Jagged Islands of Bound Entanglement and Witness-Parameterized   Probabilities

Paul B. Slater

arXiv:1905.09228·quant-ph·June 1, 2020

Jagged Islands of Bound Entanglement and Witness-Parameterized Probabilities

Paul B. Slater

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TL;DR

This paper investigates the probabilities of bound-entangled states within specific quantum state sets, using witness-parameterized families and geometric analysis, revealing precise probabilities and their relationships in two-qutrit and two-ququart systems.

Contribution

It introduces new witness-parameterized families of bound-entangled probabilities and provides exact calculations for these probabilities in high-dimensional quantum systems.

Findings

01

Bound-entangled probabilities are approximately 0.00737 (d=3) and 0.00052 (d=4).

02

Total entanglement probability is about 0.48148, combining bound and free entanglement.

03

PPT probabilities are approximately 0.5374 (d=3) and 0.4050 (d=4).

Abstract

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{\"o}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 $\frac{8 π}{27 3} \approx 0.537422$ ( $d = 3$ ) and $\frac{1}{2} + \frac{l o g ( 2 - 3 )}{8 3} \approx 0.404957$ ( $d = 4$ ). We obtain bound-entangled probabilities of $- \frac{4}{9} + \frac{4 π}{27 3} + \frac{l o g ( 3 )}{6} \approx 0.00736862$ and $\frac{- 204 + 7 l o g ( 7 ) + 168 3 c o s ^{- 1} ( \frac{11}{14} )}{1134} \approx 0.00325613$ ( $d = 3$ ) and $\frac{8 l o g ( 2 )}{27} - \frac{59}{288} \approx 0.00051583$ and $\frac{24 \text{csch}^{-1}\left(\frac{8}{\sqrt{17}}\right)}{17…

Tables1

Table 1. Table 1: Various Hilbert-Schmidt probabilities for the Hiesmayr-Löffler d = 3 𝑑 3 d=3 two-qutrit model. Notationally, ¬ \neg is the negation logic operator (NOT); ∧ \land is the conjunction logic operator (AND); and ∨ \lor is the disjunction logic operator (OR).

Constraint Imposed	Probability	Numerical Value
——-	1	1.
PPT	$\frac{8 π}{27 \sqrt{3}}$	0.537422
MUB	$\frac{1}{6}$	0.1666667
Choi	$\frac{1}{6}$	0.1666667
$PPT \land MUB$	$- \frac{4}{9} + \frac{4 π}{27 \sqrt{3}} + \frac{\log (3)}{6}$	0.00736862
$PPT \land Choi$	$- \frac{4}{9} + \frac{4 π}{27 \sqrt{3}} + \frac{\log (3)}{6}$	0.00736862
$MUB \land Choi$	$\frac{1}{9}$	0.11111
$MUB \lor Choi$	$\frac{2}{9}$	0.22222
$\neg MUB \land Choi$	$\frac{1}{18}$	0.05555
$MUB \land \neg Choi$	$\frac{1}{18}$	0.05555
$PPT \land \neg MUB$	$\frac{1}{162} (72 + 8 \sqrt{3} π - 27 \log (3))$	0.5300534
$PPT \land \neg Choi$	$\frac{1}{162} (72 + 8 \sqrt{3} π - 27 \log (3))$	0.5300534
$PPT \land MUB \land Choi$	0	0
$PPT \land (MUB \lor Choi)$	$- \frac{8}{9} + \frac{8 π}{27 \sqrt{3}} + \frac{\log (3)}{3}$	0.0147372
$\neg PPT \land MUB$	$\frac{1}{3} + \frac{22518 \sqrt{3}}{91} + \frac{3888 \sqrt{3}}{7 π} - \frac{10939 π}{27 \sqrt{3}} - \frac{\log (3)}{8}$	0.1592980
$\neg PPT \land Choi$	$\frac{1}{3} + \frac{22518 \sqrt{3}}{91} + \frac{3888 \sqrt{3}}{7 π} - \frac{10939 π}{27 \sqrt{3}} - \frac{\log (3)}{8}$	0.1592980
$\neg PPT \land \neg MUB$	$\frac{1}{162} (9 (7 + \log (27)) - 8 \sqrt{3} π)$	0.303279920
$\neg PPT \land \neg Choi$	$\frac{1}{162} (9 (7 + \log (27)) - 8 \sqrt{3} π)$	0.303279920
$\neg PPT \land \neg MUB \land \neg Choi$	$\frac{1}{9} (3 \log (3) - 1)$	0.255092985
$PPT \land \neg MUB \land \neg Choi$	$\frac{1}{9} (8 - 3 \log (3))$	0.5226847927
$PPT \lor (MUB \land Choi)$	$\frac{1}{81} (9 + 8 \sqrt{3} π)$	0.648533145

