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Characterizing the Kirkwood-Dirac positivity on second countable LCA groups

Matéo Spriet

Abstract

We define the Kirkwood-Dirac quasiprobability representation of quantum mechanics associated with the Fourier transform over second countable locally compact abelian groups. We discuss its link with the Kohn-Nirenberg quantization of the phase space $G\times \widehat{G}$. We use it to argue that in this abstract setting the Wigner-Weyl quantization, when it exists, can still be interpreted as a symmetric ordering. Then, we identify all generalized (non-normalizable) pure states having a positive Kirkwood-Dirac distribution. They are, up to the natural action of the Weyl-Heisenberg group, Haar measures on closed subgroups. This generalizes a result known for finite abelian groups. We then show that the classical fragment of quantum mechanics associated with the Kirkwood-Dirac distribution is non-trivial if and only if the group has a compact identity component. Finally, we provide for connected compact abelian groups a complete geometric description of this classical fragment.

Characterizing the Kirkwood-Dirac positivity on second countable LCA groups

Abstract

We define the Kirkwood-Dirac quasiprobability representation of quantum mechanics associated with the Fourier transform over second countable locally compact abelian groups. We discuss its link with the Kohn-Nirenberg quantization of the phase space . We use it to argue that in this abstract setting the Wigner-Weyl quantization, when it exists, can still be interpreted as a symmetric ordering. Then, we identify all generalized (non-normalizable) pure states having a positive Kirkwood-Dirac distribution. They are, up to the natural action of the Weyl-Heisenberg group, Haar measures on closed subgroups. This generalizes a result known for finite abelian groups. We then show that the classical fragment of quantum mechanics associated with the Kirkwood-Dirac distribution is non-trivial if and only if the group has a compact identity component. Finally, we provide for connected compact abelian groups a complete geometric description of this classical fragment.

Paper Structure

This paper contains 14 sections, 15 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. The Kirkwood-Dirac distribution over SCLCA groups
  3. Preliminaries
  4. The Kirkwood-Dirac distribution
  5. Weyl-Heisenberg group, Kirkwood-Dirac and Wigner-Weyl distributions and ordered quantizations of phase space
  6. Characterization of the Kirkwood-Dirac positive generalized pure states on SCLCA groups
  7. Poisson-Tate formula
  8. Proof of Theorem \ref{['KD positive on second countable LCA groups']}
  9. Identifying a classical fragment of quantum mechanics
  10. The Kirkwood-Dirac positive pure states
  11. Topological characterization of groups for which the classical fragment is non trivial
  12. Topological properties of the sets of KD-real Hilbert-Schmidt observables and of KD-positive states
  13. Complete description of the classical fragment for connected compact SCLCA groups
  14. Positivity of the Kirkwood-Dirac and Wigner distributions of quantum states: discussion

Key Result

Theorem 1.1

Let $G$ be a group. The generalized pure state $|\psi\rangle\langle\psi|$ associated with $\psi\in \mathcal{S}'(G)\setminus\{0\}$ has a positive Kirkwood-Dirac distribution if and only if there exists a closed subgroup $H\subset G$ such that, up to the action of the Weyl-Heisenberg group, $\psi$ is

Theorems & Definitions (32)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Definition 2.1
  • Lemma 2.1
  • proof
  • ...and 22 more