

$\omega(A_{f})=\tau(A_{\varphi}^{\dagger}\star
A_{f})=\frac{1}{n}\left\langle\varphi,f\right\rangle_{\mathcal{L}^{2}(\mathcal{G})}\,,$
(28)

that is, it is the normalized scalar product on $\mathcal{L}^{2}(\mathcal{G})$
of $\varphi$ and $f$.

Before coping with the problem of tomographic reconstruction, a last
ingredient is needed: A bisection is a subset $b\subset\mathcal{G}$ such that
the restriction of the source and target maps to it are bijections (see [15]
for more details in the finite case and [49] for the smooth situation).
Therefore, to any bisection $b$ there are associated two sections

$b_{s}\,:\,\Omega\,\rightarrow\,\mathcal{G}\,,\quad
b_{t}\,\colon\,\Omega\,\rightarrow\,\mathcal{G}\,,$ (29)

with respect to the source and target maps, respectively, i.e., $s\circ
b_{s}=\mathrm{id}_{\Omega}$, $t\circ b_{t}=\mathrm{id}_{\Omega}$. The set of
all bisections, we will call it $\mathcal{S}_{n}$, form a group under the
(canonical) multiplication law:

$(b^{\prime}\cdot b)_{s}(x)=b^{\prime}_{s}(t(b_{s}(x)))\circ b_{s}(x)\,,$ (30)

where $x\in\Omega$, and an analogous formula can be written for the section
$(b^{\prime}\cdot b)_{t}$. In the case of the pair-groupoid
$\mathcal{G}(\Omega)$ over a finite set the group of bisections is isomorphic
to the group $\mathbb{S}_{n}$ of all permutations of the elements in $\Omega$.
This group will play a fundamental role in the rest of the section.

###  4.3 Spin Tomography and frames on $\mathcal{G}(\Omega)$

Concerning the tomographic reconstruction procedure, it can be divided into
two steps: the first one consists in associating a family of tomograms to any
state $\omega$ and the second one consists in applying a suitable
reconstruction formula using the tomograms. The notion of frames helps in both
steps. In what follows we will shortly recall the main features of the theory
of frames, mainly referring to [35] (for more details see [21, 40]).

Given a complex separable Hilbert space $\mathcal{H}$ and a measurable space
$\mathcal{M}$ endowed with a measure $\mu$, a family of vectors
$\mathcal{F}=\left\\{\left.\mid\psi_{x}\right\rangle\,,\,\mid
x\in\mathcal{M}\right\\}$ is called a frame based on $\mathcal{M}$ if it
satisfies:

  * •

For every $\left.|\psi\right\rangle$ the function
$\mathrm{ev}_{\psi}(x)=\left\langle\psi\mid\psi_{x}\right\rangle$ is
$\mu$-measurable and belongs to $\mathcal{L}^{2}(\mathcal{M},\mu)$;

  * •

There are real numbers $0<A\leq B$ such that

$A\parallel\psi\parallel^{2}\leq\parallel\mathrm{ev}_{\psi}\parallel^{2}_{\mathcal{L}^{2}(\mathcal{M})}=\int_{\mathcal{M}}\mid\left\langle\psi,\psi_{x}\right\rangle\mid^{2}\mathrm{d}\mu(x)\,\leq
B\parallel\psi\parallel^{2}\,,$ (31)

for all $\left.\mid\psi\right\rangle\in\mathcal{H}$. In particular, when the
constants $A$ and $B$ can be chosen to be equal, the frame is called tight.

