A priori Probabilities of Separable Quantum States

Abstract

Zyczkowski, Horodecki, Sanpera, and Lewenstein (ZHSL) recently proposed a ``natural measure'' on the N-dimensional quantum systems (quant-ph/9804024), but expressed surprise when it led them to conclude that for N = 2 x 2, disentangled (separable) systems are more probable (0.632) in nature than entangled ones. We contend, however, that ZHSL's (rejected) intuition has, in fact, a sound theoretical basis, and that the a priori probability of disentangled 2 x 2 systems should more properly be viewed as (considerably) less than 0.5. We arrive at this conclusion in two quite distinct ways, the first based on classical and the second, quantum considerations. Both approaches, however, replace (in whole or part) the ZHSL (product) measure by ones based on the volume elements of monotone metrics, which in the classical case amounts to adopting the Jeffreys' prior of Bayesian theory. Only the quantum-theoretic analysis (which yields the smallest probabilities of disentanglement) uses the minimum number of parameters possible, N^2 - 1, as opposed to N^2 + N - 1 (although this "over-parameterization", as recently indicated by Byrd, should be avoidable). However, despite substantial computation, we are not able to obtain precise estimates of these probabilities, and the need for additional (possibly supercomputer) analyses is indicated (particularly so, for higher-dimensional quantum systems, such as the 2 x 3 we also study here).

14 pages, 4 tables, 4 figures, to appear in J. Phys. A

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• Hall Normalization Constants for the Bures Volumes of the n-State Quantum Systems
• Silver mean conjectures for 15-d volumes and 14-d hyperareas of the separable two-qubit systems
• Qubit-Qutrit Separability-Probability Ratios
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• Dyson Indices and Hilbert-Schmidt Separability Functions and Probabilities
• A Concise Formula for Generalized Two-Qubit Hilbert-Schmidt Separability Probabilities
• Comparing metrics for mixed quantum states: Sjoqvist and Bures
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• Moment-Based Evidence for Simple Rational-Valued Hilbert-Schmidt Generic 2 x 2 Separability Probabilities
• On the Bures Volume of Separable Quantum States
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• A priori probability that a qubit-qutrit pair is separable
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• Entanglement Classification via Witness Operators generated by Support Vector Machine
• Extended Studies of Separability Functions and Probabilities and the Relevance of Dyson Indices
• Relative volume of separable bipartite states
• Inclusion relations among separability criteria
• On the Correlations Between Quantum Entanglement and q-Information Measures
• Bures and Hilbert-Schmidt 2 x 2 Determinantal Moments
• Ratios of maximal concurrence-parameterized separability functions, and generalized Peres-Horodecki conditions
• Bipartite Bound Entanglement
• Advances in delimiting the Hilbert-Schmidt separability probability of real two-qubit systems
• Sample Complexity of Black Box Work Extraction
• Certifying nonlocal properties of noisy quantum operations
• A geometric approach to the distribution of quantum states in bipartite physical systems
• Quasirandom estimations of two-qubit operator-monotone-based separability probabilities
• Chiral symmetry: An analytic $SU(3)$ unitary matrix