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arXiv:1711.00563 · Revista Matemática Iberoamericana

Quantum mappings acting by coordinate transformations on Wigner distributions

Nuno Costa Dias, João Nuno Prata

Published Feb 5, 2019• 3 citations
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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.

DOI
10.4171/rmi/1056

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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.
We discuss two sets of conditions that are necessary and sufficient for a function defined on phase space to be a Wigner distribution function (WDF). The first set is well known and involves the function itself; the second set is less familiar and involves the function's ``symplectic'' Fourier transform. After explaining why these two sets are equivalent, we explore some properties and applications of the second one. Among other things, we show that that set includes the position-momentum uncertainty relations as a special case, and in doing so we give a new derivation of them. This derivation itself serves as the starting point for the discussion of a quantum-mechanical moment problem. It also enables us to exhibit a real-valued phase-space function that obeys the uncertainty relations but that is not a WDF.
For the continuous Wigner function and for certain discrete Wigner functions, permuting the values of the Wigner function in accordance with a symplectic linear transformation is equivalent to performing a certain unitary transformation on the state. That is, performing this unitary transformation is simply a matter of moving Wigner-function values around in phase space. This result holds in particular for the simplest discrete Wigner function defined on a d×dd \times dd×d phase space when the Hilbert-space dimension ddd is odd. It does not hold for a d×dd \times dd×d phase space if the dimension is even. Here we show, though, that a generalized version of this correspondence does apply in the case of a two-qubit phase space. In this case, a symplectic linear permutation of the points of the phase space, together with a certain reinterpretation of the Wigner function, is equivalent to a unitary transformation.
For the continuous Wigner function and for certain discrete Wigner functions, permuting the values of the Wigner function in accordance with a symplectic linear transformation is equivalent to performing a certain unitary transformation on the state. That is, performing this unitary transformation is simply a matter of moving Wigner-function values around in phase space. This result holds in particular for the simplest discrete Wigner function defined on a [Formula: see text] phase space when the Hilbert-space dimension [Formula: see text] is odd. It does not hold for a [Formula: see text] phase space if the dimension is even. Here we show, though, that a generalized version of this correspondence does apply in the case of a two-qubit phase space. In this case, a symplectic linear permutation of the points of the phase space, together with a certain reinterpretation of the Wigner function, is equivalent to a unitary transformation.
We propose a Wigner transformation of the density matrix and operators of a system with bounded coordinates and discrete momentum spectra (typically, a particle in a box) and prove that all requirements for such a transformation are satisfied. The phase-space mapping adopted here is employed to compute the quasiprobability distributions associated with the pure states of a quantum rotator and the image of the angular momentum operator. The mapping of the dynamics of these objects, both in the reversible and irreversible frame, is constructed as well, and the conditions for the existence of a classical limit in the form of a Fokker-Planck equation of motion are investigated.