Given a frame $\mathcal{F}$ one defines a bounded linear operator, called the
frame operator

$\mathfrak{F}\,\colon\,\mathcal{H}\,\rightarrow\,\mathcal{L}^{2}(\mathcal{M})\,,\quad\mathfrak{F}|\psi\rangle=\langle\psi_{(\cdot)}\mid\psi\rangle\,.$
(32)

The frame operator is injective and it admits a bounded left inverse. The
adjoint frame operator is then defined as follows

$\begin{split}\mathfrak{F}^{*}\,\colon\,\mathcal{L}^{2}(\mathcal{M})\,\rightarrow\,\mathcal{H}\,,\quad&\langle\mathfrak{F}^{*}f|\psi\rangle_{\mathcal{H}}=\langle
f,\mathfrak{F}|\psi\rangle\rangle_{\mathcal{L}^{2}(\mathcal{M})}\\\
\mathfrak{F}^{*}f=&\int_{\mathcal{M}}f(x)|\psi_{x}\rangle\mathrm{d}\mu(x)\,.\end{split}$
(33)

Using the frame operator and its adjoint one can define the metric operator
$S=\mathfrak{F}^{*}\mathfrak{F}\,\colon\,\mathcal{H}\,\rightarrow\,\mathcal{H}$
which is bounded and definite positive, so that it admits an inverse $S^{-1}$
which is still positive definite and bounded. Using the metric operator one
can introduce another frame, called the dual frame, as follows

$\mathcal{F}^{*}=\left\\{|\psi^{x}\rangle=S^{-1}|\psi_{x}\rangle\mid
x\in\mathcal{M}\right\\}\,,$ (34)

and the following property holds, which will play a major role in the rest of
this section

$\mathbb{I}=\int_{\mathcal{M}}|\psi^{x}\rangle\langle\psi_{x}|\,\mathrm{d}\mu(x)=\int_{\mathcal{M}}|\psi_{x}\rangle\langle\psi^{x}|\,\mathrm{d}\mu(x)\,.$
(35)

Therefore, a reconstruction formula for any vector
$|\psi\rangle\in\mathcal{H}$ can be written as follows

$|\psi\rangle=\int_{\mathcal{M}}\left(\mathfrak{F}|\psi\rangle\right)(x)\,|\psi^{x}\rangle\,\mathrm{d}\mu(x)\,.$
(36)

Then, summarizing the above digression, a tomographic reconstruction formula
can be obtained once a frame is introduced in the Hilbert space
$\mathcal{L}^{2}(\mathcal{G})$, the tomograms being derived from the function
$\mathfrak{F}|\psi\rangle$. In the rest of this section we will show that
actually the group of the bisections will provide us with a frame. Moreover,
when the finite set $\Omega$ is a finite field, a subgroup of the whole group
of bisection, namely the group of affine liner transformation on $\Omega$,
will determine a frame, too. Let us mention, however, that due to the
peculiarity of the permutation group of two elements the construction which we
will present cannot be applied to the q-bit case (i.e., to the case in which
$\Omega$ is a 2-element set), where the procedures outlined at the beginning
of this section are optimal, as we have already discussed.

Given a bisection $b\subset\mathcal{G}(\Omega)$, the corresponding
characteristic function $\chi_{b}$ defines a square-integrable function with
respect to the counting measure on the groupoid and the associated element in
$\mathcal{V}(\mathcal{G})$ is

$A_{b}=\sum_{(k,j)\in\mathcal{G}(\Omega)}\chi_{b}(k,j)\,(k,j)=\sum_{j\in\Omega}(t(b_{s}(j)),j)=\sum_{j\in\Omega}(\sigma_{b}(j),j)\,,$
(37)

where $\sigma_{b}$ is a bijective map on $\Omega$, i.e., a permutation. Then,
the set of all $A_{b}$, where $b$ is a bisection, defines a representation of
the permutation group $\mathbb{S}_{n}$ on the Hilbert space
$\mathcal{L}^{2}(\Omega)\cong\mathbb{C}^{n}$ which is isomorphic to the
fundamental one, and so it is unitary. A feature of this representation which
we will use next, is the fact that it is not irreducible, but for $n>2$ it is
the direct sum of the trivial representation and the $(n-1)$-dimensional
irreducible representation [5]. Moreover, let us notice that the scalar
product on $\mathcal{L}^{2}(\mathcal{G})$ can be expressed as

$\langle\psi_{1}|\psi_{2}\rangle_{\mathcal{L}^{2}(\mathcal{G})}=\sum_{j\in\Omega}\sum_{k\in\Omega}\overline{\psi_{1}(j,k)}\psi_{2}(j,k)=\mathrm{Tr}_{n}(\Psi_{1}^{\dagger}\Psi_{2})\,,$
(38)

where $\mathrm{Tr}_{n}$ is the standard trace on $n\times n$ matrices, and
$(\Psi_{a})_{j,k}=\psi_{a}(j,k)$, $a=1,2$, is a $n\times n$ matrix with
complex entries (we have used the isomorphism between
$\mathcal{V}(\mathcal{G})$ and $M_{n}(\mathbb{C})$). In order to write down a
tomographic reconstruction formula, we need the following result.

