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Wigner functions and separability for finite systems. (English) Zbl 1073.81058

Summary: A discussion of discrete Wigner functions in phase space related to mutually unbiased bases is presented. This approach requires mathematical assumptions, which limits it to systems with density matrices defined on complex Hilbert spaces of dimension \(p^n\) where \(p\) is a prime number. With this limitation, it is possible to define a phase space and Wigner functions in close analogy to the continuous case. That is, we use a phase space that is a direct sum of \(n\) two-dimensional vector spaces each containing \(p^2\) points. This is in contrast to the more usual choice of a two-dimensional phase space containing \(p^{2n}\) points. A useful aspect of this approach is that we can relate complete separability of density matrices and their Wigner functions in a natural way. We discuss this in detail for bipartite systems and present the generalization to arbitrary numbers of subsystems when \(p\) is odd. Special attention is required for two qubits \((p=2)\) and our technique fails to establish the separability property for more than two qubits. Finally, we give a brief discussion of Hamiltonian dynamics in the language developed in the paper.

MSC:

81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
81R15 Operator algebra methods applied to problems in quantum theory
82B10 Quantum equilibrium statistical mechanics (general)
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