Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2312.06322

On the hierarchy of classicality and symmetry of quantum states

A. Khvedelidze, Astghik Torosyan

Published Dec 11, 2023• 0 citations
arXivHTMLPDF

Abstract

The interrelation between classicality/quantumness and symmetry of states is discussed within the phase-space formulation of finite-dimensional quantum systems. We derive representations for classicality measures $\mathcal{Q}_N[H_{\varrho}]$ of states from the stratum of given symmetry type $[H_{\varrho}]$ for the Hilbert-Schmidt ensemble of qudits. The expressions for measures are given in terms of the permanents of matrices constructed from the vertices of the special Wigner function's positivity polytope. The supposition about the partial order of classicality indicators $\mathcal{Q}_N[H_{\varrho}]$ in accordance with the symmetry type of stratum is formulated.

DOI
10.48550/arxiv.2312.06322

Related papers

Comparing classicality of qutrits from Hilbert-Schmidt, Bures and Bogoliubov-Kubo-Mori ensembles

A. Khvedelidze, Astghik Torosyan

In the report we analyze the indicator/measure of classicality of quantum states defined as the probability to find a state with a positive Wigner function within a unitary invariant random ensemble. The indicators of classicality of three ensembles associated with the Hilbert-Schmidt, Bures and Bogoliubov-Kubo-Mori metrics on the space of quantum states of 3-level system are computed. Their dependence on a moduli parameter of the Wigner function is studied for all strata of a qutrit state space stratified in accordance with the unitary group action.

arXiv:2208.13908•Aug 29, 2022•0 citations

On Measures of Classicality/Quantumness in Quasiprobability Representations of Finite-Dimensional Quantum Systems

N. Abbasli, V. Abgaryan, M. Bures +3 more

Abstract In the present report we discuss measures of classicality/quantumness of states of finite-dimensional quantum systems, which are based on a deviation of quasiprobability distributions from true statistical distributions. Particularly, the dependence of the global indicator of classicality on the assigned geometry of a quantum state space is analysed for a whole family of Wigner quasiprobability representations. General considerations are exemplified by constructing the global indicator of classicality/quantumness for the Hilbert–Schmidt, Bures and Bogoliubov–Kubo–Mori ensembles of qubits and qutrits.

arXiv:2001.03737•Physics of Particles and Nuclei•Jul 1, 2020•2 citations

On the Nonclassicality Distance Indicator of Qudits

A. Khvedelidze, Astghik Torosyan

Abstract We consider the nonclassicality distance indicator of a state in finite-dimensional quantum systems which is evaluating a state nonclassicality by its remoteness from the set of “classical states”. The latter are identified with those states whose Wigner function is non-negative. The corresponding Wigner function’s positivity polytope in the simplex of qudit eigenvalues is introduced and the representation for the nonclassicality distance indicator as a piecewise function is derived. The results are exemplified by the qutrit case.

arXiv:2310.12173•Physics of Particles and Nuclei•Jun 1, 2024•0 citations

Метрический индикатор неклассичности кудитов : Аннотации

A. Khvedelidze, Astghik Torosyan

We consider the nonclassicality distance indicator of a state in finite-dimensional quantum systems which is evaluating a state nonclassicality by its remoteness from the set of “classical states”. The latter are identified with those states whose Wigner function is non-negative. The corresponding Wigner function’s positivity polytope in the simplex of qudit eigenvalues is introduced and the representation for the nonclassicality distance indicator as a piecewise function is derived. The results are exemplified by the qutrit case.