###### Theorem 1

Let $\mathcal{G}(\Omega)$ be the groupoid of pairs over $\Omega$, a finite set
of cardinality $n>2$. The set

$\mathcal{F}=\left\\{\chi_{b}\mid b\in\mathcal{S}_{n}\right\\}$ (39)

is a frame for the Hilbert space $\mathcal{H}=\mathcal{L}^{2}(\mathcal{G})$
over the group of bisections $\mathcal{S}_{n}$ equipped with the counting
normalized measure $\sharp$.

###### Proof 1

Since the group $\mathcal{S}_{n}$ is finite and the Hilbert space
$\mathcal{H}$ is finite dimensional, for every $\mid\psi\rangle\in\mathcal{H}$
the map $\mathrm{ev}_{\psi}$ is $\mu$-measurable and belongs to
$\mathcal{L}^{2}(\mathcal{S},\sharp)$.

Then we have to prove that there are two real numbers $0<A\leq B$ such that

$A\parallel\psi\parallel^{2}\leq\sum_{b\in\mathcal{S}_{n}}\,\frac{1}{n!}\;|\>\langle\psi|\chi_{b}\rangle_{\mathcal{L}^{2}(\mathcal{G})}\>|^{2}\,\leq
B\parallel\psi\parallel^{2}\,.$ (40)

The second inequality involving the constant $B$ is a consequence of the
Cauchy-Schwarz inequality. Using the previous scalar product it is easy to see
that $B=n^{2}$. In order to prove the first inequality let us notice that

$\begin{split}\sum_{b\in\mathcal{S}_{n}}\,\frac{1}{n!}\;|\>\langle\psi|\chi_{b}&\rangle_{\mathcal{L}^{2}(\mathcal{G})}|^{2}=\sum_{b\in\mathcal{S}_{n}}\,\frac{1}{n!}\;\langle\psi|\chi_{b}\rangle_{\mathcal{L}^{2}(\mathcal{G})}\langle\chi_{b}|\psi\rangle_{\mathcal{L}^{2}(\mathcal{G})}=\\\
=&\sum_{b\in\mathcal{S}_{n}}\frac{1}{n!}\,\mathrm{Tr}_{n}(\Psi^{\dagger}U(b))\mathrm{Tr}_{n}(U(b)^{\dagger}\Psi)\,,\end{split}$
(41)

where $U(b)$ denotes the matrix associated with the bisection $b$ in the
fundamental representation of the group $\mathbb{S}_{n}$. Let us now introduce
the projector $P$ onto the one-dimensional vector subspace of $\mathbb{C}^{n}$
supporting the trivial representation of $\mathbb{S}_{n}$. Then we have

