An important variable in the 2017 analysis of Lovas and Andai, formally establishing the Hilbert-Schmidt separability probability conjectured by Slater of $\frac{29}{64}$ for the 9-dimensional convex set of two-rebit density matrices, was the ratio ($\varepsilon =\frac{σ_2}{σ_1}$) of the two singular values ($σ_1 \geq σ_2 \geq 0$) of $D_2^{\frac{1}{2}} D_1^{-\frac{1}{2}}$. There, $D_1$ and $D_2$ were the diagonal $2 \times 2$ blocks of a $4 \times 4$ two-rebit density matrix $ρ$. Working within the Lovas-Andai "separability function" ($\tildeχ_d(\varepsilon)$) framework, Slater was able to verify further conjectures of Hilbert-Schmidt separability probabilities of $\frac{8}{33}$ and $\frac{26}{323}$ for the 15-dimensional and 26-dimensional convex sets of two-qubit and two-quater[nionic]-bit density matrices. Here, we investigate the behavior of the three singular value ratios of $V=D_2^{\frac{1}{2}} D_1^{-\frac{1}{2}}$, where now $D_1$ and $D_2$ are the $3 \times 3$ diagonal blocks of $6 \times 6$ rebit-retrit and qubit-qutrit density matrices randomly generated with respect to Hilbert-Schmidt measure. Further, we initiate a parallel study employing $8 \times 8$ density matrices. The motivation for this analysis is the conjectured relevance of these singular values in suitably extending $\tildeχ_d(\varepsilon)$ to higher dimensional systems--an issue we also approach using certain novel numeric means. Section 3.3 of the 2017 A. Lovas doctoral dissertation (written in Hungarian) appears germane to such an investigation.
An important variable in the 2017 analysis of Lovas and Andai, formally establishing the Hilbert-Schmidt separability probability conjectured by Slater of 6429 for the 9-dimensional convex set of two-rebit density matrices, was the ratio (ε=σ1σ2) of the two singular values (σ1≥σ2≥0) of D221D1−21. There, D1 and D2 were the diagonal 2×2 blocks of a 4×4 two-rebit density matrix ρ. Working within the Lovas-Andai "separability function" (χ~d(ε)) framework, Slater was able to verify further conjectures of Hilbert-Schmidt separability probabilities of 338 and 32326 for the 15-dimensional and 26-dimensional convex sets of two-qubit and two-quater[nionic]-bit density matrices. Here, we investigate the behavior of the three singular value ratios of V=D221D1−21, where now D1 and D2 are the 3×3 diagonal blocks of 6×6 rebit-retrit and qubit-qutrit density matrices randomly generated with respect to Hilbert-Schmidt measure. Further, we initiate a parallel study employing 8×8 density matrices. The motivation for this analysis is the conjectured relevance of these singular values in suitably extending χ~d(ε) to higher dimensional systems--an issue we also approach using certain novel numeric means. Section 3.3 of the 2017 A. Lovas doctoral dissertation (written in Hungarian) appears germane to such an investigation.
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 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.
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 implement a procedure-based on the Wishart-Laguerre distribution-recently outlined by {Ż}yczkowski and Khvedelidze, Rogojin and Abgaryan, for the generation of random (complex or real) N×N density matrices of rank k≤N with respect to Hilbert-Schmidt (HS) measure. In the complex case, one commences with a Ginibre matrix A of dimensions k×k+2(N−k), while for a real scenario, one employs a Ginibre matrix B of dimensions k×k+1+2(N−k). Then, the k×k product AA† or BBT is diagonalized-padded with zeros to size N×N-and rotated, obtaining a random density matrix. Implementing the procedure for rank-4 rebit-retrit states, for 800 million Ginibre-matrix realizations, 6,192,047 were found separable, for a sample probability of .00774006-suggestive of an exact value 5000387=23⋅5432⋅43=.0774. A conjecture for the HS separability probability of rebit-retrit systems of full rank is 6561860=3822⋅5⋅43≈0.1310775 (the two-rebit counterpart has been proven to be 6429=2629). Subject to these conjectures, the ratio of the rank-4 to rank-6 probabilities would be 100000059049=26⋅56310≈0.059049, with the common factor 43 cancelling. As to the intermediate rank-5 probability, a 2006 theorem of Szarek, Bengtsson and {Ż}ycskowski informs us that it must be one-half the rank-6 probability-itself conjectured to be 100027=23⋅5333, while for rank 3 or less, the associated probabilities must be 0 by a 2009 result of Ruskai and Werner. We are led to re-examine a 2005 qubit-qutrit analysis of ours, in these regards, and now find evidence for a 267370=35⋅112⋅5⋅7≈0.0261878 rank-4 to rank-6 probability ratio.
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 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.