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arXiv:2004.06745

8 Boolean Atoms Spanning the 256-Dimensional Entanglement-Probability Three-Set Algebra of the Two-Qutrit Hiesmayr-Loffler Magic Simplex of Bell States

Paul B. Slater

Published Jan 1, 2020• 1 citation
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Abstract

We obtain formulas (bot. p. 12)--including $\frac{2}{121}$ and $\frac{4 \left(242 \sqrt{3} π-1311\right)}{9801}$--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 $s> \frac{16}{9} \approx 1.7777$ and $p> \frac{2^{27}}{3^{18} \cdot 7^{15} \cdot13} \approx 5.61324 \cdot 10^{-15}$. Here, $s$ is the square of the sum (Ky Fan norm) of the eight singular values of the $8 \times 8$ correlation matrix in the Bloch representation, and $p$, the square of the product of the singular values. In the two-ququart Hiesmayr-Loffler case, one constraint is $s>\frac{9}{4} \approx 2.25$, while $\frac{3^{24}}{2^{134}} \approx 1.2968528306 \cdot 10^{-29}$ is an upper bound on the appropriate $p$ value, with an entanglement probability $\approx 0.607698$. The $S$ 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.

DOI
10.13140/rg.2.2.25790.10561

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

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•Apr 14, 2020•0 citations

Jagged Islands of Bound Entanglement and Witness-Parameterized Probabilities

Paul B. Slater

We report several witness-parameterized families of bound-entangled probabilities. Two pertain to the $d=3$ (two-qutrit) and a third to the $d=4$ (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 $\frac{8 π}{27 \sqrt{3}} \approx 0.537422$ ($d=3$) and $\frac{1}{2}+\frac{\log \left(2-\sqrt{3}\right)}{8 \sqrt{3}} \approx 0.404957$ ($d=4$). We obtain bound-entangled probabilities of $-\frac{4}{9}+\frac{4 π}{27 \sqrt{3}}+\frac{\log (3)}{6} \approx 0.00736862$ and $\frac{-204+7 \log (7)+168 \sqrt{3} \cos ^{-1}\left(\frac{11}{14}\right)}{1134} \approx 0.00325613$ ($d=3$) and $\frac{8 \log (2)}{27}-\frac{59}{288} \approx 0.00051583$ and $\frac{24 \text{csch}^{-1}\left(\frac{8}{\sqrt{17}}\right)}{17 \sqrt{17}}-\frac{91}{544} \approx 0.00218722$ ($d=4$). Thus, the total entanglement probability appears to equal $(1-\frac{8 π}{27 \sqrt{3}})+\frac{2}{81} \left(4 \sqrt{3} π-21\right) = \frac{13}{27} \approx 0.481481$.) 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 $\frac{1}{6} \approx 0.16667$, while their union and intersection gave $\frac{2}{9} \approx 0.22222$ and $\frac{1}{9} \approx 0.11111$. Further, we examine generalized Horodecki states, as well as estimating PPT-probabilities of approximately 0.39339 (very well-fitted by $\frac{7 π}{25 \sqrt{5}} \approx 0.39338962$) and 0.115732 ( for the original (8- [two-qutrit] and 15 [two-ququart]-dimensional) magic simplices themselves.

arXiv:1905.09228•Jan 1, 2019•4 citations

Archipelagos of Total Bound and Free Entanglement

Paul B. Slater

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 $\frac{1}{2}$.

arXiv:2001.01232•Jan 5, 2020•1 citation

Archipelagos of Total Bound and Free Entanglement. II

Paul B. Slater

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.

arXiv:2002.04084•Feb 10, 2020•0 citations

On the separability of Bell-diagonal states with absolutely positive partial transposition

Ayan Patra, Beatrix C. Hiesmayr

Absolutely positive partial transpose (APPT) states, which remain PPT under every global unitary, provide a spectral relaxation of absolutely separable states, which remain separable under every such transformation. Despite substantial progress, whether these two notions coincide beyond qubit-qudit systems remains unresolved, motivating the study of structured state families in which the APPT spectral constraints and separability can be analyzed exactly. We study this problem for Bell-diagonal states in two settings: the standard two-qutrit Weyl-Heisenberg basis and the two-ququart lattice basis generated by two-qubit Pauli operators. In the two-qutrit case, we show that a linearized necessary condition for APPT defines a polytope of spectra in which every associated Weyl-Heisenberg Bell-diagonal state admits an exact decomposition into explicitly separable states. This yields a separability criterion that applies to a region even larger than the APPT set itself. In the two-ququart lattice Bell-diagonal setting, we show that a state remains PPT under every permutation of its Bell coefficients if and only if its six largest Bell coefficients sum to at most 1/2, and we prove that all such states are separable. Equivalently, every entangled lattice Bell-diagonal state becomes NPT after a suitable coefficient permutation and is therefore not APPT. The proofs combine convex-polytope analysis, affine and symplectic orbit reductions, and exact rational decompositions into separable states supported on affine Lagrangian subsets. Our results rule out entangled APPT states in both Bell-diagonal families, and provide simple spectral inequalities as sufficient criteria for separability in both cases. Being basis-specific, however, they do not by themselves settle whether APPT and absolute separability coincide.

quant-pharXiv:2610.12000•Oct 9, 2026•0 citations

Qubit-Qudit Separability/PPT-Probability Analyses and Lovas-Andai Formula Extensions to Induced Measures