$\begin{split}&\frac{1}{n!}\sum_{b\in\mathcal{S}_{n}}\,\mathrm{Tr}_{n}(\Psi^{\dagger}U(b))\mathrm{Tr}_{n}(U(b)^{\dagger}\Psi)=\\\
&=\frac{1}{n!}\sum_{b\in\mathcal{S}_{n}}\,\left[\mathrm{Tr}_{n}(\Psi^{\dagger}PU(b)P)+\mathrm{Tr}_{n}(\Psi^{\dagger}(\mathbf{I}-P)U(b)(\mathbf{I}-P))\right]\cdot\\\
&\cdot\left[\mathrm{Tr}_{n}(PU(b)^{\dagger}P\Psi)+\mathrm{Tr}_{n}((\mathbf{I}-P)U(b)^{\dagger}(\mathbf{I}-P)\Psi)\right]=\\\
&=\frac{1}{n!}\sum_{b\in\mathcal{S}_{n}}\,\mathrm{Tr}_{n}(\Psi^{\dagger}PU(b)P)\mathrm{Tr}_{n}(PU(b)^{\dagger}P\Psi)+\\\
&\frac{1}{n!}\sum_{b\in\mathcal{S}_{n}}\,\mathrm{Tr}_{n}(\Psi^{\dagger}(\mathbf{I}-P)U(b)(\mathbf{I}-P))\mathrm{Tr}_{n}((\mathbf{I}-P)U(b)^{\dagger}(\mathbf{I}-P)\Psi)=\\\
&=\mathrm{Tr}_{n}(P\Psi^{\dagger}P\Psi)+\frac{1}{n-1}\mathrm{Tr}_{n}((\mathbf{I}-P)\Psi^{\dagger}(\mathbf{I}-P)\Psi)>\frac{1}{n-1}\parallel\psi\parallel^{2}_{\mathcal{L}^{2}(\mathcal{G})}\,,\end{split}$
(42)

where in the second equation we have used the orthogonality properties of the
irreducible unitaries representations of the permutation group for $n>2$ [37].
Since $A=\frac{1}{n-1}>0$ the family $\mathcal{F}$ is a frame, but it is not
tight.

In order to find the dual frame we need the inverse of the metric operator
$S$. It can be easily computed using the same chain of equalities in the proof
of Th.1:

$\begin{split}\langle\psi_{1}|S|\psi_{2}&\rangle_{\mathcal{L}^{2}(\mathcal{G})}=\frac{1}{n!}\sum_{b\in\mathcal{S}_{n}}\,\mathrm{Tr}_{n}(\Psi_{1}^{\dagger}U(b))\mathrm{Tr}_{n}(U(b)^{\dagger}\Psi_{2})=\\\
\frac{1}{n-1}&\left[\mathrm{Tr}_{n}(\Psi_{1}^{\dagger}\Psi_{2})+(d-2)\mathrm{Tr}_{n}(P\Psi_{1}^{\dagger}P\Psi_{2})\right]\,,\end{split}$
(43)

so that

$S=\frac{1}{d-1}\left[\mathbb{I}+(d-2)\Pi\right]$ (44)

where
$\Pi\,\colon\,\mathcal{L}^{2}(\mathcal{G})\,\rightarrow\,\mathcal{L}^{2}(\mathcal{G})$
is the projection operator

$(\Pi\psi)(k,j)=\sum_{l\in\Omega}\sum_{m\in\Omega}\,P(k,l)\psi(l,m)P(m,j)\,.$
(45)

Since $\Pi$ is a projection operator, the inverse of the metric operator
exists and it is equal to

$S^{-1}=(d-1)\left[\mathbb{I}-\frac{(d-2)}{(d-1)}\Pi\right]\,.$ (46)

Then, the dual frame is written as

$\mathcal{F}^{*}=\left\\{|\chi^{b}\rangle=S^{-1}|\chi_{b}\rangle\;|\;b\in\mathcal{S}_{n}\right\\}\,,$
(47)

and we have that

$\mathbb{I}=\frac{1}{n!}\sum_{b\in\mathcal{S}_{n}}|\chi_{b}\rangle\langle\chi^{b}|\,.$
(48)

Therefore, we can write a state as follows

$\begin{split}\omega(A_{f})=\frac{1}{n}\langle\varphi|f\rangle_{\mathcal{L}^{2}(\mathcal{G})}&=\frac{1}{n!}\sum_{b\in\mathcal{S}_{n}}\frac{1}{n}\langle\varphi|\chi_{b}\rangle_{\mathcal{L}^{2}(\mathcal{G})}\langle\chi^{b}|f\rangle_{\mathcal{L}^{2}(\mathcal{G})}=\\\
=&\frac{1}{n!}\sum_{b\in\mathcal{S}_{n}}F_{\varphi}(b)\langle\chi^{b}|f\rangle_{\mathcal{L}^{2}(\mathcal{G})}\,,\end{split}$
(49)

and if one looks more carefully at the term $F_{\varphi}(b)$ one can notice
that it can be rewritten as follows

