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arXiv:1803.10680

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

Paul B. Slater

Published Mar 28, 2018• 1 citation
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Abstract

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".

DOI
10.48550/arxiv.1803.10680

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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

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.

•May 1, 2017•0 citations
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.
Master Lovas–Andai and equivalent formulas verifying the 833\frac{8}{33}338​ 8 33 two-qubit Hilbert–Schmidt separability probability and companion rational-valued conjectures
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.
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.