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Construction of the Lovas-Andai Two-Qubit Function $\tilde{\chi}_2 (\varepsilon )=\frac{1}{3} \varepsilon ^2 \left(4-\varepsilon ^2\right)$ Verifies the $\frac{8}{33}$-Hilbert Schmidt Separability Probability Conjecture

Paul B. Slater

Published May 1, 2017• 0 citations
OpenAlex

Abstract

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 $\varepsilon \in [0,1]$ is the ratio $\sigma(V)$ of the singular values of the $2 \times 2$ matrix $V=D_2^{1/2} D_1^{-1/2}$ formed from the two $2 \times 2$ diagonal blocks ($D_1, D_2$) of a randomly generated $4 \times 4$ density matrix $D$. In the Slater setting, the independent variable $\mu$ is the diagonal-entry ratio $\sqrt{\frac{d_ {11} d_ {44}}{d_ {22} d_ {33}}}$--with, importantly, $\mu=\varepsilon$ or $\mu=\frac{1}{\varepsilon}$ when both $D_1$ and $D_2$ are themselves diagonal. Lovas and Andai established that their two-rebit function $\tilde{\chi}_1 (\varepsilon )$ ($\approx \varepsilon$) yields the previously conjectured Hilbert-Schmidt separability probability of $\frac{29}{64}$. 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, $\tilde{\chi}_2(\varepsilon) =\frac{1}{3} \varepsilon ^2 \left(4-\varepsilon ^2\right)$. Verification of the companion conjecture of a Hilbert-Schmidt separability probability of $\frac{8}{33}$ immediately follows in the Lovas-Andai framework. We obtain the formulas for $\tilde{\chi}_1(\varepsilon)$ and $\tilde{\chi}_2(\varepsilon)$ by taking $D_1$ and $D_2$ 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 $\mu=\varepsilon$. We also investigate extensions of these analyses to rebit-retrit and qubit-qutrit ($6 \times 6$) settings.

Related papers

Density Matrix Diagonal-Block Lovas-Andai-type singular-value ratios for qubit-qudit separability/PPT probability analyses

Paul B. Slater

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.

arXiv:2106.16104•Jun 30, 2021•0 citations

Qubit-Qudit Separability/PPT-Probability Analyses and Lovas-Andai Formula Extensions to Induced Measures

Paul B. Slater

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 $\frac{8}{33} = \frac{2^3}{3 \cdot 11} \approx 0.242424$ and (the 9-dimensional) two-rebits of $\frac{29}{64} =\frac{29}{2^6} \approx 0.453125$. Based in part on extensive numerical computations, we advance the possibilities of a qubit-qutrit value of $\frac{27}{1000} = (\frac{3}{10})^3 =\frac{3^3}{2^3 \cdot 5^3} = 0.027$ and a rebit-retrit one of $\frac{860}{6561} =\frac{2^2 \cdot 5 \cdot 43}{3^8} \approx 0.131078$. These four values for $2 \times m$ 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 $\frac{16}{3 π^2} \approx 0.54038$, 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 $\frac{1}{3} \varepsilon^2 (4 -\varepsilon^2)$, yielding $\frac{8}{33}$, its $k=1$ induced measure counterpart is $\frac{1}{4} \varepsilon ^2 \left(3-\varepsilon ^2\right)^2$, yielding $\frac{61}{143} =\frac{61}{11 \cdot 13} \approx 0.426573$, where $\varepsilon$ is a singular-value ratio. We investigate, in these regards, the possibility of extending the previously-obtained "Lovas-Andai master formula".

arXiv:1803.10680•Mar 28, 2018•1 citation

Qubit-qudit separability/PPT-probability investigations, including Lovas-Andai formula advancements