$F_{\varphi}(b)=\frac{1}{n}\langle\varphi|\chi_{b}\rangle_{\mathcal{L}^{2}(\mathcal{G})}=\frac{1}{n}\mathrm{Tr}_{n}(\Phi
U(b))=\sum_{m=1}^{n}\frac{e^{i\theta_{m}(b)}}{n}\langle
m|\Phi|m\rangle_{\mathbb{C}^{n}}\,,$ (50)

where $|m\rangle$ is a basis of eigenfunctions of the matrix $U(b)$ with
eigenvalues $\mathrm{e}^{i\theta_{m}(b)}$. The vector with components

$p_{m}=\frac{1}{n}\langle m|\Phi|m\rangle_{\mathbb{C}^{n}}$ (51)

is a probability vector, which is the tomogram associated via the bisection
$b$ to the state $\Phi$. As a final comment, let us notice that the function
$F_{\varphi}(b)$ can be expressed in terms of the scalar product

$F_{\varphi}(b)=\frac{1}{n}\langle\varphi|\chi_{b}\rangle_{\mathcal{L}^{2}(\mathcal{G})}=\frac{1}{n}\sum_{(k,j)\in\mathcal{G}}\overline{\varphi}(j,k)\chi_{b}(j,k)\,,$
(52)

and since $\chi_{b}$ is a characteristic function, the previous formula can be
interpreted as the “integral” over the subset $b$ which provides a clear
analogy with the classical Radon transform.

Such analogy becomes even more evident if one consider the case in which
$\Omega$ is the set $\mathbb{Z}_{n}$, of integers modulo $n$, with $n$ an odd
prime number, such that $\Omega$ is a finite field. In this case, one can
consider a subgroup of the group of all permutation, which is the group
$\mathrm{Aff}_{n}$ of affine linear transformation, an element of which is
denoted by the pair $(\mu,l)=g$, with $\mu,l\in\Omega$, $\mu\neq 0$. This
group acts on the set $\Omega$ itself as $\sigma_{(\mu,l)}(j)=\mu j+l$, and
the action is transitive, since given $k,j\in\Omega$ there exists
$(\mu,l)\in\mathrm{Aff}_{n}$ such that $k=\mu j+l$ (take, for instance,
$\mu=1$ and $l=k-j$). With an abuse of notation the set of bisections
associated with the graph of this action will be denoted by
$\mathrm{Aff}_{n}$. Its elements, analogously, will be denoted by the symbol
$g\subset\mathcal{G}(\Omega)$ and the corresponding characteristic functions
by $\chi_{g}$. The matrices associated with these characteristic functions
will be written as $U(g)$ and form a representation of the group
$\mathrm{Aff}_{n}$ which is unitary and not irreducible. However it is the
direct sum of the trivial representation and the unique $n-1$-dimensional
irreducible representations [34]. This can be proven by looking at the
characters of the irreducible representation of the group $\mathrm{Aff}_{n}$
and noticing that

$\begin{split}\mathrm{Tr}(U(g))=0\quad\quad\mathrm{for}\,\,g=(1,l)\\\
\mathrm{Tr}(U(g))=1\quad\quad\mathrm{for}\,\,g=(\mu,l),\;\mu\neq 1\end{split}$
(53)

where the last equality comes from the fact that the equation
$\sigma_{(\mu,l)}(j)=j$ has a unique solution, $j=-\frac{l}{\mu-1}$ and
consequently there is a unique fixed point under the action defined by
$\sigma$.

