Formulas for Generalized Two-Qubit Separability Probabilities

TL;DR

This paper derives formulas for generalized two-qubit separability probabilities using hypergeometric functions, revealing new mathematical structures and potential universal formulas for quantum state separability.

Contribution

It introduces explicit formulas for separability probabilities involving hypergeometric functions and difference equations, extending previous results to generalized measures and parameters.

Findings

Formulas for $Q(k,lpha)$ involving hypergeometric functions are derived.

Number-theoretic formulas for hypergeometric parameters are established.

A compact hypergeometric form for $Q(k,lpha)$ is constructed.

Abstract

To begin, we find certain formulas $Q(k,\alpha)= G_1^k(\alpha) G_2^k(\alpha)$, for $k = -1, 0, 1,...,9$. These yield that part of the total separability probability, $P(k,\alpha)$, for generalized (real, complex, quaternionic,\ldots) two-qubit states endowed with random induced measure, for which the determinantal inequality $|\rho^{PT}| >|\rho|$ holds. Here $\rho$ denotes a $4 \times 4$ density matrix, obtained by tracing over the pure states in $4 \times (4 +k)$-dimensions, and $\rho^{PT}$, its partial transpose. Further, $\alpha$ is a Dyson-index-like parameter with $\alpha = 1$ for the standard (15-dimensional) convex set of (complex) two-qubit states. For $k=0$, we obtain the previously reported Hilbert-Schmidt formulas, with (the real case) $Q(0,\frac{1}{2}) = \frac{29}{128}$, (the standard complex case) $Q(0,1)=\frac{4}{33}$, and (the quaternionic case) $Q(0,2)= \frac{13}{323}$---the three simply equalling $ P(0,\alpha)/2$. The factors $G_2^k(\alpha)$ are sums of polynomial-weighted generalized hypergeometric functions $_{p}F_{p-1}$, $p \geq 7$, all with argument $z=\frac{27}{64} =(\frac{3}{4})^3$. We find number-theoretic-based formulas for the upper ($u_{ik}$) and lower ($b_{ik}$) parameter sets of these functions and, then, equivalently express $G_2^k(\alpha)$ in terms of first-order difference equations. Applications of Zeilberger's algorithm yield "concise" forms, parallel to the one obtained previously for $P(0,\alpha) =2 Q(0,\alpha)$. For nonnegative half-integer and integer values of $\alpha$, $Q(k,\alpha)$ has descending roots starting at $k=-\alpha-1$. Then, we (C. Dunkl and I) construct a remarkably compact (hypergeometric) form for $Q(k,\alpha)$ itself. The possibility of an analogous "master" formula for $P(k,\alpha)$ is, then, investigated, and a number of interesting results found.