Paul B. Slater

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 ($2 \times 3$) qubit-qutrit and ($2 \times 4$) 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 $(\frac{3}{10})^3 =\frac{3^3}{2^3 \cdot 5^3} = \frac{27}{1000} =0.027$, with the counterpart two-qubit separability probablity being $\frac{8}{33}=\frac{2^3}{3 \cdot 11} \approx 0.242424$. In the ($2 \times 4$) qubit-qudit case, we advance $\frac{16}{12375} =\frac{4^2}{3^2 \cdot 5^3 \cdot 11} \approx 0.001292929$ as the PPT-probability. The numerators of the probabilities for $2 \times m$ scenarios are, thus, possibly simply powers of $m$. We also search for the rebit-retrit counterpart to the established two-rebit result of $\frac{29}{64} =\frac{29}{2^6} \approx 0.453125$. Additionally, we find that the Hilbert-Schmidt separability/PPT-probabilities for the two-rebit, rebit-retrit and two-retrit $X$-states all equal $\frac{16}{3 \pi^2} \approx 0.54038$, 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.

•Mar 28, 2018•0 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.

arXiv:1612.02798•Dec 8, 2016•0 citations

Quasirandom Estimation of Bures Two-Qubit and Two-Rebit Separability Probabilities

Paul B. Slater

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 $\frac{25}{341} =\frac{5^2}{11 \cdot 31} \approx 0.07331378299$. The corresponding probabilities based on the Hilbert-Schmidt and operator monotone function $\sqrt{x}$ measures are (still subject to formal proof) essentially known to be $\frac{8}{33} = \frac{2^3}{3 \cdot 11}$ and $1-\frac{256}{27 \pi^2}=1-\frac{4^4}{3^3 \pi^2}$, respectively. Further, the analogous pair of two-rebit probabilities has been proven by Lovas and Andai to be $\frac{29}{64} = \frac{29}{2^6}$ 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 $\frac{27}{1000}=\frac{3^3}{2^3 \cdot 5^3}$ and $\frac{860}{6561}= \frac{2^2 \cdot 5 \cdot 43}{3^8}$, respectively.

•Jan 25, 2019•0 citations

Qubit-Qubit and Qubit-Qutrit Separability Functions and Probabilities

Paul B. Slater

We list in increasing order -- 1/3, 3/8, 2/5, 135 pi/1024, 16/(3 pi^2), 3 pi/16, 5/8, 105 pi/512, 2 - 435 pi/1024, 11/16, 1 -- a number of exact two-qubit Hilbert-Schmidt (HS) separability probabilities, we are able to compute. Each probability corresponds to a specific scenario -- a class of 4 x 4 density matrices (rho) with an m-subset (m<6) of its six off-diagonal pairs of symmetrically located entries set to zero. Initially, we consider only scenarios in which the (6-m) non-nullified pairs are real, but then permit them to be complex (as well as quaternionic) in nature also. The general analytical strategy we implement is based on the Bloore density matrix parameterization (J. Phys. A, 9, 2059 [1976]), allowing us to conveniently reduce the dimensionalities of required integrations. For each scenario, we identify a certain univariate ``separability function'' S_{scenario}(nu), where nu = rho_{11} rho_{44}/(rho_{22} rho_{33}). The integral over nu in [0,infty] of the product of this function with a scenario-specific (marginal) jacobian function J_{scenario}(nu) yields the HS {separable} volume (V^{HS}_{sep}). The ratio of V^{HS}_{sep} to the HS total (entangled and non-entangled) volume, gives us the HS scenario-specific separability probability. Among the possible forms that we have so far determined for S_{scenario}(nu) are piecewise combinations of c, c sqrt{nu} and c nu for nu in [0,1] and (in a dual manner) c, c/sqrt{nu} and c/nu for nu in [1,infty]. We also obtain bivariate separability functions S_{scenario)^{6 x 6}(nu_{1},nu_{2}} in the qubit-qutrit case, involving 6 x 6 density matrices, with ratio variables, nu_{1}= rho_{11} rho_{55}/(rho_{22} rho_{44}), and nu_{2}= rho_{22} rho_{66}/(rho_{33} rho_{55}). Additionally, we investigate parallel two-qutrit and three-qubit problems.

