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arXiv:2002.04084

Archipelagos of Total Bound and Free Entanglement. II

Paul B. Slater

Published Feb 10, 2020• 0 citations
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Abstract

In the indicated preceding preprint (I), we reported the results of, in particular interest here, certain three-parameter qubit-ququart ($2 \times 4$) and two-ququart ($4 \times 4$) analyses. In them, we relied upon entanglement constraints given by Li and Qiao. However, further studies of ours conclusively show--using the well-known necessary and sufficient conditions for positive-semidefiniteness that all leading minors (of separable components, in this context) be nonnegative--that certain of the constraints given are flawed and need to be replaced (by weaker ones). Doing so, leads to a new set of results, somewhat qualitatively different and, in certain respects, simpler in nature. For example, bound-entanglement probabilities of $\frac{2}{3} \left(\sqrt{2}-1\right) \approx 0.276142$, $\frac{1}{4} \left(3-2 \log ^2(2)-\log (4)\right) \approx 0.1632$, $\frac{1}{2}-\frac{2}{3 π^2} \approx 0.432453$ and $\frac{1}{6}$, are reported for various implementations of constraints. We also adopt the Li-Qiao three-parameter framework to a two-parameter one, with interesting visual results.

DOI
10.48550/arxiv.2002.04084

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Archipelagos of Total Bound and Free Entanglement

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Exact entanglement bases and general bound entanglement

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In this paper has been withrawn by the author due the error in the proof of theoem 1.

