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

Journal of Mathematical Physics

On informational completeness of covariant localization observables and Wigner coefficients

D.M. Healy, Franklin E. Schroeck

Published Jan 1, 1995• 32 citations
Publisher

Abstract

An orthogonality theorem is obtained for square integrable representations on homogeneous spaces of a locally compact group. From this, a series of lemmas is derived showing the informational completeness of natural covariant localization operators, as well as of the generalized Wigner distributions (matrix elements of the group). Some of these results give explicit reconstruction formulas for the quantum state from its expectation values against these families of operators. The results are applied to special (phase space) representations of the Heisenberg, affine, and Galilei groups.

DOI
10.1063/1.531318

Related papers

A class of affine Wigner functions with extended covariance properties

J. Bertrand, P. Bertrand

Affine Wigner functions are phase space representations based on the affine group in place of the usual Weyl–Heisenberg group of quantum mechanics. Such representations are relevant to the time–frequency analysis of real signals. An interesting family is singled out by the requirement of covariance with respect to each solvable three-parameter group containing the affine group. Explicit forms are given in each case and properties such as unitarity and localization are discussed. Some particular distributions are recovered.

Journal of Mathematical Physics•Jul 1, 1992•118 citations

A general theorem on square-integrability: Vector coherent states

S. Twareque Ali

We derive a generalization of the well-known theorem for the square integrability of a unitary irreducible representation of a locally compact group. The generalization covers the case of representations admitting vector coherent states. The result is illustrated by an example drawn from the isochronous Galilei group. The construction yields a wide variety of coherent states, labeled by phase space points, which satisfy a resolution of the identity condition, and incorporate spin degrees of freedom.

Journal of Mathematical Physics•Aug 1, 1998•27 citations

On Constructing Informationally Complete Covariant Positive Operator-Valued Measures

Grigori Gennadievich Amosov

We study a projective unitary representation of the product G=G˜×G, where G is a locally compact Abelian group and G^ is its dual consisting of characters on G. It is proven that the representation is irreducible, which allows us to define a covariant positive operator-valued measure (covariant POVM) generated by orbits of projective unitary representations of G. The quantum tomography associated with the representation is discussed. It is shown that the integration over such a covariant POVM defines a family of contractions which are multiples of unitary operators from the representation. Using this fact, it is proven that the measure is informationally complete. The obtained results are illustrated by optical tomography on groups and by a measure with a density that has a value in the set of coherent states.

arXiv:2301.12492•Entropy•May 11, 2023•3 citations

Symplectically covariant Schrödinger equation in phase space

Maurice A. de Gosson

A classical theorem of Stone and von Neumann says that the Schrödinger representation is, up to unitary equivalences, the only irreducible representation of the Heisenberg group on the Hilbert space of square-integrable functions on configuration space. Using the Wigner-Moyal transform we construct an irreducible representation of the Heisenberg group on a certain Hilbert space of square-integrable functions defined on phase space. This allows us to extend the usual Weyl calculus into a phase-space calculus and leads us to a quantum mechanics in phase space, equivalent to standard quantum mechanics. We also briefly discuss the extension of metaplectic operators to phase space and the probabilistic interpretation of the solutions of the phase space Schrödinger equation

arXiv:math-ph/0505073•Journal of Physics A Mathematical and General•Oct 5, 2005•47 citations

Quantum Tomography with Phase Space Measurements

Jussi Schultz

This thesis addresses the use of covariant phase space observables in quantum tomography. Necessary and sufficient conditions for the informational completeness of covariant phase space observables are proved, and some state reconstruction formulae are derived. Different measurement schemes for measuring phase space observables are considered. Special emphasis is given to the quantum optical eight-port homodyne detection scheme and, in particular, on the effect of non-unit detector efficiencies on the measured observable. It is shown that the informational completeness of the observable does not depend on the efficiencies. As a related problem, the possibility of reconstructing the position and momentum distributions from the marginal statistics of a phase space observable is considered. It is shown that informational completeness for the phase space observable is neither necessary nor sufficient for this procedure. Two methods for determining the distributions from the marginal statistics are presented. Finally, two alternative methods for determining the state are considered. Some of their shortcomings when compared to the phase space method are discussed.

