Two-Qubit Separabilities as Piecewise Continuous Functions of Maximal Concurrence. II--The Relevance of Dyson Indices

TL;DR

This paper investigates the relationship between two-qubit separability functions and Dyson indices, providing numerical evidence that eigenvalue-parameterized functions follow Dyson patterns in the upper concurrence range, with implications for quantum entanglement analysis.

Contribution

It demonstrates that eigenvalue-parameterized separability functions adhere to Dyson index patterns in the upper concurrence range, extending previous findings to a new parameterization.

Findings

Eigenvalue-parameterized separability functions follow Dyson patterns for 1/2 <= C <= 1.

Real ESF is proportional to (2-2 C)^(3/2).

Complex ESF is proportional to the square of the real ESF, i.e., (2-2 C)^3.

Abstract

We importantly amend a certain parenthetical remark made in Part I (arXiv:0806.3294), to the effect that although two-qubit diagonal-entry-parameterized separability functions had been shown (arXiv:0704.3723) to clearly conform to a pattern dictated by the "Dyson indices" (beta = 1 [real], 2 [complex], 4 [quaternionic]) of random matrix theory, this did not appear to be the case with regard to eigenvalue-parameterized separability functions (ESFs). But upon further examination of the extensive numerical analyses reported in Part I, we find quite convincing evidence that adherence to the Dyson-index pattern does also hold for ESFs, at least as regards the upper half-range 1/2 <= C <= 1 of the maximal concurrence over spectral orbits, C. To be specific, it strongly appears that in this upper half-range, the real two-qubit ESF is simply proportional to (2-2 C)^(3/2), and its complex counterpart--in conformity to the Dyson-index pattern--proportional to the square of the real ESF, that is, (2-2 C)^3. The previously documented piecewise continuous ("semilinear") behavior in the lower half-range still appears, however, to lack any particular Dyson-index-related interpretation.