arXiv:quant-ph/0702134•Feb 14, 2007•1 citation
Construction of the Lovas-Andai Two-Qubit Function χ~2(ε)=13ε2(4−ε2)\tilde{\chi}_2 (\varepsilon )=\frac{1}{3} \varepsilon ^2 \left(4-\varepsilon ^2\right)χ~​2​(ε)=31​ε2(4−ε2) Verifies the 833\frac{8}{33}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]\varepsilon \in [0,1]ε∈[0,1] is the ratio σ(V)\sigma(V)σ(V) of the singular values of the 2×22 \times 22×2 matrix V=D21/2D1−1/2V=D_2^{1/2} D_1^{-1/2}V=D21/2​D1−1/2​ formed from the two 2×22 \times 22×2 diagonal blocks (D1,D2D_1, D_2D1​,D2​) of a randomly generated 4×44 \times 44×4 density matrix DDD. In the Slater setting, the independent variable μ\muμ is the diagonal-entry ratio d11d44d22d33\sqrt{\frac{d_ {11} d_ {44}}{d_ {22} d_ {33}}}d22​d33​d11​d44​​​--with, importantly, μ=ε\mu=\varepsilonμ=ε or μ=1ε\mu=\frac{1}{\varepsilon}μ=ε1​ when both D1D_1D1​ and D2D_2D2​ are themselves diagonal. Lovas and Andai established that their two-rebit function χ~1(ε)\tilde{\chi}_1 (\varepsilon )χ~​1​(ε) (≈ε\approx \varepsilon≈ε) yields the previously conjectured Hilbert-Schmidt separability probability of 2964\frac{29}{64}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)=13ε2(4−ε2)\tilde{\chi}_2(\varepsilon) =\frac{1}{3} \varepsilon ^2 \left(4-\varepsilon ^2\right)χ~​2​(ε)=31​ε2(4−ε2). Verification of the companion conjecture of a Hilbert-Schmidt separability probability of 833\frac{8}{33}338​ immediately follows in the Lovas-Andai framework. We obtain the formulas for χ~1(ε)\tilde{\chi}_1(\varepsilon)χ~​1​(ε) and χ~2(ε)\tilde{\chi}_2(\varepsilon)χ~​2​(ε) by taking D1D_1D1​ and D2D_2D2​ 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\mu=\varepsilonμ=ε. We also investigate extensions of these analyses to rebit-retrit and qubit-qutrit (6×66 \times 66×6) settings.
An important variable in the 2017 analysis of Lovas and Andai, formally establishing the Hilbert-Schmidt separability probability conjectured by Slater of 2964\frac{29}{64}6429​ for the 9-dimensional convex set of two-rebit density matrices, was the ratio (ε=σ2σ1\varepsilon =\frac{σ_2}{σ_1}ε=σ1​σ2​​) of the two singular values (σ1≥σ2≥0σ_1 \geq σ_2 \geq 0σ1​≥σ2​≥0) of D212D1−12D_2^{\frac{1}{2}} D_1^{-\frac{1}{2}}D221​​D1−21​​. There, D1D_1D1​ and D2D_2D2​ were the diagonal 2×22 \times 22×2 blocks of a 4×44 \times 44×4 two-rebit density matrix ρρρ. Working within the Lovas-Andai "separability function" (χ~d(ε)\tildeχ_d(\varepsilon)χ~​d​(ε)) framework, Slater was able to verify further conjectures of Hilbert-Schmidt separability probabilities of 833\frac{8}{33}338​ and 26323\frac{26}{323}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=D212D1−12V=D_2^{\frac{1}{2}} D_1^{-\frac{1}{2}}V=D221​​D1−21​​, where now D1D_1D1​ and D2D_2D2​ are the 3×33 \times 33×3 diagonal blocks of 6×66 \times 66×6 rebit-retrit and qubit-qutrit density matrices randomly generated with respect to Hilbert-Schmidt measure. Further, we initiate a parallel study employing 8×88 \times 88×8 density matrices. The motivation for this analysis is the conjectured relevance of these singular values in suitably extending χ~d(ε)\tildeχ_d(\varepsilon)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.
