Two-qubit X-matrices have been the subject of considerable recent attention, as they lend themselves more readily to analytical investigations than two-qubit density matrices of arbitrary nature. Here, we maximally exploit this relative ease of analysis to formally derive an exhaustive collection of results pertaining to the separability probabilities of generalized two-qubit X-matrices endowed with Hilbert–Schmidt and, more broadly, induced measures. Further, the analytical results obtained exhibit interesting parallels to corresponding earlier (but, contrastingly, not yet fully rigorous) results for general two-qubit states — deduced on the basis of determinantal moment formulas. Geometric interpretations can be given to arbitrary positive values of the random-matrix Dyson-index-like parameter [Formula: see text] employed.
Two-qubit X-matrices have been the subject of considerable recent attention, as they lend themselves more readily to analytical investigations than two-qubit density matrices of arbitrary nature. Here, we maximally exploit this relative ease of analysis to formally derive an exhaustive collection of results pertaining to the separability probabilities of generalized two-qubit X-matrices endowed with Hilbert–Schmidt and, more broadly, induced measures. Further, the analytical results obtained exhibit interesting parallels to corresponding earlier (but, contrastingly, not yet fully rigorous) results for general two-qubit states — deduced on the basis of determinantal moment formulas. Geometric interpretations can be given to arbitrary positive values of the random-matrix Dyson-index-like parameter [Formula: see text] employed.
Previously, a formula, incorporating a hypergeometric function, for the Hilbert-Schmidt-averaged determinantal moments of density-matrices ( ) and their partial transposes ( ), was applied with to the generalized two-qubit separability probability question. The formula can, furthermore, be viewed, as we note here, as an averaging over “induced measures in the space of mixed quantum states.” The associated induced-measure separability probabilities ( ) are found— via a high-precision density approximation procedure—to assume interesting, relatively simple rational values in the two-re[al]bit ( ), (standard) two-qubit ( ), and two-quater[nionic]bit ( ) cases. We deduce rather simple companion (rebit, qubit, quaterbit, …) formulas that successfully reproduce the rational values assumed for general . These formulas are observed to share certain features, possibly allowing them to be incorporated into a single master formula.
We find equivalent hypergeometric- and difference-equation-based formulas, Q(k,α)=G1k(α)G2k(α), for k=−1,0,1,…,9, for that (rational-valued) portion of the total separability probability for generalized two-qubit states endowed with random induced measure, for which the determinantal inequality ∣ρPT∣>∣ρ∣ holds. Here ρ denotes a 4×4 density matrix and ρPT, its partial transpose, while α is a Dyson-index-like parameter with α=1 for the standard (15-dimensional) convex set of two-qubit states. The dimension of the space in which these density matrices is embedded is 4×(4+k). For the symmetric case of k=0, we obtain the previously reported Hilbert-Schmidt formulas, with (the two-re[al]bit case) Q(0,21)=12829, (the standard two-qubit case) Q(0,1)=334, and (the two-quater[nionic]bit case) Q(0,2)=32313. The factors G2k(α) can be written as the sum of weighted hypergeometric functions pFp−1, p≥7, all with argument 6427=(43)3. We find formulas for the upper and lower parameter sets of these functions and, then, equivalently express G2k(α) in terms of first-order difference equations. The factors G1k(α) are equal to (6427)α−1 times ratios of products of six Pochhammer symbols involving the indicated parameters. Some remarkable α− and k-specific invariant asymptotic properties (again, involving 6427 and related quantities) of separability probability formulas emerge.
To begin, we find certain formulas Q(k,α)=G1k(α)G2k(α) , for k=-1,0,1,…,9 . These yield that part of the total separability probability, P(k,α) , for generalized (real, complex, quaternionic , etc.) two-qubit states endowed with random induced measure, for which the determinantal inequality ρPT>ρ holds. Here ρ denotes a 4×4 density matrix, obtained by tracing over the pure states in 4×(4+k) -dimensions, and ρPT denotes its partial transpose. Further, α is a Dyson-index-like parameter with α=1 for the standard (15-dimensional) convex set of (complex) two-qubit states. For k=0 , we obtain the previously reported Hilbert-Schmidt formulas, with Q(0,1/2)=29/128 (the real case), Q(0,1)=4/33 (the standard complex case), and Q(0,2)=13/323 (the quaternionic case), the three simply equalling P(0,α)/2 . The factors G2k(α) are sums of polynomial-weighted generalized hypergeometric functions pFp-1 , p≥7 , all with argument z=27/64=(3/4)3 . We find number-theoretic-based formulas for the upper ( uik ) and lower ( bik ) parameter sets of these functions and, then, equivalently express G2k(α) in terms of first-order difference equations. Applications of Zeilberger’s algorithm yield “concise” forms of Q(-1,α) , Q(1,α) , and Q(3,α) , parallel to the one obtained previously (Slater 2013) for P(0,α)=2Q(0,α) . For nonnegative half-integer and integer values of α , Q(k,α
Previously, a formula, incorporating a 5F4 hypergeometric function, for the Hilbert-Schmidt-averaged determinantal moments ρPTnρk/ρk of 4×4 density-matrices ( ρ ) and their partial transposes ( |ρPT| ), was applied with k=0 to the generalized two-qubit separability probability question. The formula can, furthermore, be viewed, as we note here, as an averaging over “induced measures in the space of mixed quantum states.” The associated induced-measure separability probabilities ( k=1,2,… ) are found—viaa high-precision density approximation procedure—to assume interesting, relatively simple rational values in the two-re[al]bit ( α=1/2 ), (standard) two-qubit ( α=1 ), and two-quater[nionic]bit ( α=2 ) cases. We deduce rather simple companion (rebit, qubit, quaterbit, …) formulas that successfully reproduce the rational values assumed forgeneral k . These formulas are observed to share certain features, possibly allowing them to be incorporated into a single master formula.
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.
The nonnegativity of the determinant of the partial transpose of a two-qubit (4 x 4) density matrix is both a necessary and sufficient condition for its separability. While the determinant is restricted to the interval [0,1/256], the determinant of the partial transpose can range over [-1/16,1/256], with negative values corresponding to entangled states. We report here the exact values of the first nine moments of the probability distribution of the partial transpose over this interval, with respect to the Hilbert-Schmidt (metric volume element) measure on the nine-dimensional convex set of real two-qubit density matrices. Rational functions C_{2 j}(m), yielding the coefficients of the 2j-th power of even polynomials occurring at intermediate steps in our derivation of the m-th moment, emerge. These functions possess poles at finite series of consecutive half-integers (m=-3/2,-1/2,...,(2j-1)/2), and certain (trivial) roots at finite series of consecutive natural numbers (m=0, 1,...). Additionally, the (nontrivial) dominant roots of C_{2 j}(m) approach the same half-integer values (m = (2 j-1)/2, (2 j-3)/2,...), as j increases. The first two moments (mean and variance) found--when employed in the one-sided Chebyshev inequality--give an upper bound of 30397/34749 = 0.874759 on the separability probability of real two-qubit density matrices. We are able to report general formulas for the m-th moment of the Hilbert-Schmidt probability distribution of the density matrix determinant over [0,1/256], in the real, complex and quaternionic two-qubit cases.