A Concise Formula for Generalized Two-Qubit Hilbert-Schmidt Separability Probabilities

Abstract

We report major advances in the research program initiated in "Moment-Based Evidence for Simple Rational-Valued Hilbert-Schmidt Generic 2 x 2 Separability Probabilities" (J. Phys. A, 45, 095305 [2012]). A highly succinct separability probability function P(alpha) is put forth, yielding for generic (9-dimensional) two-rebit systems, P(1/2) = 29/64, (15-dimensional) two-qubit systems, P(1) = 8/33 and (27-dimensional) two-quater(nionic)bit systems, P(2)=26/323. This particular form of P(alpha) was obtained by Qing-Hu Hou and colleagues by applying Zeilberger's algorithm ("creative telescoping") to a fully equivalent--but considerably more complicated--expression containing six 7F6 hypergeometric functions (all with argument 27/64 = (3/4)^3). That hypergeometric form itself had been obtained using systematic, high-accuracy probability-distribution-reconstruction computations. These employed 7,501 determinantal moments of partially transposed four-by-four density matrices, parameterized by alpha = 1/2, 1, 3/2,...,32. From these computations, exact rational-valued separability probabilities were discernible. The (integral/half-integral) sequences of 32 rational values, then, served as input to the Mathematica FindSequenceFunction command, from which the initially obtained hypergeometric form of P(alpha) emerged.

20 pages, 6 figures, final version to appear in J. Physics A. arXiv admin note: substantial text overlap with arXiv:1203.4498, arXiv:1209.1613

References in corpus (8)

• The volume of separable states is super-doubly-exponentially small
• Entanglement between two subsystems, the Wigner semicircle and extreme value statistics
• Volume of the quantum mechanical state space
• Dyson Indices and Hilbert-Schmidt Separability Functions and Probabilities
• Visualizing Two Qubits
• Extended Studies of Separability Functions and Probabilities and the Relevance of Dyson Indices
• Bures and Hilbert-Schmidt 2 x 2 Determinantal Moments
• A statistical portrait of the entanglement decay of two-qubit memories

Cited by in corpus (19)

• Creative Telescoping for Holonomic Functions
• Volumes of conditioned bipartite state spaces
• Invariance of separability probability over reduced states in 4x4 bipartite systems
• Relative volume of separable bipartite states
• Numerical Computations of Separability Probabilities
• Generalized Two-Qubit Whole and Half Hilbert-Schmidt Separability Probabilities
• Formulas for Rational-Valued Separability Probabilities of Random Induced Generalized Two-Qubit States
• Incompatibility probability of random quantum measurements
• Separability Probability Formulas and Their Proofs for Generalized Two-Qubit X-Matrices Endowed with Hilbert-Schmidt and Induced Measures
• Numerical and Exact Analyses of Bures and Hilbert-Schmidt Separability and PPT-Probabilities
• Invariance of Bipartite Separability and PPT-Probabilities over Casimir Invariants of Reduced States
• Master Lovas-Andai and Equivalent Formulas Verifying the $\frac{8}{33}$ Two-Qubit Hilbert-Schmidt Separability Probability and Companion Rational-Valued Conjectures
• Extensions of Generalized Two-Qubit Separability Probability Analyses to Higher Dimensions, Additional Measures and New Methodologies
• Formulas for Generalized Two-Qubit Separability Probabilities
• Two-Qubit Separability Probabilities as Joint Functions of the Bloch Radii of the Qubit Subsystems
• Quasirandom estimations of two-qubit operator-monotone-based separability probabilities
• Entanglement in bipartite X-states: Analytical results for the volume of states with positive partial transpose
• One application of Duistermaat-Heckman measure in quantum information theory
• Confident entanglement detection via separable numerical range