Octonionic two-qubit separability probability conjectures

TL;DR

This paper investigates conjectured formulas for the probability that two-qubit states are separable under Hilbert-Schmidt measure, extending previous results to the octonionic case and testing the validity of these conjectures through numerical analysis.

Contribution

It proposes and tests a conjectured octonionic two-qubit separability probability formula, extending known results for real and complex cases to the octonionic setting.

Findings

Numerical computations support the conjectured octonionic probability value.

The study explores the relation between determinantal-power parameter and induced measure.

Specific probability values are proposed for different parameter settings.

Abstract

We study, further, a conjectured formula for generalized two-qubit Hilbert-Schmidt separability probabilities that has recently been proven by Lovas and Andai (https://arxiv.org/pdf/1610.01410.pdf) for its real (two-rebit) asserted value ($\frac{29}{64}$), and that has also been very strongly supported numerically for its complex ($\frac{8}{33}$), and quaternionic ($\frac{26}{323}$) counterparts. Now, we seek to test the presumptive octonionic value of $\frac{44482}{4091349} \approx 0.0108722$. We are somewhat encouraged by certain numerical computations, indicating that this (51-dimensional) instance of the conjecture might be fulfilled by setting a certain determinantal-power parameter $a$, introduced by Forrester (https://arxiv.org/pdf/1610.08081.pdf), to 0 (or possibly near to 0). Hilbert-Schmidt measure being the case $k=0$ of random induced measure, for $k=1$, the corresponding octonionic separability probability conjecture is $\frac{7612846}{293213345} \approx 0.0259635$, while for $k=2$, it is $\frac{4893392}{95041567} \approx 0.0514869, \ldots$. The relation between the parameters $a$ and $k$ is explored.