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 833=233⋅11≈0.242424\frac{8}{33} = \frac{2^3}{3 \cdot 11} \approx 0.242424338​=3⋅1123​≈0.242424 and (the 9-dimensional) two-rebits of 2964=2926≈0.453125\frac{29}{64} =\frac{29}{2^6} \approx 0.4531256429​=2629​≈0.453125. Based in part on extensive numerical computations, we advance the possibilities of a qubit-qutrit value of 271000=(310)3=3323⋅53=0.027\frac{27}{1000} = (\frac{3}{10})^3 =\frac{3^3}{2^3 \cdot 5^3} = 0.027100027​=(103​)3=23⋅5333​=0.027 and a rebit-retrit one of 8606561=22⋅5⋅4338≈0.131078\frac{860}{6561} =\frac{2^2 \cdot 5 \cdot 43}{3^8} \approx 0.1310786561860​=3822⋅5⋅43​≈0.131078. These four values for 2×m2 \times m2×m systems (m=2,3m=2,3m=2,3) suggest certain numerator/denominator sequences involving powers of mmm, which we further investigate for m>3m>3m>3. Additionally, we find that the Hilbert-Schmidt separability/PPT-probabilities for the two-rebit, rebit-retrit and two-retrit XXX-states all equal 163π2≈0.54038\frac{16}{3 π^2} \approx 0.540383π216​≈0.54038, as well as more generally, that the probabilities based on induced measures are equal across these three sets of XXX-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 13ε2(4−ε2)\frac{1}{3} \varepsilon^2 (4 -\varepsilon^2)31​ε2(4−ε2), yielding 833\frac{8}{33}338​, its k=1k=1k=1 induced measure counterpart is 14ε2(3−ε2)2\frac{1}{4} \varepsilon ^2 \left(3-\varepsilon ^2\right)^241​ε2(3−ε2)2, yielding 61143=6111⋅13≈0.426573\frac{61}{143} =\frac{61}{11 \cdot 13} \approx 0.42657314361​=11⋅1361​≈0.426573, where ε\varepsilonε 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 (2×32 \times 32×3) qubit-qutrit and (2×42 \times 42×4) 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 (310)3=3323⋅53=271000=0.027(\frac{3}{10})^3 =\frac{3^3}{2^3 \cdot 5^3} = \frac{27}{1000} =0.027(103​)3=23⋅5333​=100027​=0.027, with the counterpart two-qubit separability probablity being 833=233⋅11≈0.242424\frac{8}{33}=\frac{2^3}{3 \cdot 11} \approx 0.242424338​=3⋅1123​≈0.242424. In the (2×42 \times 42×4) qubit-qudit case, we advance 1612375=4232⋅53⋅11≈0.001292929\frac{16}{12375} =\frac{4^2}{3^2 \cdot 5^3 \cdot 11} \approx 0.0012929291237516​=32⋅53⋅1142​≈0.001292929 as the PPT-probability. The numerators of the probabilities for 2×m2 \times m2×m scenarios are, thus, possibly simply powers of mmm. We also search for the rebit-retrit counterpart to the established two-rebit result of 2964=2926≈0.453125\frac{29}{64} =\frac{29}{2^6} \approx 0.4531256429​=2629​≈0.453125. Additionally, we find that the Hilbert-Schmidt separability/PPT-probabilities for the two-rebit, rebit-retrit and two-retrit XXX-states all equal 163π2≈0.54038\frac{16}{3 \pi^2} \approx 0.540383π216​≈0.54038, as well as more generally, that the probabilities based on induced measures are equal across these three sets of XXX-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 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 (2964\frac{29}{64}6429​), and that has also been very strongly supported numerically for its complex (833\frac{8}{33}338​), and quaternionic (26323\frac{26}{323}32326​) counterparts. Now, we seek to test the presumptive octonionic value of 444824091349≈0.0108722\frac{44482}{4091349} \approx 0.0108722409134944482​≈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 