ФИЗИКА ЭЛЕМЕНТАРНЫХ ЧАСТИЦ И АТОМНОГО ЯДРА•Jun 20, 2024•0 citations

The classical limit of the quantum baker's map

Mark Morgan-Tracy

This work concerns the finding of the semi-classical form of the coherent state representation for the class of quantum baker’s maps defined by Schack and Caves. It begins by introducing the finite-dimensional Hilbert space on which the quantum baker’s map is defined. Its pertinent features including the all important symmetry operators are introduced and given a full explanation. We also introduce the finite-dimensional phase space which will give the semi-classical limit a geometrical interpretation. For a D dimensional Hilbert space, the finite-dimensional phase space is found to be a grid with D2 points. Each point corresponds to a particular pair of position and momentum displacement operator eigenphases. We then detail the derivation of the finite-dimension version of the Wigner function, a quasi-distribution for the finite-dimensional phase space. We show that its most “irregular” feature, mainly its property of having more values than was thought necessary, can be explained by its correct behavior under the symmetry operations, a feature lacking in other Wigner candidates. However, even this special choice for the Wigner function proves unusable in the semi-classical limit as it is found to have a non-convergent limit. We then turn to another possible phase space function: the Q-function. It being necessary to find a suitable coherent state for the finite-dimensional Hilbert space, we begin by studying the properties of the periodically continued Gaussian states. These are the typical Weyl coherent states made periodic in both position and momentum such as to make them legitimate finite-dimensional states. Developing certain mathematical techniques allows us to show that they have compatible position and momentum representations, that a subset of them are complete and can be used to define a Q-function, and that this function obeys all of the symmetry properties. Finally, we use these coherent states to find a representation for the propagator of the quantum baker’s map. In the semi-classical limit, i.e. the large dimension limit, this representation is found, for most of the maps, to take a form of the exponentiation of the classical map’s generating function. This form was predicted long ago by Van Vleck as indicator of an operator’s classical limit. Therefore, we assert that these maps limit to the classically chaotic baker’s map. In certain limiting schemes, however, the Schack-Caves maps do not reach this form and must be given a different interpretation.

•Jan 1, 2002•1 citation

Local unitary symmetries and entanglement invariants

Markus Johansson

We investigate the relation between local unitary symmetries and entanglement invariants of multi-qubit systems. The Hilbert space of such systems can be stratified in terms of states with different types of symmetry. We review the connection between this stratification and the ring of entanglement invariants and the corresponding geometric description in terms of algebraic varieties. On a stratum of a non-trivial symmetry group the invariants of the symmetry preserving operations gives a sufficient description of entanglement. Finding these invariants is often a simpler problem than finding the invariants of the local unitary group. The conditions, as given by the Luna–Richardson theorem, for when the ring of such invariants is isomorphic to the ring of local unitary invariants on the stratum are discussed. As an example we consider symmetry groups that can be diagonalized by local unitary operations and for which the group action on each qubit is non-trivial. On the stratum of such a symmetry the entanglement can be described in terms of a canonical form and the invariants of the symmetry preserving operations. This canonical form and the invariants are directly determined by the symmetry group. Further, we briefly discuss how some recently proposed entanglement classification schemes capture symmetry properties.

