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arXiv:1311.4447

A Balanced $_{10}F_{9}$ Hypergeometric Hybrid Hilbert-Schmidt/Bures Two-Qubit Function and Related Constructions

Paul B. Slater

Published Nov 18, 2013• 0 citations
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Abstract

We seek to develop a Bures (minimal monotone/statistical distinguishability) metric-based series of formulas for the moments of probability distributions over the determinants $|ρ|$ and $|ρ^{PT}|$ of $4 \times 4$ density matrices, $ρ$, for generalized (rebit, quater[nionic]bit,\ldots) two-qubit systems, analogous to a series that has been obtained for the Hilbert-Schmidt (HS) metric. In particular, we desire--using moment-inversion procedures--to be able to closely test the previously-developed conjecture (J. Geom. Phys., 53, 74 [2005]) that the Bures separability probability over the (standard, fifteen-dimensional convex set of) two-qubit states is $\frac{1680 \left(\sqrt{2}-1\right)}{π^8} \approx 0.0733389$--while, in the HS context, strong evidence has been adduced, along the indicated analytical lines, that the counterpart of this value is $\frac{8}{33}$ (J. Phys. A}, 45, 095305 [2012]). Working within the "utility function" framework of Dunkl employed in that latter study, we obtain an interesting $_{10}F_{9}$ balanced hypergeometric function based on a "hybridization" of known Bures and HS terms. This exercise appears to provide an upper bound on the Bures two-qubit separability probability of 0.0798218. We also examine the yet unresolved HS qubit-qutrit scenario. Mathematica calculations indicate that if the same form of hypergeometric paradigm as has been established for the generalized two-qubit HS moments is followed in either the HS qubit-qutrit or Bures two-qubit cases, then a balanced hypergeometric function $_{p}F_{p-1}$ with $p>9$ would be required.

