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

TL;DR

This paper presents a simple, exact formula for the probability of separability in various two-qubit quantum systems, derived through advanced computational and mathematical techniques, improving understanding of quantum entanglement probabilities.

Contribution

The authors derive a concise, exact formula for generalized two-qubit Hilbert-Schmidt separability probabilities using hypergeometric functions and Zeilberger's algorithm, advancing previous computational methods.

Findings

Exact rational-valued separability probabilities for different quantum systems.

A hypergeometric function-based formula for P(alpha) was obtained.

The formula applies to systems with dimensions 9, 15, and 27.

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.