Two-Qubit Hilbert-Schmidt Separability Functions and Probabilities for Full-Dimensional Even-Dyson-Index Scenarios

TL;DR

This paper estimates Hilbert-Schmidt separability functions for quaternionic two-qubit systems, confirming a Dyson-index-based relationship with complex and real cases, and computes their separability probabilities.

Contribution

It provides numerical estimates of quaternionic two-qubit separability functions and confirms their proportionality to powers of complex and real cases based on Dyson indices.

Findings

Quaternionic separability function proportional to the square of the complex function.

Quaternionic separability probability estimated at approximately 0.0774.

Supports Dyson-index-based relationships among separability functions.

Abstract

We extend the findings and analyses of our two recent studies (Phys. Rev. A, 75, 032326 [2007] and arXiv:0704.3723) by, first, obtaining numerical estimates of the separability function based on the (Euclidean, flat) Hilbert-Schmidt (HS) metric for the 27-dimensional convex set of quaternionic two-qubit systems. The estimated function appears to be strongly consistent with our previously-formulated Dyson-index (beta = 1, 2, 4) ansatz, dictating that the quaternionic (beta=4) separability function should be exactly proportional to the square of the separability function for the 15-dimensional convex set of two-qubit complex (beta=2) systems, as well as the fourth power of the separability function for the 9-dimensional convex set of two-qubit real (beta=1) systems. In particular, we conclude that S_{quat}(mu) =(6/71)^2 ((3-mu^2) mu)^4 =(S_{complex}(mu))^2. Here, mu =(rho_{11} rho_{44}/(rho_{22} rho_{33})^(1/2), where rho is a 4 x 4 two-qubit density matrix. We can, thus, supplement (and fortify) our previous assertion that the HS separability probability of the two-qubit complex states is 8/33= 0.242424, by claiming that its quaternionic counterpart is 72442944/936239725 = 0.0773765. We also comment on and analyze the odd beta=1 and 3 cases.