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arXiv:1506.08739

Bloch Radii Repulsion in Separable Two-Qubit Systems

Paul B. Slater

Published Jun 29, 2015• 2 citations
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Abstract

Milz and Strunz recently reported substantial evidence to further support the previously conjectured separability probability of $\frac{8}{33}$ for two-qubit systems ($ρ$) endowed with Hilbert-Schmidt measure. Additionally, they found that along the radius ($r$) of the Bloch ball representing either of the two single-qubit subsystems, this value appeared constant (but jumping to unity at the locus of the pure states, $r=1$). Further, they also observed (personal communication) such separability probability $r$-invariance, when using, more broadly, random induced measure ($K=3,4,5,\ldots$), with $K=4$ corresponding to the (symmetric) Hilbert-Schmidt case. Among the findings here is that this invariance is maintained even after splitting the separability probabilities into those parts arising from the determinantal inequality $|ρ^{PT}| >|ρ|$ and those from $|ρ| > |ρ^{PT}| >0$, where the partial transpose is indicated. The nine-dimensional set of generic two-re[al]bit states endowed with Hilbert-Schmidt measure is also examined, with similar $r$-invariance conclusions. Contrastingly, two-qubit separability probabilities based on the Bures (minimal monotone) measure {\it diminish} with $r$. Moreover, we study the forms that the separability probabilities take as joint (bivariate) functions of the radii ($r_A, r_B$) of the Bloch balls of {\it both} single-qubit subsystems. Here, a form of Bloch radii {\it repulsion} for separable two-qubit systems emerges in {\it all} our several analyses. Separability probabilities tend to be smaller when the lengths of the two radii are closer. In Appendix A, we report certain companion analytic results for the much-investigated, more amenable (7-dimensional) $X$-states model.

