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Communications in Theoretical Physics

Wigner Function: from Ensemble Average of Density Operator to Its One Matrix Element in Entangled Pure States

Fan Hong‐Yi

Published Nov 15, 2002• 2 citations
Publisher

Abstract

We show that the Wigner function (an ensemble average of the density operator ρ, Δ is the Wigner operator) can be expressed as a matrix element of ρ in the entangled pure states. In doing so, converting from quantum master equations to time-evolution equation of the Wigner functions seems direct and concise. The entangled states are defined in the enlarged Fock space with a fictitious freedom.

DOI
10.1088/0253-6102/38/5/533

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We show that the Wigner function (an ensemble average of the density operator ρ, Δ is the Wigner operator) can be expressed as a matrix element of ρ in the entangled pure states. In doing so, converting from quantum master equations to time-evolution equation of the Wigner functions seems direct and concise. The entangled states are defined in the enlarged Fock space with a fictitious freedom.
The Lindblad master equation governs the general Markovian evolution of a density operator for an open quantum system. Semiclassical Wigner functions represent density operators in phase space in terms of chords on a classical manifold, so that the amplitude and phase of each chord contribution is classically defined. Inserting such a Wigner function into a phase space version of the master equation, its explicit evolution is derived in the absence of dissipation. There results a simple extension of the unitary evolution of the semiclassical Wigner function, which does not affect the phase of each chord contribution, while dampening its amplitude exponentially. Projecting the Wigner function on to an orthogonal position or momentum basis, the dampening of long chords emerges as the exponential decay of off-diagonal elements of the density matrix.
As is well known, the evolution of quantum state can be replaced by its Wigner function’s time evolution. The Wigner function of a quantum state is the same as the density matrix of a quantum state, because they both contain many messages, such as the probability distribution and phases. Thus, the important information about the quantum state in the evolution process can be obtained more quickly and effectively by studying the Wigner function of a quantum state. In this paper, based on the classical diffusion equation, the diffusion equation of the quantum state density operator is derived by using the P representation of the density operator. Furthermore, by introducing the Weyl ordering symbol of the quantum operator, the corresponding Weyl quantization scheme is given. In addition, the evolution equation of Wigner operator in diffusion channel is established by using another phase space representation of density operator—Wigner function, and the solution form of Wigner operator is given. In this paper, we derive the evolution law of Wigner operator in quantum diffusion channel for the first time, that is, the form of Wigner operator at any time in the evolution process. Based on this conclusion, the evolution of coherent states through quantum diffusion channels is discussed.