Exact Bures Probabilities that Two Quantum Bits are Classically Correlated

Abstract

In previous studies, we have explored the ansatz that the volume elements of the Bures metrics over quantum systems might serve as prior distributions, in analogy to the (classical) Bayesian role of the volume elements ("Jeffreys' priors") of Fisher information metrics. Continuing this work, we obtain exact Bures probabilities that the members of certain low-dimensional subsets of the fifteen-dimensional convex set of 4 x 4 density matrices are separable or classically correlated. The main analytical tools employed are symbolic integration and a formula of Dittmann (quant-ph/9908044) for Bures metric tensors. This study complements an earlier one (quant-ph/9810026) in which numerical (randomization) --- but not integration --- methods were used to estimate Bures separability probabilities for unrestricted 4 x 4 or 6 x 6 density matrices. The exact values adduced here for pairs of quantum bits (qubits), typically, well exceed the estimate (.1) there, but this disparity may be attributable to our focus on special low-dimensional subsets. Quite remarkably, for the q = 1 and q = 1/2 states inferred using the principle of maximum nonadditive (Tsallis) entropy, the separability probabilities are both equal to 2^{1/2} - 1. For the Werner qubit-qutrit and qutrit-qutrit states, the probabilities are vanishingly small, while in the qubit-qubit case it is 1/4.

Seventeen pages, LaTeX, eleven postscript figures. In this version, subsequent (!) to publication in European Physical Journal B, we correct the (1,1)-entries of the 4 x 4 matrices given in formulas (6) and (7), that is, the numerators should both read v^2 - x^2 - y^2 - z^2, rather than v^2 - x^2 + y^2 + z^2

References in corpus (3)

• Nonlocal properties of two-qubit gates and mixed states and optimization of quantum computations
• On the geometry of entangled states
• Essentially All Gaussian Two-Party Quantum States are a priori Nonclassical but Classically Correlated

Cited by in corpus (25)

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• Bures Measures over the Spaces of Two and Three-Dimensional Density Matrices
• Silver mean conjectures for 15-d volumes and 14-d hyperareas of the separable two-qubit systems
• Qubit-Qutrit Separability-Probability Ratios
• Dyson Indices and Hilbert-Schmidt Separability Functions and Probabilities
• Volumes of conditioned bipartite state spaces
• A Concise Formula for Generalized Two-Qubit Hilbert-Schmidt Separability Probabilities
• Eigenvalues, Separability and Absolute Separability of Two-Qubit States
• Moment-Based Evidence for Simple Rational-Valued Hilbert-Schmidt Generic 2 x 2 Separability Probabilities
• On the Bures Volume of Separable Quantum States
• Hilbert-Schmidt Separability Probabilities and Noninformativity of Priors
• Two-Qubit Separability Probabilities and Beta Functions
• A priori probability that a qubit-qutrit pair is separable
• Extended Studies of Separability Functions and Probabilities and the Relevance of Dyson Indices
• Inclusion relations among separability criteria
• On the Correlations Between Quantum Entanglement and q-Information Measures
• Bures Geometry of the Three-Level Quantum Systems. I
• Bures and Hilbert-Schmidt 2 x 2 Determinantal Moments
• Numerical and Exact Analyses of Bures and Hilbert-Schmidt Separability and PPT-Probabilities
• Ratios of maximal concurrence-parameterized separability functions, and generalized Peres-Horodecki conditions
• Extensions of Generalized Two-Qubit Separability Probability Analyses to Higher Dimensions, Additional Measures and New Methodologies
• Advances in delimiting the Hilbert-Schmidt separability probability of real two-qubit systems
• Mixed State Holonomies
• A geometric approach to the distribution of quantum states in bipartite physical systems
• Chiral symmetry: An analytic $SU(3)$ unitary matrix