arXiv:quant-ph/0409080•Sep 14, 2004•0 citations
In the indicated preceding preprint (I), we reported the results of, in particular interest here, certain three-parameter qubit-ququart (2×42 \times 42×4) and two-ququart (4×44 \times 44×4) analyses. In them, we relied upon entanglement constraints given by Li and Qiao. However, further studies of ours conclusively show--using the well-known necessary and sufficient conditions for positive-semidefiniteness that all leading minors (of separable components, in this context) be nonnegative--that certain of the constraints given are flawed and need to be replaced (by weaker ones). Doing so, leads to a new set of results, somewhat qualitatively different and, in certain respects, simpler in nature. For example, bound-entanglement probabilities of 23(2−1)≈0.276142\frac{2}{3} \left(\sqrt{2}-1\right) \approx 0.27614232​(2​−1)≈0.276142, 14(3−2log⁡2(2)−log⁡(4))≈0.1632\frac{1}{4} \left(3-2 \log ^2(2)-\log (4)\right) \approx 0.163241​(3−2log2(2)−log(4))≈0.1632, 12−23π2≈0.432453\frac{1}{2}-\frac{2}{3 π^2} \approx 0.43245321​−3π22​≈0.432453 and 16\frac{1}{6}61​, are reported for various implementations of constraints. We also adopt the Li-Qiao three-parameter framework to a two-parameter one, with interesting visual results.
First, we considerably simplify an initially quite complicated formula -- involving dilogarithms. It yields the total bound entanglement probability (≈0.0865542\approx 0.0865542≈0.0865542) for a qubit-ququart (2×42 \times 42×4) three-parameter model, recently analyzed for its separability properties by Li and Qiao. An "archipelago" of disjoint bound-entangled regions appears in the space of parameters, somewhat similarly to those recently found in our preprint, "Jagged Islands of Bound Entanglement and Witness-Parameterized Probabilities". There, two-qutrit and two-ququart Hiesmayr-L{ö}ffler "magic simplices" and generalized Horodecki states had been examined. However, contrastingly, in the present study, the entirety of bound entanglement--given by the formula obtained--is clearly captured in the archipelago found. Further, we "upgrade" the qubit-ququart model to a two-ququart one, for which we again find a bound-entangled archipelago, with its total probability simply being now 1729(473−512log⁡(2716)(1+log⁡(2716)))≈0.0890496\frac{1}{729} \left(473-512 \log \left(\frac{27}{16}\right) \left(1+\log \left(\frac{27}{16}\right)\right)\right) \approx 0.08904967291​(473−512log(1627​)(1+log(1627​)))≈0.0890496. Then, "downgrading" the qubit-ququart model to a two-qubit one, we find an archipelago of total non-bound/free entanglement probability 12\frac{1}{2}21​.
We report several witness-parameterized families of bound-entangled probabilities. Two pertain to the d=3d=3d=3 (two-qutrit) and a third to the d=4d=4d=4 (two-ququart) subsets analyzed by Hiesmayr and L{ö}ffler of "magic" simplices of Bell states. The Hilbert-Schmidt probabilities of positive-partial-transpose (PPT) states--within which we search for bound-entangled states--are 8π273≈0.537422\frac{8 π}{27 \sqrt{3}} \approx 0.537422273​8π​≈0.537422 (d=3d=3d=3) and 12+log⁡(2−3)83≈0.404957\frac{1}{2}+\frac{\log \left(2-\sqrt{3}\right)}{8 \sqrt{3}} \approx 0.40495721​+83​log(2−3​)​≈0.404957 (d=4d=4d=4). We obtain bound-entangled probabilities of −49+4π273+log⁡(3)6≈0.00736862-\frac{4}{9}+\frac{4 π}{27 \sqrt{3}}+\frac{\log (3)}{6} \approx 0.00736862−94​+273​4π​+6log(3)​≈0.00736862 and −204+7log⁡(7)+1683cos⁡−1(1114)1134≈0.00325613\frac{-204+7 \log (7)+168 \sqrt{3} \cos ^{-1}\left(\frac{11}{14}\right)}{1134} \approx 0.003256131134−204+7log(7)+1683​cos−1(1411​)​≈0.00325613 (d=3d=3d=3) and 8log⁡(2)27−59288≈0.00051583\frac{8 \log (2)}{27}-\frac{59}{288} \approx 0.00051583278log(2)​−28859​≈0.00051583 and 24csch−1(817)1717−91544≈0.00218722\frac{24 \text{csch}^{-1}\left(\frac{8}{\sqrt{17}}\right)}{17 \sqrt{17}}-\frac{91}{544} \approx 0.002187221717​24csch−1(17​8​)​−54491​≈0.00218722 (d=4d=4d=4). Thus, the total entanglement probability appears to equal (1−8π273)+281(43π−21)=1327≈0.481481(1-\frac{8 π}{27 \sqrt{3}})+\frac{2}{81} \left(4 \sqrt{3} π-21\right) = \frac{13}{27} \approx 0.481481(1−273​8π​)+812​(43​π−21)=2713​≈0.481481.) The families, encompassing these results, are parameterized using generalized Choi and Jafarizadeh-Behzadi-Akbari witnesses. The same bound-entangled probability was achieved with both--the sets ("jagged islands") detected having void intersection. The entanglement (bound and "non-bound"/"free") probability for both was 16≈0.16667\frac{1}{6} \approx 0.1666761​≈0.16667, while their union and intersection gave 29≈0.22222\frac{2}{9} \approx 0.2222292​≈0.22222 and 19≈0.11111\frac{1}{9} \approx 0.1111191​≈0.11111. Further, we examine generalized Horodecki states, as well as estimating PPT-probabilities of approximately 0.39339 (very well-fitted by 7π255≈0.39338962\frac{7 π}{25 \sqrt{5}} \approx 0.39338962255​7π​≈0.39338962) and 0.115732 ( for the original (8- [two-qutrit] and 15 [two-ququart]-dimensional) magic simplices themselves.
We obtain formulas (bot. p. 12)--including 2121\frac{2}{121}1212​ and 4(2423π−1311)9801\frac{4 \left(242 \sqrt{3} π-1311\right)}{9801}98014(2423​π−1311)​--for the eight atoms (Fig. 11), summing to 1, which span a 256-dimensional three-set (P, S, PPT) entanglement-probability boolean algebra for the two-qutrit Hiesmayr-Loffler states. PPT denotes positive partial transpose, while P and S provide the Li-Qiao necessary and}sufficient conditions for entanglement. The constraints ensuring entanglement are s>169≈1.7777s> \frac{16}{9} \approx 1.7777s>916​≈1.7777 and p>227318⋅715⋅13≈5.61324⋅10−15p> \frac{2^{27}}{3^{18} \cdot 7^{15} \cdot13} \approx 5.61324 \cdot 10^{-15}p>318⋅715⋅13227​≈5.61324⋅10−15. Here, sss is the square of the sum (Ky Fan norm) of the eight singular values of the 8×88 \times 88×8 correlation matrix in the Bloch representation, and ppp, the square of the product of the singular values. In the two-ququart Hiesmayr-Loffler case, one constraint is s>94≈2.25s>\frac{9}{4} \approx 2.25s>49​≈2.25, while 3242134≈1.2968528306⋅10−29\frac{3^{24}}{2^{134}} \approx 1.2968528306 \cdot 10^{-29}2134324​≈1.2968528306⋅10−29 is an upper bound on the appropriate ppp value, with an entanglement probability ≈0.607698\approx 0.607698≈0.607698. The SSS constraints, in both cases, prove equivalent to the well-known CCNR/realignment criteria. Further, we detect and verify--using software of A. Mandilara--pseudo-one-copy undistillable (POCU) negative partial transposed two-qutrit states distributed over the surface of the separable states. Additionally, we study the best separable approximation problem within this two-qutrit setting, and obtain explicit decompositions of separable states into the sum of eleven product states. Numerous quantities of interest--including the eight atoms--were, first, estimated using a quasirandom procedure.
We seek to derive the probability—expressed in terms of the Hilbert–Schmidt (Euclidean or flat) metric—that a generic (nine-dimensional) real two-qubit system is separable, by implementing the well-known Peres–Horodecki test on the partial transposes (PTs) of the associated 4 × 4 density matrices (ρ). But the full implementation of the test—requiring that the determinant of the PT be nonnegative for separability to hold—appears to be, at least presently, computationally intractable. So, we have previously implemented—using the auxiliary concept of a diagonal-entry-parameterized separability function (DESF)—the weaker implied test of nonnegativity of the six 2 × 2 principal minors of the PT. This yielded an exact upper bound on the separability probability of . Here, we piece together (reflection-symmetric) results obtained by requiring that each of the four 3 × 3 principal minors of the PT, in turn, be nonnegative, giving an improved/reduced upper bound of . Then, we conclude that a still further improved upper bound of can be found by similarly piecing together the (reflection-symmetric) results of enforcing the simultaneous nonnegativity of certain pairs of the four 3 × 3 principal minors. Numerical simulations—as opposed to exact symbolic calculations—indicate, on the other hand, that the true probability is certainly less than . Our analyses lead us to suggest a possible form for the true DESF, yielding a separability probability of , while the absolute separability probability of provides the best exact lower bound established so far. In deriving our improved upper bounds, we rely repeatedly upon the use of certain integrals over cubes that arise. Finally, we apply an independence assumption to a pair of DESFs that comes close to reproducing our numerical estimate of the true separability function.