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 ) =
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 ) =
Milz and Strunz recently reported substantial evidence to further support the previously conjectured separability probability of 338 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,…), 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 ∣ρPT∣>∣ρ∣ and those from ∣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 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 (rA,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) X-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.
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 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.
Abstract Many works have been devoted to determining the volume of the set of separable states and the separability probability both analytically and numerically. However, computing the exact separability probability remains a highly nontrivial open problem. In these notes, we examine this problem in the simplest context of two-qubit states. A novel aspect of our method is the use of the Duistermaat-Heckman measure, which is the push-forward of Liouville measure on the coadjoint orbit along the moment map. On each coadjoint orbit, the Hilbert-Schmidt measure differs from the Duistermaat-Heckman measure by a constant. The latter measure can be explicitly computed by using tools from symplectic geometry. We confirm the conjecture that the separability probability of two-qubit states with Hilbert-Schmidt measure is 8 33 .
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.