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 (6429), and that has also been very strongly supported numerically for its complex (338), and quaternionic (32326) counterparts. Now, we seek to test the presumptive octonionic value of 409134944482≈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 2932133457612846≈0.0259635, while for k=2, it is 950415674893392≈0.0514869,…. The relation between the parameters a and k is explored.
We begin by seeking the qubit-qutrit and rebit-retrit counterparts to the now well-established Hilbert-Schmidt separability probabilities for (the 15-dimensional convex set of) two-qubits of 338=3⋅1123≈0.242424 and (the 9-dimensional) two-rebits of 6429=2629≈0.453125. Based in part on extensive numerical computations, we advance the possibilities of a qubit-qutrit value of 100027=(103)3=23⋅5333=0.027 and a rebit-retrit one of 6561860=3822⋅5⋅43≈0.131078. These four values for 2×m systems (m=2,3) suggest certain numerator/denominator sequences involving powers of m, which we further investigate for m>3. Additionally, we find that the Hilbert-Schmidt separability/PPT-probabilities for the two-rebit, rebit-retrit and two-retrit X-states all equal 3π216≈0.54038, as well as more generally, that the probabilities based on induced measures are equal across these three sets of X-states. Then, we extend the generalized two-qubit framework introduced by Lovas and Andai from Hilbert-Schmidt measures to induced ones. For instance, while the Lovas-Andai two-qubit function is 31ε2(4−ε2), yielding 338, its k=1 induced measure counterpart is 41ε2(3−ε2)2, yielding 14361=11⋅1361≈0.426573, where ε is a singular-value ratio. We investigate, in these regards, the possibility of extending the previously-obtained "Lovas-Andai master formula".
We begin by advancing conjectures, closely fitting numerical estimates, for the Hilbert-Schmidt separability/PPT probabilities of the 35- and 63-dimensional convex sets of (2×3) qubit-qutrit and (2×4) qubit-qudit states, as well as certain related scenarios. We reason that the underlying volume formulas might closely parallel those now well-established--via a Lovas-Andai formula--in their lower (15)-dimensional two-qubit analogue. The qubit-qutrit conjecture we obtain is (103)3=23⋅5333=100027=0.027, with the counterpart two-qubit separability probablity being 338=3⋅1123≈0.242424. In the (2×4) qubit-qudit case, we advance 1237516=32⋅53⋅1142≈0.001292929 as the PPT-probability. The numerators of the probabilities for 2×m scenarios are, thus, possibly simply powers of m. We also search for the rebit-retrit counterpart to the established two-rebit result of 6429=2629≈0.453125. Additionally, we find that the Hilbert-Schmidt separability/PPT-probabilities for the two-rebit, rebit-retrit and two-retrit X-states all equal 3π216≈0.54038, as well as more generally, that the probabilities based on induced measures are equal across these three sets of X-states. Then, we extend the Lovas-Andai two-qubit framework from Hilbert-Schmidt measures to induced ones, raising the possibility of a still more general master formula.
Construction of the Lovas-Andai Two-Qubit Function χ~2(ε)=31ε2(4−ε2) Verifies the 338-Hilbert Schmidt Separability Probability Conjecture
We investigate relationships between two forms of Hilbert-Schmidt two-re[al]bit and two-qubit functions''--those recently advanced by Lovas and Andai (arXiv:1610.01410), and those earlier presented by Slater ({\it J. Phys. A} {\bf{40}} [2007] 14279). In the Lovas-Andai framework, the independent variable ε∈[0,1] is the ratio σ(V) of the singular values of the 2×2 matrix V=D21/2D1−1/2 formed from the two 2×2 diagonal blocks (D1,D2) of a randomly generated 4×4 density matrix D. In the Slater setting, the independent variable μ is the diagonal-entry ratio d22d33d11d44--with, importantly, μ=ε or μ=ε1 when both D1 and D2 are themselves diagonal. Lovas and Andai established that their two-rebit function χ~1(ε) (≈ε) yields the previously conjectured Hilbert-Schmidt separability probability of 6429. We are able, in the Slater framework (using cylindrical algebraic decompositions [CAD] to enforce positivity constraints), to reproduce this result. Further, we similarly obtain its new (much simpler) two-qubit counterpart, χ~2(ε)=31ε2(4−ε2). Verification of the companion conjecture of a Hilbert-Schmidt separability probability of 338 immediately follows in the Lovas-Andai framework. We obtain the formulas for χ~1(ε) and χ~2(ε) by taking D1 and D2 to be diagonal, allowing us to proceed in lower (7 and 11), rather than the full (9 and 15) dimensions occupied by the convex sets of two-rebit and two-qubit states. The CAD's themselves involve 4 and 8 variables, in addition to μ=ε. We also investigate extensions of these analyses to rebit-retrit and qubit-qutrit (6×6) settings.
