Quantum mappings acting by coordinate transformations on Wigner distributions

Abstract

We prove two results about Wigner distributions. Firstly, that the Wigner transform is the only sesquilinear map $\mathcal{S}({\mathbb{R}}^n)\times \mathcal{S}({\mathbb{R}}^n)\to \mathcal{S}({\mathbb{R}}^{2n})$ which is bounded and covariant under phase-space translations and linear symplectomorphisms. Consequently, the Wigner distributions form the only set of quasidistributions which is invariant under linear symplectic transformations. Secondly, we prove that the maximal group of (linear or non-linear) coordinate transformations that preserves the set of (pure or mixed) Wigner distributions consists of the translations and the linear symplectic and antisymplectic transformations.

26 pages, to appear in Revista Matemática Iberoamericana

References in corpus (3)

• Fundamentals of Quantum Information Theory
• Admissible states in quantum phase space
• The Narcowich-Wigner spectrum of a pure state

Cited by in corpus (1)

• Symplectic Transformations on Wigner Distributions and Time Frequency Signal Design