aaa, introduced by Forrester (https://arxiv.org/pdf/1610.08081.pdf), to 0 (or possibly near to 0). Hilbert-Schmidt measure being the case k=0k=0k=0 of random induced measure, for k=1k=1k=1, the corresponding octonionic separability probability conjecture is 7612846293213345≈0.0259635\frac{7612846}{293213345} \approx 0.02596352932133457612846​≈0.0259635, while for k=2k=2k=2, it is 489339295041567≈0.0514869,…\frac{4893392}{95041567} \approx 0.0514869, \ldots950415674893392​≈0.0514869,…. The relation between the parameters aaa and kkk 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 25341=5211⋅31≈0.07331378299\frac{25}{341} =\frac{5^2}{11 \cdot 31} \approx 0.0733137829934125​=11⋅3152​≈0.07331378299. The corresponding probabilities based on the Hilbert-Schmidt and operator monotone function x\sqrt{x}x​ measures are (still subject to formal proof) essentially known to be 833=233⋅11\frac{8}{33} = \frac{2^3}{3 \cdot 11}338​=3⋅1123​ and 1−25627π2=1−4433π21-\frac{256}{27 \pi^2}=1-\frac{4^4}{3^3 \pi^2}1−27π2256​=1−33π244​, respectively. Further, the analogous pair of two-rebit probabilities has been proven by Lovas and Andai to be 2964=2926\frac{29}{64} = \frac{29}{2^6}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 271000=3323⋅53\frac{27}{1000}=\frac{3^3}{2^3 \cdot 5^3}100027​=23⋅5333​ and 8606561=22⋅5⋅4338\frac{860}{6561}= \frac{2^2 \cdot 5 \cdot 43}{3^8}6561860​=3822⋅5⋅43​, respectively.
We list in increasing order -- 1/3, 3/8, 2/5, 135 pi/1024, 16/(3 pi^2), 3 pi/16, 5/8, 105 pi/512, 2 - 435 pi/1024, 11/16, 1 -- a number of exact two-qubit Hilbert-Schmidt (HS) separability probabilities, we are able to compute. Each probability corresponds to a specific scenario -- a class of 4 x 4 density matrices (rho) with an m-subset (m<6) of its six off-diagonal pairs of symmetrically located entries set to zero. Initially, we consider only scenarios in which the (6-m) non-nullified pairs are real, but then permit them to be complex (as well as quaternionic) in nature also. The general analytical strategy we implement is based on the Bloore density matrix parameterization (J. Phys. A, 9, 2059 [1976]), allowing us to conveniently reduce the dimensionalities of required integrations. For each scenario, we identify a certain univariate ``separability function'' S_{scenario}(nu), where nu = rho_{11} rho_{44}/(rho_{22} rho_{33}). The integral over nu in [0,infty] of the product of this function with a scenario-specific (marginal) jacobian function J_{scenario}(nu) yields the HS {separable} volume (V^{HS}_{sep}). The ratio of V^{HS}_{sep} to the HS total (entangled and non-entangled) volume, gives us the HS scenario-specific separability probability. Among the possible forms that we have so far determined for S_{scenario}(nu) are piecewise combinations of c, c sqrt{nu} and c nu for nu in [0,1] and (in a dual manner) c, c/sqrt{nu} and c/nu for nu in [1,infty]. We also obtain bivariate separability functions S_{scenario)^{6 x 6}(nu_{1},nu_{2}} in the qubit-qutrit case, involving 6 x 6 density matrices, with ratio variables, nu_{1}= rho_{11} rho_{55}/(rho_{22} rho_{44}), and nu_{2}= rho_{22} rho_{66}/(rho_{33} rho_{55}). Additionally, we investigate parallel two-qutrit and three-qubit problems.