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AIP conference proceedings

Geometrical and group properties of discrete analogs of the center-of-mass and the cluster tomograms

Ashot S. Avanesov, Vladimir I. Man’ko

Published Jan 1, 2021• 0 citations
Publisher

Abstract

We adopt methods of symplectic tomography and discrete phase space for the description of states of discrete variable quantum systems (qudits). The proposed tomographic functions are constructed as generalized analogs of the center-of-mass and the cluster tomograms and associated with finite linear manifolds in the discrete phase space. Hilbert spaces of considered qudits must have the power of a prime dimension, so the corresponding phase spaces are the vector spaces over finite fields. We find conditions for the nonnegativity of the constructed functions and obtain formulae for the density matrix restoration.

DOI
10.1063/5.0055480

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We adopt methods of symplectic tomography and discrete phase space for the description of states of discrete variable quantum systems (qudits). The proposed tomographic functions are constructed as generalized analogs of the center-of-mass and the cluster tomograms and associated with finite linear manifolds in the discrete phase space. Hilbert spaces of considered qudits must have the power of a prime dimension, so the corresponding phase spaces are the vector spaces over finite fields. We find conditions for the nonnegativity of the constructed functions and obtain formulae for the density matrix restoration.
The existing relation between the tomographic description of quantum states and the convolution algebra of certain discrete groupoids represented on Hilbert spaces will be discussed. The realizations of groupoid algebras based on qudit, photon-number (Fock) states and symplectic tomography quantizers and dequantizers will be constructed. Conditions for identifying the convolution product of groupoid functions and the star--product arising from a quantization--dequantization scheme will be given. A tomographic approach to construct quasi--distributions out of suitable immersions of groupoids into Hilbert spaces will be formulated and, finally, intertwining kernels for such generalized symplectic tomograms will be evaluated explicitly.
Quantum tomography for continuous variables is based on the symplectic transformation group acting in the phase space. A particular case of symplectic tomography is optical tomography re-lated to the action of a special orthogonal group. In the tomographic description of spin states, the connection between special unitary and special orthogonal groups is used. We analyze the represen-tation for spin tomography using the Cayley–Klein parameters and discuss an analog of symplectic tomography for discrete variables. We propose a representation for tomograms of discrete variables through quaternions and employ the qubit-state tomogram to illustrate the method elaborated.
It is now well established that quantum tomography provides an alternative picture of quantum mechanics. It is common to introduce tomographic concepts starting with the Schrödinger–Dirac picture of quantum mechanics on Hilbert spaces. In this picture, states are a primary concept and observables are derived from them. On the other hand, the Heisenberg picture, which has evolved in the C ⋆ -algebraic approach to quantum mechanics, starts with the algebra of observables and introduces states as a derived concept. The equivalence between these two pictures amounts, essentially, to the Gelfand–Naimark–Segal construction. In this construction, the abstract C ⋆ -algebra is realized as an algebra of operators acting on a constructed Hilbert space. The representation that is defined may be reducible or irreducible, but in either case it allows us to identify a unitary group associated with the C ⋆ -algebra by means of its invertible elements. In this picture both states and observables are appropriate functions on the group; it also follows that quantum tomograms are strictly related with appropriate functions (positive-type) on the group. In this paper we present, using very simple examples, a tomographic description emerging from the set of ideas connected with the C ⋆ -algebra picture of quantum mechanics. In particular, we introduce the tomographic probability distributions for finite and compact groups, and formulate an autonomous criterion to recognize a given probability distribution as a tomogram of quantum state.
We present a short review of the general principles of constructing tomograms of quantum states. We derive a general tomographic reconstruction formula for the quantum density operator of a system with a dynamical Lie group. In the reconstruction formula, the multiplicity of irreducible representation in Clebsch–Gordan decomposition is taken into account. Various approaches to spin tomography are discussed. An integral representation for the tomographic probability is found and a contraction of the spin tomogram to the photon-number tomography distribution is considered. The case of SU (3) tomography is discussed with the examples of quark states (related to the simplest triplet representations) and octet states.
Some non-linear generalizations of classical Radon tomography were recently introduced by M. Asorey et al [Phys. Rev. A 77, 042115 (2008), where the straight lines of the standard Radon map are replaced by quadratic curves (ellipses, hyperbolas, circles) or quadratic surfaces (ellipsoids, hyperboloids, spheres). We consider here the quantum version of this novel non-linear approach and obtain, by systematic use of the Weyl map, a tomographic encoding approach to quantum states. Non-linear quantum tomograms admit a simple formulation within the framework of the star-product quantization scheme and the reconstruction formulae of the density operators are explicitly given in a closed form, with an explicit construction of quantizers and dequantizers. The role of symmetry groups behind the generalized tomographic maps is analyzed in some detail. We also introduce new generalizations of the standard singular dequantizers of the symplectic tomographic schemes, where the Dirac delta-distributions of operator-valued arguments are replaced by smooth window functions, giving rise to the new concept of "thick" quantum tomography. Applications for quantum state measurements of photons and matter waves are discussed.
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 focussed 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 continuous case.