Eigenvalues, Separability and Absolute Separability of Two-Qubit States

TL;DR

This paper derives exact separability functions for two real two-qubit systems using eigenvalues and Euler angles, and calculates probabilities of absolute separability employing bounds and trigonometric identities.

Contribution

It provides explicit eigenvalue-parameterized separability functions for two real two-qubit states and computes absolute separability probabilities using bounds and geometric identities.

Findings

Exact separability functions for two real two-qubit states derived

Calculated Hilbert-Schmidt probabilities for absolute separability

Utilized Euler-angle parameterization and geometric identities in analysis

Abstract

Substantial progress has recently been reported in the determination of the Hilbert-Schmidt (HS) separability probabilities for two-qubit and qubit-qutrit (real, complex and quaternionic) systems. An important theoretical concept employed has been that of a separability function. It appears that if one could analogously obtain separability functions parameterized by the eigenvalues of the density matrices in question--rather than the diagonal entries, as originally used--comparable progress could be achieved in obtaining separability probabilities based on the broad, interesting class of monotone metrics (the Bures, being its most prominent [minimal] member). Though large-scale numerical estimations of such eigenvalue-parameterized functions have been undertaken, it seems desirable also to study them in lower-dimensional specialized scenarios in which they can be exactly obtained. In this regard, we employ an Euler-angle parameterization of SO(4) derived by S. Cacciatori (reported in an Appendix)--in the manner of the SU(4)-density matrix parameterization of Tilma, Byrd and Sudarshan. We are, thus, able to find simple exact separability (inverse-sine-like) functions for two real two-qubit (rebit) systems, both having three free eigenvalues and one free Euler angle. We also employ the important Verstraete-Audenaert-de Moor bound to obtain exact HS probabilities that a generic two-qubit state is absolutely separable (that is, can not be entangled by unitary transformations). In this regard, we make copious use of trigonometric identities involving the tetrahedral dihedral angle arccos(1/3).