•Aug 17, 2012•0 citations

Phase-space representations of general statistical physical theories

Matthias Singer, Werner Stulpe

It is shown that Hilbert-space quantum mechanics and many other statistical theories can be represented on some phase space, in the sense that states can be identified with probability measures and observables can be described by functions. In the general context of statistical dualities, informationally complete observables are introduced and a theorem on their existence is proven. The correspondence between these observables and the injective affine mappings from the states into the probability measures on phase space, i.e., the phase-space representations, is pointed out. In particular, a description of all observables by functions is presented, such that all expectation values can, in arbitrarily good physical approximation, be calculated as integrals. Moreover, some new aspects of the particular case of those phase-space representations of quantum mechanics that are related to certain joint position-momentum observables are discussed.

Journal of Mathematical Physics•Jan 1, 1992•52 citations
An orthogonality theorem is obtained for square integrable representations on homogeneous spaces of a locally compact group. From this, a series of lemmas is derived showing the informational completeness of natural covariant localization operators, as well as of the generalized Wigner distributions (matrix elements of the group). Some of these results give explicit reconstruction formulas for the quantum state from its expectation values against these families of operators. The results are applied to special (phase space) representations of the Heisenberg, affine, and Galilei groups.
Affine Wigner functions are phase space representations based on the affine group in place of the usual Weyl–Heisenberg group of quantum mechanics. Such representations are relevant to the time–frequency analysis of real signals. An interesting family is singled out by the requirement of covariance with respect to each solvable three-parameter group containing the affine group. Explicit forms are given in each case and properties such as unitarity and localization are discussed. Some particular distributions are recovered.
We derive a generalization of the well-known theorem for the square integrability of a unitary irreducible representation of a locally compact group. The generalization covers the case of representations admitting vector coherent states. The result is illustrated by an example drawn from the isochronous Galilei group. The construction yields a wide variety of coherent states, labeled by phase space points, which satisfy a resolution of the identity condition, and incorporate spin degrees of freedom.
We study a projective unitary representation of the product G=G˜×G, where G is a locally compact Abelian group and G^ is its dual consisting of characters on G. It is proven that the representation is irreducible, which allows us to define a covariant positive operator-valued measure (covariant POVM) generated by orbits of projective unitary representations of G. The quantum tomography associated with the representation is discussed. It is shown that the integration over such a covariant POVM defines a family of contractions which are multiples of unitary operators from the representation. Using this fact, it is proven that the measure is informationally complete. The obtained results are illustrated by optical tomography on groups and by a measure with a density that has a value in the set of coherent states.
A classical theorem of Stone and von Neumann says that the Schrödinger representation is, up to unitary equivalences, the only irreducible representation of the Heisenberg group on the Hilbert space of square-integrable functions on configuration space. Using the Wigner-Moyal transform we construct an irreducible representation of the Heisenberg group on a certain Hilbert space of square-integrable functions defined on phase space. This allows us to extend the usual Weyl calculus into a phase-space calculus and leads us to a quantum mechanics in phase space, equivalent to standard quantum mechanics. We also briefly discuss the extension of metaplectic operators to phase space and the probabilistic interpretation of the solutions of the phase space Schrödinger equation
This thesis addresses the use of covariant phase space observables in quantum tomography. Necessary and sufficient conditions for the informational completeness of covariant phase space observables are proved, and some state reconstruction formulae are derived. Different measurement schemes for measuring phase space observables are considered. Special emphasis is given to the quantum optical eight-port homodyne detection scheme and, in particular, on the effect of non-unit detector efficiencies on the measured observable. It is shown that the informational completeness of the observable does not depend on the efficiencies. As a related problem, the possibility of reconstructing the position and momentum distributions from the marginal statistics of a phase space observable is considered. It is shown that informational completeness for the phase space observable is neither necessary nor sufficient for this procedure. Two methods for determining the distributions from the marginal statistics are presented. Finally, two alternative methods for determining the state are considered. Some of their shortcomings when compared to the phase space method are discussed.
It is shown that Hilbert-space quantum mechanics and many other statistical theories can be represented on some phase space, in the sense that states can be identified with probability measures and observables can be described by functions. In the general context of statistical dualities, informationally complete observables are introduced and a theorem on their existence is proven. The correspondence between these observables and the injective affine mappings from the states into the probability measures on phase space, i.e., the phase-space representations, is pointed out. In particular, a description of all observables by functions is presented, such that all expectation values can, in arbitrarily good physical approximation, be calculated as integrals. Moreover, some new aspects of the particular case of those phase-space representations of quantum mechanics that are related to certain joint position-momentum observables are discussed.