Now the same proof as in Th.1 can be applied to show that the sets

$\begin{split}&\mathcal{F}=\left\\{|\chi_{g}\rangle\mid
g\in\mathrm{Aff}_{n}\right\\}\\\
\mathcal{F}^{*}=&\left\\{|\chi^{g}\rangle=S^{-1}|\chi_{g}\rangle\;|\;g\in\mathrm{Aff}_{n}\right\\}\,,\end{split}$
(54)

form a pair of dual frames which can be used to provide the following
tomographic reconstruction formula

$\begin{split}\omega(A_{f})=\frac{1}{n}\langle\varphi|f\rangle_{\mathcal{L}^{2}(\mathcal{G})}&=\frac{1}{n^{2}(n-1)}\sum_{g\in\mathrm{Aff}_{n}}\langle\varphi|\chi_{g}\rangle_{\mathcal{L}^{2}(\mathcal{G})}\langle\chi^{g}|f\rangle_{\mathcal{L}^{2}(\mathcal{G})}=\\\
=&\frac{1}{n(n-1)}\sum_{g\in\mathrm{Aff}_{n}}F_{\varphi}(g)\langle\chi^{g}|f\rangle_{\mathcal{L}^{2}(\mathcal{G})}\,.\end{split}$
(55)

In this case the correspondence with the classical Radon transform for
$\mathbb{R}\times\mathbb{R}$ is more evident, since the subset $g$ is the
analogue of a linear submanifold of the plane in a discrete and finite setting
and, accordingly, the term $F_{\varphi}(g)$ can be interpreted as the
“integral” over this linear submanifold. Moreover, we have that, for
$g=(1,l)$, it holds

$\begin{split}&nF_{\varphi}((\mu=1,l))=\sum_{(k,j)\in\mathcal{G}}\overline{\varphi}((k,j))\delta(k-j-l)=\\\
&=\frac{1}{n}\sum_{(k,j)\in\mathcal{G}}\sum_{m=0}^{n-1}\overline{\varphi}((k,j))\exp\left(i\frac{2\pi}{n}m(k-j-l)\right)=\\\
&=\sum_{m=0}^{n-1}e^{-i\frac{2\pi}{n}ml}\sum_{(k,j)\in\mathcal{G}}\overline{\varphi}((k,j))\frac{e^{-i\frac{2\pi}{n}mj}}{\sqrt{n}}\frac{e^{i\frac{2\pi}{n}mk}}{\sqrt{n}}=\\\
&=\sum_{m=0}^{n-1}e^{-i\frac{2\pi}{n}ml}\sum_{(k,j)\in\mathcal{G}}\overline{\varphi}((k,j))\psi_{m}(k)\overline{\psi}_{m}(j)=\\\
&=\sum_{m=0}^{n-1}e^{-i\frac{2\pi}{n}ml}\langle
m|\Phi|m\rangle_{\mathbb{C}^{n}}\end{split}$

where $\delta$ denotes the function which is 1 whenever its argument is 0.
From the above equation we have that the function $F_{\varphi}(1,l)$ is the
discrete Fourier transform of the tomogram associated with $\varphi$ via the
bisection $g$, and this is in agreement with Bochner’s theorem [28]. On the
other hand, if $g=(\mu,l)$ with $\mu\neq 1$ we can replace the indices $k,j$
in Eq.(4.3) with the new indices $r,s$ both ranging in $\Omega$ such that

$j=-\frac{l}{\mu-1}+r\,,\quad\quad k=-\frac{l}{\mu-1}+s\,.$ (56)

