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arXiv:2106.14056 · Advances in Theoretical and Mathematical Physics

On the Wigner distribution of the reduced density matrix

Maurice A. de Gosson, Charlyne de Gosson

Published Jan 1, 2022• 2 citations
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Abstract

CConsider a bipartite quantum system consisting of two subsystems A and B. The reduced density matrix ofA a is obtained by taking the partial trace with respect to B. In this work, we will show that the Wigner distribution of this reduced density matrix is obtained by integrating the total Wigner distribution with respect to the phase space variables corresponding to subsystem B. The proof we give is rigorous (as opposed to those found in the literature) and makes use of the Weyl--Wigner--Moyal phase space formalism. Our main result is applied to general Gaussian mixed states, of which it gives a particularly simple and precise description. We also briefly discuss the purification of a mixed state from the Wigner formalism point of view.

DOI
10.4310/atmp.2022.v26.n9.a5

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CConsider a bipartite quantum system consisting of two subsystems A and B. The reduced density matrix ofA a is obtained by taking the partial trace with respect to B. In this work, we will show that the Wigner distribution of this reduced density matrix is obtained by integrating the total Wigner distribution with respect to the phase space variables corresponding to subsystem B. The proof we give is rigorous (as opposed to those found in the literature) and makes use of the Weyl--Wigner--Moyal phase space formalism. Our main result is applied to general Gaussian mixed states, of which it gives a particularly simple and precise description. We also briefly discuss the purification of a mixed state from the Wigner formalism point of view.
Quantum phase‐space distributions offer a royal road into the fascinating quantum–classical interface; the Wigner function being the first and best example. However, the subject is frequent with subtleties and textbook‐level misinformation; e.g. “The Wigner distribution can give wrong answers for some operator expectation values” . Since the Wigner distribution is just another representation of the density matrix, it must yield correct answers. To that end, Marlan Scully has asked at several international conferences (the 2015 Prague conference being one of them) the following question: “Starting with the density matrix (not the Moyal characteristic function), could you give me a simple direct derivation of the Wigner distribution?” Section contains his answer. In Appendix D , we give a related treatment and make contact with other approaches. We hope that as a result of our studies, the Wigner distribution will become more deeply appreciated.
Time-symmetric quantum mechanics can be described in the usual Weyl--Wigner--Moyal formalism (WWM) by using the properties of the Wigner distribution, and its generalization, the cross-Wigner distribution. The use of the latter makes clear a strongly oscillating interference between the pre- and post-selected states. This approach allows us to give explicit formulas for the state reconstruction problem, thus generalizing known results to the case of arbitrary observables. In a forthcoming paper we will extend these results to other quantization schemes.
This thesis describes quantum mechanics in the phase space formulation. We introduce quantization and in particular the Weyl quantization. We study a general class of phase space distribution functions on phase space. The Wigner distribution function is one such distribution function. The Wigner distribution function in general attains negative values and thus can not be interpreted as a real probability density, as opposed to for example the Husimi distribution function. The Husimi distribution however does not yield the correct marginal distribution functions known from quantum mechanics. Properties of the Wigner and Husimi distribution function are studied to more extent. We calculate the Wigner and Husimi distribution function for the energy eigenstates of a particle trapped in a box. We then look at the semi classical limit for this example. The time evolution of Wigner functions are studied by making use of the Moyal bracket. The Moyal bracket and the Poisson bracket are compared in the classical limit. The phase space formulation of quantum mechanics has as advantage that classical concepts can be studied and compared to quantum mechanics. For certain quantum mechanical systems the time evolution of Wigner distribution functions becomes equivalent to the classical time evolution stated in the exact Egerov theorem. Another advantage of using Wigner functions is when one is interested in systems involving mixed states. A disadvantage of the phase space formulation is that for most problems it quickly loses its simplicity and becomes hard to calculate.
Abstract Coherent states, in Wigner representation of quantum mechanics which is classical in structure with Wigner function playing the role of classical phase space probability distribution, and standard quantum mechanical expression for average values of corresponding Weyl quantum operators, general expression for matrix elements of the operator corresponding to phase of the oscillator by Weyl procedure, are obtained in the |n〉 basis. The general expression for matrix elements in the same basis of any other operator is also obtained. As the only mathematical technique necessary in the proposed procedure is a simple change of variables to polar coordinates in the corresponding integrals, this way to introduce Weyl phase operator, and some other operators in Weyl quantization, greatly simplifies necessary derivations and calculations.
In order that a bivariate normal probability distribution in phase space with variances ${\mathrm{\ensuremath{\sigma}}}_{\mathit{q}}$,${\mathrm{\ensuremath{\sigma}}}_{\mathit{p}}$ and covariance ${\mathrm{\ensuremath{\sigma}}}_{\mathit{q},}$p may correspond to a Wigner distribution of a pure or a mixed state, it is necessary and sufficient that Heisenberg's uncertainty relation in Schr\"odinger form ${\mathrm{\ensuremath{\sigma}}}_{\mathit{q}}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{p}}$-${\mathrm{\ensuremath{\sigma}}}_{\mathit{q},}$p2{\mathit{p}}^{2}p2\ensuremath{\ge}${\mathrm{\ensuremath{\Elzxh}}}^{2}$/4 should be satisfied. The diagonalization of the corresponding density matrix entails a correspondence between the statistical and the physical properties of temperature-dependent oscillator states; the expansion of the density matrix into coherent states allows a physical interpretation in phase space.
The Wigner function W(x,p)W(x,p)W(x,p) is related to the Weyl function $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{W}(X,P)$ through a two-dimensional Fourier transform. Several properties for these functions are proved which lead naturally to an extended phase space $x\ensuremath{-}p\ensuremath{-}X\ensuremath{-}P.$ Uncertainty relations for $\ensuremath{\delta}x\ensuremath{\delta}P$ and for $\ensuremath{\delta}p\ensuremath{\delta}X$ are studied. They reduce to the usual uncertainty relation in the special case of pure states, but are different for mixed states. Thermal density matrices obey these uncertainty relations as equalities.