DOI
10.48550/arxiv.1311.4447

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A Balanced 10F9_{10}F_{9}10​F9​ Hypergeometric Hybrid Hilbert-Schmidt/Bures Two-Qubit Function and Related Constructions
We seek to develop a Bures (minimal monotone/statistical distinguishability) metric-based series of formulas for the moments of probability distributions over the determinants ∣ρ∣|ρ|∣ρ∣ and ∣ρPT∣|ρ^{PT}|∣ρPT∣ of 4×44 \times 44×4 density matrices, ρρρ, for generalized (rebit, quater[nionic]bit,\ldots) two-qubit systems, analogous to a series that has been obtained for the Hilbert-Schmidt (HS) metric. In particular, we desire--using moment-inversion procedures--to be able to closely test the previously-developed conjecture (J. Geom. Phys., 53, 74 [2005]) that the Bures separability probability over the (standard, fifteen-dimensional convex set of) two-qubit states is 1680(2−1)π8≈0.0733389\frac{1680 \left(\sqrt{2}-1\right)}{π^8} \approx 0.0733389π81680(2​−1)​≈0.0733389--while, in the HS context, strong evidence has been adduced, along the indicated analytical lines, that the counterpart of this value is 833\frac{8}{33}338​ (J. Phys. A}, 45, 095305 [2012]). Working within the "utility function" framework of Dunkl employed in that latter study, we obtain an interesting 10F9_{10}F_{9}10​F9​ balanced hypergeometric function based on a "hybridization" of known Bures and HS terms. This exercise appears to provide an upper bound on the Bures two-qubit separability probability of 0.0798218. We also examine the yet unresolved HS qubit-qutrit scenario. Mathematica calculations indicate that if the same form of hypergeometric paradigm as has been established for the generalized two-qubit HS moments is followed in either the HS qubit-qutrit or Bures two-qubit cases, then a balanced hypergeometric function pFp−1_{p}F_{p-1}p​Fp−1​ with p>9p>9p>9 would be required.
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.
We report major advances in the research program initiated in 'Moment-based evidence for simple rational-valued Hilbert–Schmidt generic 2 × 2 separability probabilities' (Slater and Dunkl 2012 J. Phys. A: Math. Theor. 45 095305). A highly succinct separability probability function P (α) is put forth, yielding for generic (nine-dimensional) two-rebit systems, , (15-dimensional) two-qubit systems, and (27-dimensional) two-quater(nionic)bit systems, . This particular form of P (α) was obtained by Qing-Hu Hou by applying Zeilberger's algorithm ('creative telescoping') to a fully equivalent—but considerably more complicated—expression containing six 7 F 6 hypergeometric functions (all with argument ). That hypergeometric form itself had been obtained using systematic, high-accuracy probability-distribution-reconstruction computations. These employed 7501 determinantal moments of partially transposed 4 × 4 density matrices, parameterized by . From these computations, exact rational-valued separability probabilities were discernible. The (integral/half-integral) sequences of 32 rational values then served as input to the Mathematica FindSequenceFunction command, from which the initially obtained hypergeometric form of P (α) emerged.
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 investigate the possibility of deriving analytical formulas for the 15-dimensional separable volumes, in terms of any of a number of metrics of interest (Hilbert-Schmidt [HS], Bures,...), of the two-qubit (four-level) systems. This would appear to require 15-fold symbolic integrations over a complicated convex body (defined by both separability and feasibility constraints). The associated 15-dimensional integrands -- in terms of the Tilma-Byrd-Sudarshan Euler-angle-based parameterization of the 4 x 4 density matrices ρ(math-ph/0202002) -- would be the products of 12-dimensional Haar measure μ_{Haar} (common to each metric) and 3-dimensional measures μ_{metric} (specific to each metric) over the 3d-simplex formed by the four eigenvalues of ρ. We attempt here to estimate/determine the 3-dimensional integrands (the products of the various [known] μ_{metric}'s and an unknown symmetric weighting function W) remaining after the (putative) 12-fold integration of μ_{Haar} over the twelve Euler angles. We do this by fitting W so that the conjectured HS separable volumes and hyperareas (quant-ph/0410238; cf. quant-ph/0609006) are reproduced. We further evaluate a number of possible choices of W by seeing how well they also yield the conjectured separable volumes for the Bures, Kubo-Mori, Wigner-Yanase and (arithmetic) average monotone metrics and the conjectured separable Bures hyperarea (quant-ph/0308037,Table VI). We, in fact, find two such exact (rather similar) choices that give these five conjectured (non-HS) values all within 5%. In addition to the above-mentioned Euler angle parameterization of ρ, we make extensive use of the Bloore parameterization (J. Phys. A 9 [1976], 2059) in a companion set of two-qubit separability analyses.
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 find equivalent hypergeometric- and difference-equation-based formulas, Q(k,α)=G1k(α)G2k(α)Q(k,α)= G_1^k(α) G_2^k(α)Q(k,α)=G1k​(α)G2k​(α), for k=−1,0,1,…,9k = -1, 0, 1,\ldots,9k=−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∣|ρ^{PT}| >|ρ|∣ρPT∣>∣ρ∣ holds. Here ρρρ denotes a 4×44 \times 44×4 density matrix and ρPTρ^{PT}ρPT, its partial transpose, while ααα is a Dyson-index-like parameter with α=1α= 1α=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)4 \times (4 +k)4×(4+k). For the symmetric case of k=0k=0k=0, we obtain the previously reported Hilbert-Schmidt formulas, with (the two-re[al]bit case) Q(0,12)=29128Q(0,\frac{1}{2}) = \frac{29}{128}Q(0,21​)=12829​, (the standard two-qubit case) Q(0,1)=433Q(0,1)=\frac{4}{33}Q(0,1)=334​, and (the two-quater[nionic]bit case) Q(0,2)=13323Q(0,2)= \frac{13}{323}Q(0,2)=32313​. The factors G2k(α)G_2^k(α)G2k​(α) can be written as the sum of weighted hypergeometric functions pFp−1_{p}F_{p-1}p​Fp−1​, p≥7p \geq 7p≥7, all with argument 2764=(34)3\frac{27}{64} =(\frac{3}{4})^36427​=(43​)3. We find formulas for the upper and lower parameter sets of these functions and, then, equivalently express G2k(α)G_2^k(α)G2k​(α) in terms of first-order difference equations. The factors G1k(α)G_1^k(α)G1k​(α) are equal to (2764)α−1(\frac{27}{64})^{α-1}(6427​)α−1 times ratios of products of six Pochhammer symbols involving the indicated parameters. Some remarkable α−α-α− and kkk-specific invariant asymptotic properties (again, involving 2764\frac{27}{64}6427​ and related quantities) of separability probability formulas emerge.