Quantum Tomography and Schwinger’s Picture of Quantum Mechanics

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.

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 $\mathcal{B}(\mathcal{H})$.

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.