DOI
10.48550/arxiv.1506.08739

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Milz and Strunz recently reported substantial evidence to further support the previously conjectured separability probability of 833\frac{8}{33}338​ for two-qubit systems (ρρρ) endowed with Hilbert-Schmidt measure. Additionally, they found that along the radius (rrr) of the Bloch ball representing either of the two single-qubit subsystems, this value appeared constant (but jumping to unity at the locus of the pure states, r=1r=1r=1). Further, they also observed (personal communication) such separability probability rrr-invariance, when using, more broadly, random induced measure (K=3,4,5,…K=3,4,5,\ldotsK=3,4,5,…), with K=4K=4K=4 corresponding to the (symmetric) Hilbert-Schmidt case. Among the findings here is that this invariance is maintained even after splitting the separability probabilities into those parts arising from the determinantal inequality ∣ρPT∣>∣ρ∣|ρ^{PT}| >|ρ|∣ρPT∣>∣ρ∣ and those from ∣ρ∣>∣ρPT∣>0|ρ| > |ρ^{PT}| >0∣ρ∣>∣ρPT∣>0, where the partial transpose is indicated. The nine-dimensional set of generic two-re[al]bit states endowed with Hilbert-Schmidt measure is also examined, with similar rrr-invariance conclusions. Contrastingly, two-qubit separability probabilities based on the Bures (minimal monotone) measure {\it diminish} with rrr. Moreover, we study the forms that the separability probabilities take as joint (bivariate) functions of the radii (rA,rBr_A, r_BrA​,rB​) of the Bloch balls of {\it both} single-qubit subsystems. Here, a form of Bloch radii {\it repulsion} for separable two-qubit systems emerges in {\it all} our several analyses. Separability probabilities tend to be smaller when the lengths of the two radii are closer. In Appendix A, we report certain companion analytic results for the much-investigated, more amenable (7-dimensional) XXX-states model.
We detect a certain pattern of behavior of separability probabilities [Formula: see text] for two-qubit systems endowed with Hilbert–Schmidt (HS), and more generally, random induced measures, where [Formula: see text] and [Formula: see text] are the Bloch radii ([Formula: see text]) of the qubit reduced states ([Formula: see text]). We observe a relative repulsion of radii effect, that is [Formula: see text], except for rather narrow “crossover” intervals [Formula: see text]. Among the seven specific cases we study are, firstly, the “toy” seven-dimensional [Formula: see text]-states model and, then, the fifteen-dimensional two-qubit states obtained by tracing over the pure states in [Formula: see text]-dimensions, for [Formula: see text], with [Formula: see text] corresponding to HS (flat/Euclidean) measure. We also examine the real (two-rebit) [Formula: see text], the [Formula: see text]-states [Formula: see text], and Bures (minimal monotone)–for which no nontrivial crossover behavior is observed–instances. In the two [Formula: see text]-states cases, we derive analytical results; for [Formula: see text], we propose formulas that well-fit our numerical results; and for the other scenarios, rely presently upon large numerical analyses. The separability probability crossover regions found expand in length (lower [Formula: see text]) as [Formula: see text] increases. This report continues our efforts [P. B. Slater, arXiv:1506.08739] to extend the recent work of [S. Milz and W. T. Strunz, J. Phys. A 48 (2015) 035306.] from a univariate ([Formula: see text]) framework — in which they found separability probabilities to hold constant with [Formula: see text] — to a bivariate ([Formula: see text]) one. We also analyze the two-qutrit and qubit–qutrit counterparts reported in Quantum Inform. Process. 15 (2016) 3745 in this context, and study two-qubit separability probabilities of the form [Formula: see text]. A physics.stack.exchange link to a contribution by Mark Fischler addressing, in considerable detail, the construction of suitable bivariate distributions is indicated at the end of the paper.
Abstract The geometric separability probability of composite quantum systems has been extensively studied in the recent decades. One of the simplest but strikingly difficult problem is to compute the separability probability of qubit–qubit and rebit–rebit quantum states with respect to the Hilbert–Schmidt measure. A lot of numerical simulations confirm the P r e b i t − r e b i t = 29 64 and P q u b i t − q u b i t = 8 33 conjectured probabilities. We provide a rigorous proof for the separability probability in the real case and we give explicit integral formulas for the complex and quaternionic case. Milz and Strunz studied the separability probability with respect to given subsystems. They conjectured that the separability probability of qubit–qubit (and qubit-qutrit) states of the form of ( D 1 C C ∗ D 2 ) depends on D = D 1 + D 2 (on single qubit subsystems), moreover it depends only on the Bloch radii ( r ) of D and it is constant in r . Using the Peres–Horodecki criterion for separability we give a mathematical proof for the 29 64 probability and we present an integral formula for the complex case which hopefully will help to prove the 8 33 probability, too. We prove Milz and Strunz’s conjecture for rebit–rebit and qubit–qubit states. The case, when the state space is endowed with the volume form generated by the operator monotone function f ( x ) =
Compelling evidence-though yet no formal proof--has been adduced that the probability that a generic two-qubit state (ρρρ) is separable is 833\frac{8}{33}338​ (arXiv:1301.6617, arXiv:1109.2560, arXiv:0704.3723). Proceeding in related analytical frameworks, using a further determinantal moment formula of C. Dunkl (Appendix), we reach the conclusion that one-half of this probability arises when the determinantal inequality ∣ρPT∣>∣ρ∣|ρ^{PT}|>|ρ|∣ρPT∣>∣ρ∣, where PTPTPT denotes the partial transpose, is satisfied, and, the other half, when ∣ρ∣>∣ρPT∣|ρ|>|ρ^{PT}|∣ρ∣>∣ρPT∣. These probabilities are taken with respect to the flat, Hilbert-Schmidt measure on the fifteen-dimensional convex set of 4×44 \times 44×4 density matrices. We find fully parallel bisection/equipartition results for the previously adduced, as well, two-"re[al]bit" and two-"quater[nionic]bit"separability probabilities of 2964\frac{29}{64}6429​ and 26323\frac{26}{323}32326​, respectively. The computational results reported lend strong support to those obtained earlier--including the "concise formula" that yields them--most conspicuously amongst those findings being the 2964,833\frac{29}{64}, \frac{8}{33}6429​,338​ and 26323\frac{26}{323}32326​ probabilities noted.
We report a concise answer--in the case of 2 x 2 systems--to the fundamental quantum-information-theoretic question as to "the volume of separable states" posed by Zyczkowski, Horodecki, Sanpera and Lewenstein (Phys. Rev. A, 58, 883 [1998]). We proceed by applying the Mathematica command FindSequenceFunction to a series of conjectured Hilbert-Schmidt generic 2 x 2 (rational-valued) separability probabilities p(a), a = 1, 2,...,32, with a = 1 indexing standard two-qubit systems, and a = 2, two-quater(nionic)bit systems. These 32 inputted values of p(a)--as well as 32 companion non-inputted values for the half-integers, a = 1/2 (two-re[al]bit) systems), 3/2,..., 63/2, are advanced on the basis of high-precision probability-distribution-reconstruction computations, employing 7,501 determinantal moments of partially transposed 4 x 4 density matrices. The function P(a) given by application of the command fully reproduces both of these 32-length sequences, and an equivalent outcome is obtained if the half-integral series is the one inputted. The lengthy expression (containing six hypergeometric functions) obtained for P(a) is, then, impressively condensed (by Qing-Hu Hou and colleagues), using Zeilberger's algorithm. For generic (9-dimensional) two-rebit systems, P(1/2) = 29/64, (15-dimensional) two-qubit systems, P(1) = 8/33, (27-dimensional) two-quaterbit systems, P(2) = 26/323, while for generic classical (3-dimensional) systems, P(0)=1.
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 seek to gain insight into the nature of the determinantal moments 〈|ρ PT | n |ρ| k 〉 Bures of generic (nine-dimensional) two-rebit and (fifteen-dimensional) two-qubit systems (ρ), PT denoting partial transpose. Such information—as it has proved to be in the Hilbert–Schmidt counterpart—should be useful, employing probability-distribution reconstruction (inverse) procedures, in obtaining Bures 2 × 2 separability probabilities. The (regularizing) strategy we first adopt is to plot the ratio of numerically-generated (Ginibre ensemble) estimates of the Bures moments to the corresponding (apparently) exactly-known Hilbert–Schmidt moments (Slater and Dunkl 2012 J. Phys. A: Math. Theor. 45 095305). Then, through a combination of symbolic and numerical computations, we obtain strong evidence as to the exact values (and underlying patterns) of certain Bures determinantal moments. In particular, the first moment (average) of |ρ PT | (where ) for the two-qubit systems is, remarkably, . The analogous value for the two-rebit systems is . While in the Hilbert–Schmidt case is the ratio of 3 n -degree polynomials in k , it appears to be the ratio of 5 n -degree polynomials in k in the Bures case.