arXiv:1401.2110•Journal of Physics A Mathematical and Theoretical•Oct 30, 2014•4 citations
The interrelation between classicality/quantumness and symmetry of states is discussed within the phase-space formulation of finite-dimensional quantum systems. We derive representations for classicality measures QN[Hϱ]\mathcal{Q}_N[H_{\varrho}]QN​[Hϱ​] of states from the stratum of given symmetry type [Hϱ][H_{\varrho}][Hϱ​] for the Hilbert-Schmidt ensemble of qudits. The expressions for measures are given in terms of the permanents of matrices constructed from the vertices of the special Wigner function's positivity polytope. The supposition about the partial order of classicality indicators QN[Hϱ]\mathcal{Q}_N[H_{\varrho}]QN​[Hϱ​] in accordance with the symmetry type of stratum is formulated.
In the report we analyze the indicator/measure of classicality of quantum states defined as the probability to find a state with a positive Wigner function within a unitary invariant random ensemble. The indicators of classicality of three ensembles associated with the Hilbert-Schmidt, Bures and Bogoliubov-Kubo-Mori metrics on the space of quantum states of 3-level system are computed. Their dependence on a moduli parameter of the Wigner function is studied for all strata of a qutrit state space stratified in accordance with the unitary group action.
Abstract In the present report we discuss measures of classicality/quantumness of states of finite-dimensional quantum systems, which are based on a deviation of quasiprobability distributions from true statistical distributions. Particularly, the dependence of the global indicator of classicality on the assigned geometry of a quantum state space is analysed for a whole family of Wigner quasiprobability representations. General considerations are exemplified by constructing the global indicator of classicality/quantumness for the Hilbert–Schmidt, Bures and Bogoliubov–Kubo–Mori ensembles of qubits and qutrits.
Abstract We consider the nonclassicality distance indicator of a state in finite-dimensional quantum systems which is evaluating a state nonclassicality by its remoteness from the set of “classical states”. The latter are identified with those states whose Wigner function is non-negative. The corresponding Wigner function’s positivity polytope in the simplex of qudit eigenvalues is introduced and the representation for the nonclassicality distance indicator as a piecewise function is derived. The results are exemplified by the qutrit case.
We consider the nonclassicality distance indicator of a state in finite-dimensional quantum systems which is evaluating a state nonclassicality by its remoteness from the set of “classical states”. The latter are identified with those states whose Wigner function is non-negative. The corresponding Wigner function’s positivity polytope in the simplex of qudit eigenvalues is introduced and the representation for the nonclassicality distance indicator as a piecewise function is derived. The results are exemplified by the qutrit case.
This work concerns the finding of the semi-classical form of the coherent state representation for the class of quantum baker’s maps defined by Schack and Caves. It begins by introducing the finite-dimensional Hilbert space on which the quantum baker’s map is defined. Its pertinent features including the all important symmetry operators are introduced and given a full explanation. We also introduce the finite-dimensional phase space which will give the semi-classical limit a geometrical interpretation. For a D dimensional Hilbert space, the finite-dimensional phase space is found to be a grid with D2 points. Each point corresponds to a particular pair of position and momentum displacement operator eigenphases. We then detail the derivation of the finite-dimension version of the Wigner function, a quasi-distribution for the finite-dimensional phase space. We show that its most “irregular” feature, mainly its property of having more values than was thought necessary, can be explained by its correct behavior under the symmetry operations, a feature lacking in other Wigner candidates. However, even this special choice for the Wigner function proves unusable in the semi-classical limit as it is found to have a non-convergent limit. We then turn to another possible phase space function: the Q-function. It being necessary to find a suitable coherent state for the finite-dimensional Hilbert space, we begin by studying the properties of the periodically continued Gaussian states. These are the typical Weyl coherent states made periodic in both position and momentum such as to make them legitimate finite-dimensional states. Developing certain mathematical techniques allows us to show that they have compatible position and momentum representations, that a subset of them are complete and can be used to define a Q-function, and that this function obeys all of the symmetry properties. Finally, we use these coherent states to find a representation for the propagator of the quantum baker’s map. In the semi-classical limit, i.e. the large dimension limit, this representation is found, for most of the maps, to take a form of the exponentiation of the classical map’s generating function. This form was predicted long ago by Van Vleck as indicator of an operator’s classical limit. Therefore, we assert that these maps limit to the classically chaotic baker’s map. In certain limiting schemes, however, the Schack-Caves maps do not reach this form and must be given a different interpretation.
We investigate the relation between local unitary symmetries and entanglement invariants of multi-qubit systems. The Hilbert space of such systems can be stratified in terms of states with different types of symmetry. We review the connection between this stratification and the ring of entanglement invariants and the corresponding geometric description in terms of algebraic varieties. On a stratum of a non-trivial symmetry group the invariants of the symmetry preserving operations gives a sufficient description of entanglement. Finding these invariants is often a simpler problem than finding the invariants of the local unitary group. The conditions, as given by the Luna–Richardson theorem, for when the ring of such invariants is isomorphic to the ring of local unitary invariants on the stratum are discussed. As an example we consider symmetry groups that can be diagonalized by local unitary operations and for which the group action on each qubit is non-trivial. On the stratum of such a symmetry the entanglement can be described in terms of a canonical form and the invariants of the symmetry preserving operations. This canonical form and the invariants are directly determined by the symmetry group. Further, we briefly discuss how some recently proposed entanglement classification schemes capture symmetry properties.