Relative volume of separable bipartite states

Abstract

Every choice of an orthonormal frame in the d-dimensional Hilbert space of a system corresponds to one set of all mutually commuting density matrices or, equivalently, a classical statistical state space of the system; the quantum state space itself can thus be profitably viewed as an SU(d) orbit of classical state spaces, one for each orthonormal frame. We exploit this connection to study the relative volume of separable states of a bipartite quantum system. While the two-qubit case is studied in considerable analytic detail, for higher dimensional systems we fall back on Monte Carlo. Several new insights seem to emerge from our study.

Essentially the published version

References in corpus (5)

• The volume of separable states is super-doubly-exponentially small
• Geometry of the set of mixed quantum states: An apophatic approach
• A Concise Formula for Generalized Two-Qubit Hilbert-Schmidt Separability Probabilities
• Moment-Based Evidence for Simple Rational-Valued Hilbert-Schmidt Generic 2 x 2 Separability Probabilities
• Dissecting the qutrit

Cited by in corpus (8)

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• Numerical and Exact Analyses of Bures and Hilbert-Schmidt Separability and PPT-Probabilities
• Extensions of Generalized Two-Qubit Separability Probability Analyses to Higher Dimensions, Additional Measures and New Methodologies
• Master Lovas-Andai and Equivalent Formulas Verifying the $\frac{8}{33}$ Two-Qubit Hilbert-Schmidt Separability Probability and Companion Rational-Valued Conjectures
• Polytope structures for Greenberger-Horne-Zeilinger diagonal states