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arXiv:2001.01232

Archipelagos of Total Bound and Free Entanglement

Paul B. Slater

Published Jan 5, 2020• 1 citation
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Abstract

First, we considerably simplify an initially quite complicated formula -- involving dilogarithms. It yields the total bound entanglement probability ($\approx 0.0865542$) for a qubit-ququart ($2 \times 4$) 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 $\frac{1}{729} \left(473-512 \log \left(\frac{27}{16}\right) \left(1+\log \left(\frac{27}{16}\right)\right)\right) \approx 0.0890496$. Then, "downgrading" the qubit-ququart model to a two-qubit one, we find an archipelago of total non-bound/free entanglement probability $\frac{1}{2}$.

DOI
10.48550/arxiv.2001.01232

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The Hilbert Series of Measures of Entanglement for 4 Qubits

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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​.
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.
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.
Li-Qiao Entanglement Constraints Confirming the Entanglement Probability of 1327\frac{13}{27}2713​ for the Two-Qutrit Hiesmayr-Löffler Magic Simplex of Bell States
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.