In this way we can write

$\begin{split}nF_{\varphi}((\mu,l))=\sum_{(k,j)\in\mathcal{G}}\overline{\varphi}((\frac{-l}{\mu-1}+&s,\frac{-l}{\mu-1}+r))\chi_{g}(\frac{-l}{\mu-1}+s,\frac{-l}{\mu-1}+r))=\\\
=\varphi((\frac{-l}{\mu-1},\frac{-l}{\mu-1}))+\sum_{(r,s)\neq
0}&\sum_{m=1}^{n-1}\overline{\varphi}((\frac{-l}{\mu-1}+s,\frac{-l}{\mu-1}+r))\delta(s-\mu
r)=\\\ =\varphi((\frac{-l}{\mu-1},\frac{-l}{\mu-1}))+\sum_{(r,s)\neq
0}&\sum_{m=1}^{n-1}\overline{\varphi}((\frac{-l}{\mu-1}+s,\frac{-l}{\mu-1}+r))\cdot\\\
&\cdot\delta(\log(s)-\log(r)-\log(\mu))=\\\
=\varphi((\frac{-l}{\mu-1},\frac{-l}{\mu-1}))+\sum_{(r,s)\neq
0}&\sum_{m=1}^{n-1}\overline{\varphi}((\frac{-l}{\mu-1}+s,\frac{-l}{\mu-1}+r))\cdot\\\
&\cdot\frac{\exp\left(i\frac{2\pi}{n}m(\log(s)-\log(r)-\log(\mu))\right)}{n-1}=\\\
=\varphi((\frac{-l}{\mu-1},\frac{-l}{\mu-1}))+&\sum_{m=1}^{n-1}e^{-i\frac{2\pi}{n-1}m\log(\mu)}\sum_{(r,s)\neq
0}\overline{\varphi}((\frac{-l}{\mu-1}+s,\frac{-l}{\mu-1}+r))\cdot\\\
&\cdot\frac{e^{-i\frac{2\pi}{n-1}m\log(r)}}{\sqrt{n-1}}\frac{e^{i\frac{2\pi}{n-1}m\log(s)}}{\sqrt{n-1}}=\\\
=\varphi((\frac{-l}{\mu-1},\frac{-l}{\mu-1}))&+\sum_{m=1}^{n-1}e^{-i\frac{2\pi}{n-1}m\log(\mu)}\langle
m_{l/\mu}|\Phi|m_{l/\mu}\rangle_{\mathbb{C}^{n}}=\\\
=\sum_{m=0}^{n-1}e^{-i\frac{2\pi}{n-1}m\log(\mu)}&\langle
m_{l/\mu}|\Phi|m_{l/\mu}\rangle_{\mathbb{C}^{n}}\,,\end{split}$

where $\log$ denotes the discrete logarithm which exists whenever $\Omega$ is
a finite field. Therefore, also in this case the function $F_{\varphi}(\mu,l)$
is the discrete Fourier transform of the tomogram associated with $\varphi$
via the bisection $g$. Let us notice that using the group of affine linear
transformations we found $n+1$ different orthonormal basis of the vector space
$\mathbb{C}^{n}$

$\left\\{|m\rangle\right\\}\,,\;\left\\{|m_{l/\mu}\rangle\;\mid\;l=0,1,\cdots,n-1\rangle\right\\}$
(57)

but they are not mutually unbiased.

The concept of mutually unbiased measurements, indeed, is connected to the
concept of complementarity and in this sense the Heisenberg-Weyl group seems
to plays a more important role, rather than the affine one. A family of
mutually unbiased bases can be obtained as follows. Let us consider two
operators $X$, $Y$ satisfying $XY=\omega YX$, where $\omega$ is a $n$-root of
the identity. Then the eigenbases of the operators $X$, $Y$, $XY^{k}$,
$k=1,\cdots,n-1$, are mutually unbiased. If we look at the characteristic
functions supported on the graph of bisections, we can obtain the same
eigenbases if we consider the DSF functions

$V_{r}(k,j)=\exp(i(S_{r}(k)-S_{r}(j)))\,,\quad
S_{r}(j)=\frac{r}{2}j(j-1)\,,\;r=1,\cdots,n-1$ (58)

and the operators associated with the functions on the groupoid

$\tilde{U}(r)(k,j)=(V_{r}\chi_{(1,1)})(k,j)=\exp(i(S_{r}(j)-S_{r}(k)))\delta(k-j-1)\,.$
(59)

The new functions $\tilde{U}(r)$ are unitarily equivalent representations of
the unit translation operator, and their powers provide representations of the
group of integers modulo $n$, under the addition operation. Due to the
connection between DSF functions and the variational principle in Quantum
Mechanics, further work is required to understand whether some features of
mutually unbiased bases could be derived directly from some variational
problem. We will address this issue in forthcoming works.

On the other hand, $POVM$s are a generalization of the concept of quantum
measurements not involving orthogonal projectors. In the groupoid approach to
Quantum Mechanics we have presented in this work, pieces of information about
the state have been obtained by looking at bisections, leading us to families
of orthonormal bases. According to the authors’ point of view, a generalized
procedure which would allow to include $POVM$s in the picture should be based
on a notion of inference procedure which roots back at Cencov’s theory [7] on
statistical decision rules. Enter the details of this formulation is far
beyond the scope of this work and it will be addressed somewhere else.