We extend the findings and analyses of our two recent studies (Phys. Rev. A, 75, 032326 [2007] and arXiv:0704.3723) by, first, obtaining numerical estimates of the separability function based on the (Euclidean, flat) Hilbert-Schmidt (HS) metric for the 27-dimensional convex set of quaternionic two-qubit systems. The estimated function appears to be strongly consistent with our previously-formulated Dyson-index (beta = 1, 2, 4) ansatz, dictating that the quaternionic (beta=4) separability function should be exactly proportional to the square of the separability function for the 15-dimensional convex set of two-qubit complex (beta=2) systems, as well as the fourth power of the separability function for the 9-dimensional convex set of two-qubit real (beta=1) systems. In particular, we conclude that S_{quat}(mu) =(6/71)^2 ((3-mu^2) mu)^4 =(S_{complex}(mu))^2. Here, mu =(rho_{11} rho_{44}/(rho_{22} rho_{33})^(1/2), where rho is a 4 x 4 two-qubit density matrix. We can, thus, supplement (and fortify) our previous assertion that the HS separability probability of the two-qubit complex states is 8/33= 0.242424, by claiming that its quaternionic counterpart is 72442944/936239725 = 0.0773765. We also comment on and analyze the odd beta=1 and 3 cases.
We employ a quasirandom methodology, recently developed by Martin Roberts, to estimate the separability probabilities, with respect to the Bures (minimal monotone/statistical distinguishability) measure, of generic two-qubit and two-rebit states. This procedure, based on generalized properties of the golden ratio, yielded, in the course of almost fifteen billion iterations, two-qubit estimates repeatedly agreeing to close to nine decimal places with 34125=11⋅3152≈0.07331378299. The corresponding probabilities based on the Hilbert-Schmidt and operator monotone function x measures are (still subject to formal proof) essentially known to be 338=3⋅1123 and 1−27π2256=1−33π244, respectively. Further, the analogous pair of two-rebit probabilities has been proven by Lovas and Andai to be 6429=2629 and approximately 0.26223. In the Bures two-rebit case, we do not presently perceive an exact value corresponding to our quasirandom estimate of 0.15709715, based on over twenty billion iterations. The quasirandom methodology can also be applied to test recent conjectures that the Hilbert-Schmidt qubit-qutrit and rebit-retrit separability probabilities are 100027=23⋅5333 and 6561860=3822⋅5⋅43, respectively.
To begin, we find certain formulas Q(k,α)=G1k(α)G2k(α) , for k=-1,0,1,…,9 . These yield that part of the total separability probability, P(k,α) , for generalized (real, complex, quaternionic , etc.) two-qubit states endowed with random induced measure, for which the determinantal inequality ρPT>ρ holds. Here ρ denotes a 4×4 density matrix, obtained by tracing over the pure states in 4×(4+k) -dimensions, and ρPT denotes its partial transpose. Further, α is a Dyson-index-like parameter with α=1 for the standard (15-dimensional) convex set of (complex) two-qubit states. For k=0 , we obtain the previously reported Hilbert-Schmidt formulas, with Q(0,1/2)=29/128 (the real case), Q(0,1)=4/33 (the standard complex case), and Q(0,2)=13/323 (the quaternionic case), the three simply equalling P(0,α)/2 . The factors G2k(α) are sums of polynomial-weighted generalized hypergeometric functions pFp-1 , p≥7 , all with argument z=27/64=(3/4)3 . We find number-theoretic-based formulas for the upper ( uik ) and lower ( bik ) parameter sets of these functions and, then, equivalently express G2k(α) in terms of first-order difference equations. Applications of Zeilberger’s algorithm yield “concise” forms of Q(-1,α) , Q(1,α) , and Q(3,α) , parallel to the one obtained previously (Slater 2013) for P(0,α)=2Q(0,α) . For nonnegative half-integer and integer values of α , Q(k,α