Paul B. Slater

We begin by seeking the qubit-qutrit and rebit-retrit counterparts to the now well-established Hilbert-Schmidt separability probabilities for (the 15-dimensional convex set of) two-qubits of $\frac{8}{33} = \frac{2^3}{3 \cdot 11} \approx 0.242424$ and (the 9-dimensional) two-rebits of $\frac{29}{64} =\frac{29}{2^6} \approx 0.453125$. Based in part on extensive numerical computations, we advance the possibilities of a qubit-qutrit value of $\frac{27}{1000} = (\frac{3}{10})^3 =\frac{3^3}{2^3 \cdot 5^3} = 0.027$ and a rebit-retrit one of $\frac{860}{6561} =\frac{2^2 \cdot 5 \cdot 43}{3^8} \approx 0.131078$. These four values for $2 \times m$ systems ($m=2,3$) suggest certain numerator/denominator sequences involving powers of $m$, which we further investigate for $m>3$. Additionally, we find that the Hilbert-Schmidt separability/PPT-probabilities for the two-rebit, rebit-retrit and two-retrit $X$-states all equal $\frac{16}{3 π^2} \approx 0.54038$, 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 generalized two-qubit framework introduced by Lovas and Andai from Hilbert-Schmidt measures to induced ones. For instance, while the Lovas-Andai two-qubit function is $\frac{1}{3} \varepsilon^2 (4 -\varepsilon^2)$, yielding $\frac{8}{33}$, its $k=1$ induced measure counterpart is $\frac{1}{4} \varepsilon ^2 \left(3-\varepsilon ^2\right)^2$, yielding $\frac{61}{143} =\frac{61}{11 \cdot 13} \approx 0.426573$, where $\varepsilon$ is a singular-value ratio. We investigate, in these regards, the possibility of extending the previously-obtained "Lovas-Andai master formula".

arXiv:1803.10680•Mar 28, 2018•1 citation
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.
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 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.
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.
Absolutely positive partial transpose (APPT) states, which remain PPT under every global unitary, provide a spectral relaxation of absolutely separable states, which remain separable under every such transformation. Despite substantial progress, whether these two notions coincide beyond qubit-qudit systems remains unresolved, motivating the study of structured state families in which the APPT spectral constraints and separability can be analyzed exactly. We study this problem for Bell-diagonal states in two settings: the standard two-qutrit Weyl-Heisenberg basis and the two-ququart lattice basis generated by two-qubit Pauli operators. In the two-qutrit case, we show that a linearized necessary condition for APPT defines a polytope of spectra in which every associated Weyl-Heisenberg Bell-diagonal state admits an exact decomposition into explicitly separable states. This yields a separability criterion that applies to a region even larger than the APPT set itself. In the two-ququart lattice Bell-diagonal setting, we show that a state remains PPT under every permutation of its Bell coefficients if and only if its six largest Bell coefficients sum to at most 1/2, and we prove that all such states are separable. Equivalently, every entangled lattice Bell-diagonal state becomes NPT after a suitable coefficient permutation and is therefore not APPT. The proofs combine convex-polytope analysis, affine and symplectic orbit reductions, and exact rational decompositions into separable states supported on affine Lagrangian subsets. Our results rule out entangled APPT states in both Bell-diagonal families, and provide simple spectral inequalities as sufficient criteria for separability in both cases. Being basis-specific, however, they do not by themselves settle whether APPT and absolute separability coincide.
We begin by seeking the qubit-qutrit and rebit-retrit counterparts to the now well-established Hilbert-Schmidt separability probabilities for (the 15-dimensional convex set of) two-qubits of 833=233⋅11≈0.242424\frac{8}{33} = \frac{2^3}{3 \cdot 11} \approx 0.242424338​=3⋅1123​≈0.242424 and (the 9-dimensional) two-rebits of 2964=2926≈0.453125\frac{29}{64} =\frac{29}{2^6} \approx 0.4531256429​=2629​≈0.453125. Based in part on extensive numerical computations, we advance the possibilities of a qubit-qutrit value of 271000=(310)3=3323⋅53=0.027\frac{27}{1000} = (\frac{3}{10})^3 =\frac{3^3}{2^3 \cdot 5^3} = 0.027100027​=(103​)3=23⋅5333​=0.027 and a rebit-retrit one of 8606561=22⋅5⋅4338≈0.131078\frac{860}{6561} =\frac{2^2 \cdot 5 \cdot 43}{3^8} \approx 0.1310786561860​=3822⋅5⋅43​≈0.131078. These four values for 2×m2 \times m2×m systems (m=2,3m=2,3m=2,3) suggest certain numerator/denominator sequences involving powers of mmm, which we further investigate for m>3m>3m>3. Additionally, we find that the Hilbert-Schmidt separability/PPT-probabilities for the two-rebit, rebit-retrit and two-retrit XXX-states all equal 163π2≈0.54038\frac{16}{3 π^2} \approx 0.540383π216​≈0.54038, as well as more generally, that the probabilities based on induced measures are equal across these three sets of XXX-states. Then, we extend the generalized two-qubit framework introduced by Lovas and Andai from Hilbert-Schmidt measures to induced ones. For instance, while the Lovas-Andai two-qubit function is 13ε2(4−ε2)\frac{1}{3} \varepsilon^2 (4 -\varepsilon^2)31​ε2(4−ε2), yielding 833\frac{8}{33}338​, its k=1k=1k=1 induced measure counterpart is 14ε2(3−ε2)2\frac{1}{4} \varepsilon ^2 \left(3-\varepsilon ^2\right)^241​ε2(3−ε2)2, yielding 61143=6111⋅13≈0.426573\frac{61}{143} =\frac{61}{11 \cdot 13} \approx 0.42657314361​=11⋅1361​≈0.426573, where ε\varepsilonε is a singular-value ratio. We investigate, in these regards, the possibility of extending the previously-obtained "Lovas-Andai master formula".