As a final remark let us notice that given a family of probability
distributions we are not ensured that the function we are going to build using
the reconstruction formula Eq.(49) will define a quantum state. However, an
answer to this question can be obtained by looking back at Eq.(26)-(27), where
we observed that a quantum state is associated with a semi-definite positive
function on the groupoid $\mathcal{G}(\Omega)$. Therefore, a family of
probability distributions
$\mathcal{P}=\left\\{p^{(b)}_{m}\,,\;m=0,1,\cdots,n-1|\>b\in\mathcal{S}\right\\}$
can be interpreted as a family of tomograms of a quantum state iff the
function

$\varphi=\frac{1}{n!}\sum_{b\in\mathcal{S}}\left(\sum_{m=0}^{n-1}p^{(b)}_{m}e^{i\theta_{m}(b)}\right)|\chi^{b}\rangle\,,$
(60)

is a semi-definite positive function on the groupoid $\mathcal{G}(\Omega)$.

##  5 Conclusions

In this paper we have started to address the problem of Quantum State
Tomography from the point of view of Schwingers picture of Quantum Mechanics
based on groupoids and their algebras. In particular, this should be
considered as a preliminary work where some questions related to exact
tomography for finite dimensional quantum systems have been investigated. We
have presented a tomographic reconstruction procedure for states of the
groupoid-algebra of the pair groupoid $\mathcal{G}(\Omega)$ over a finite set
$\Omega$. This procedure is inspired by the classical Radon transform, since
tomograms are associated to the “integral” over discrete analogues of the
graphs of homeomorphisms. In the particular instance of $n$ being an odd prime
number, the set $\Omega$ of integers modulo $n$ is a field and we can restrict
the integration to the discrete analogue of Lagrangian subspaces, which are
images of the field $\Omega$ via affine linear maps. This point of view can be
immediately generalized to the case where $n$ is a prime power, such that one
can consider $\Omega$ to be the corresponding finite Galois field. Then, we
can consider again the group of affine linear transformations. However,
another question arises: in the case where $\Omega$ is not a field, is it
possible to select a subset of bisections which would provide us with a frame
and a reconstruction procedure? This would correspond to a reconstruction
procedure where, instead of considering the integration over the graph of
linear maps, one is taking the graphs of non-linear functions (the continuous
analogue has been analyzed in [2] for instance). On the other hand, Real State
Tomography is a currently very active research field due to its fundamental
role in the development of reliable Quantum Technologies. In conclusion,
according to the autors’ point of view, a deeper investigation of the problem
of real tomography in the groupoidal framework of Schwinger’s picture of
Quantum Mechanics requires a formulation of inference and statistical decision
in said context, and is postponed to future works.

## Acknowledgments

The authors acknowledge financial support from the Spanish Ministry of Economy
and Competitiveness, through the Severo Ochoa Programme for Centres of
Excellence in RD (SEV-2015/0554), the MINECO research project
PID2020-117477GB-I00, and Comunidad de Madrid project QUITEMAD++,
S2018/TCS-A4342. GM would like to thank partial financial support provided by
the Santander/UC3M Excellence Chair Program 2019/2020, and he is also a member
of the Gruppo Nazionale di Fisica Matematica (INDAM), Italy. FDC acknowledges
support from the CONEX-Plus programme funded by Universidad Carlos III de
Madrid and the European Union’s Horizon 2020 research and innovation programme
under the Marie Sklodowska-Curie grant agreement No. 801538. FMC acknowledges
that the work has been supported by the Madrid Government (Comunidad de
Madrid-Spain) under the Multiannual Agreement with UC3M in the line of
“Research Funds for Beatriz Galindo Fellowships” (C$\setminus$&QIG-BG-CM-
UC3M), and in the context of the V PRICIT (Regional Programme of Research and
Technological Innovation).

The authors would like to thank the anonymous referees for many comments which
lead to a substantial improvement of the manuscript.

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