Two-Qubit Separabilities as Piecewise Continuous Functions of Maximal Concurrence

TL;DR

This paper investigates how two-qubit state separability can be described as piecewise functions of maximal concurrence, providing insights into the geometry of quantum states and their separability probabilities.

Contribution

It introduces the concept of eigenvalue-parameterized separability functions (EPSFs) as functions of maximal concurrence, supported by numerical evidence for their piecewise nature.

Findings

EPSFs exhibit pronounced jumps at C=1/2

Numerical results support the hypothesis of EPSFs as functions of maximal concurrence

Discontinuities in EPSFs align with theoretical expectations

Abstract

The generic real (b=1) and complex (b=2) two-qubit states are 9-dimensional and 15-dimensional in nature, respectively. The total volumes of the spaces they occupy with respect to the Hilbert-Schmidt and Bures metrics are obtainable as special cases of formulas of Zyczkowski and Sommers. We claim that if one could determine certain metric-independent 3-dimensional "eigenvalue-parameterized separability functions" (EPSFs), then these formulas could be readily modified so as to yield the Hilbert-Schmidt and Bures volumes occupied by only the separable two-qubit states (and hence associated separability probabilities). Motivated by analogous earlier analyses of "diagonal-entry-parameterized separability functions", we further explore the possibility that such 3-dimensional EPSFs might, in turn, be expressible as univariate functions of some special relevant variable--which we hypothesize to be the maximal concurrence (0 < C <1) over spectral orbits. Extensive numerical results we obtain are rather closely supportive of this hypothesis. Both the real and complex estimated EPSFs exhibit clearly pronounced jumps of magnitude roughly 50% at C=1/2, as well as a number of additional matching discontinuities.