Abstract Many works have been devoted to determining the volume of the set of separable states and the separability probability both analytically and numerically. However, computing the exact separability probability remains a highly nontrivial open problem. In these notes, we examine this problem in the simplest context of two-qubit states. A novel aspect of our method is the use of the Duistermaat-Heckman measure, which is the push-forward of Liouville measure on the coadjoint orbit along the moment map. On each coadjoint orbit, the Hilbert-Schmidt measure differs from the Duistermaat-Heckman measure by a constant. The latter measure can be explicitly computed by using tools from symplectic geometry. We confirm the conjecture that the separability probability of two-qubit states with Hilbert-Schmidt measure is 8 33 .Abstract Many works have been devoted to determining the volume of the set of separable states and the separability probability both analytically and numerically. However, computing the exact separability probability remains a highly nontrivial open problem. In these notes, we examine this problem in the simplest context of two-qubit states. A novel aspect of our method is the use of the Duistermaat-Heckman measure, which is the push-forward of Liouville measure on the coadjoint orbit along the moment map. On each coadjoint orbit, the Hilbert-Schmidt measure differs from the Duistermaat-Heckman measure by a constant. The latter measure can be explicitly computed by using tools from symplectic geometry. We confirm the conjecture that the separability probability of two-qubit states with Hilbert-Schmidt measure is 8 33 .

Journal of Physics A Mathematical and TheoreticalOct 8, 20243 citations

An algorithm based on negative probabilities for a separability criterion

M. A. de Ponte, Salomon Sylvain Mizrahi, M. H. Y. Moussa

Quantum Information ProcessingJul 7, 20150 citations

Octonionic two-qubit separability probability conjectures

Paul B. Slater

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.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.

arXiv:1612.02798Dec 8, 20160 citations

Separability Probabilities from Density-Matrix Moments

Paul B. Slater

Sep 12, 20110 citations

Dynamical simulations of probabilities

Michail A. Zak

Chaos Solitons & FractalsMay 1, 199716 citations

\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.