Equations144

ρ_{d} = \frac{q _{4} ( 1 - δ ( d - 3 )) \sum _{z = 2}^{d - 2} ( \sum _{i = 0}^{d - 1} P _{i, z} )}{d} + \frac{q _{2} \sum _{i = 1}^{d - 1} P _{i, 0}}{( d - 1 ) ( d + 1 )} + \frac{q _{3} \sum _{i = 0}^{d - 1} P _{i, 1}}{d} +

ρ_{d} = \frac{q _{4} ( 1 - δ ( d - 3 )) \sum _{z = 2}^{d - 2} ( \sum _{i = 0}^{d - 1} P _{i, z} )}{d} + \frac{q _{2} \sum _{i = 1}^{d - 1} P _{i, 0}}{( d - 1 ) ( d + 1 )} + \frac{q _{3} \sum _{i = 0}^{d - 1} P _{i, 1}}{d} +

\frac{( - \frac{q _{1}}{d ^{2} - d - 1} - \frac{q _{2}}{d + 1} - ( d - 3 ) q _{4} - q _{3} + 1 ) IdentityMatrix [ d ^{2} ]}{d ^{2}} + \frac{q _{1} P _{0, 0}}{d ^{2} - d - 1},

\frac{( - \frac{q _{1}}{d ^{2} - d - 1} - \frac{q _{2}}{d + 1} - ( d - 3 ) q _{4} - q _{3} + 1 ) IdentityMatrix [ d ^{2} ]}{d ^{2}} + \frac{q _{1} P _{0, 0}}{d ^{2} - d - 1},

c [0, 0] = \frac{1}{180} (32 q_{1} - 5 q_{2} - 20 q_{3} + 20) \equiv Q_{1},

c [0, 0] = \frac{1}{180} (32 q_{1} - 5 q_{2} - 20 q_{3} + 20) \equiv Q_{1},

c [k, l] = \frac{1}{180} (- 4 q_{1} - 5 q_{2} + 40 q_{3} + 20) \equiv Q_{2}

c [k, l] = \frac{1}{180} (- 4 q_{1} - 5 q_{2} + 40 q_{3} + 20) \equiv Q_{2}

c [k, l] = \frac{1}{360} (- 8 q_{1} + 35 q_{2} - 40 q_{3} + 40) \equiv Q_{3}

c [k, l] = \frac{1}{360} (- 8 q_{1} + 35 q_{2} - 40 q_{3} + 40) \equiv Q_{3}

{q_{1}, q_{2}, q_{3}} = {\frac{5}{3} (4 Q_{1} + 3 Q_{2} + 2 Q_{3} - 1), \frac{8}{3} (Q_{1} + 3 Q_{2} + 5 Q_{3} - 1), Q_{1} + 6 Q_{2} + 2 Q_{3} - 1},

{q_{1}, q_{2}, q_{3}} = {\frac{5}{3} (4 Q_{1} + 3 Q_{2} + 2 Q_{3} - 1), \frac{8}{3} (Q_{1} + 3 Q_{2} + 5 Q_{3} - 1), Q_{1} + 6 Q_{2} + 2 Q_{3} - 1},

4 q_{1} + 5 q_{2} < 40 q_{3} + 20 \land 10 q_{3} < 4 q_{1} + 5 q_{2} + 10 \land 4 q_{1} + 5 q_{2} + 20 q_{3} < 20

4 q_{1} + 5 q_{2} < 40 q_{3} + 20 \land 10 q_{3} < 4 q_{1} + 5 q_{2} + 10 \land 4 q_{1} + 5 q_{2} + 20 q_{3} < 20

\land 5 (q_{2} + 4 q_{3} - 4) < 32 q_{1} \land (8 q_{1} - 5 (7 q_{2} - 8 q_{3} + 8)) (8 q_{1} + q_{2} - 8 q_{3} + 8) < 0

\land 5 (q_{2} + 4 q_{3} - 4) < 32 q_{1} \land (8 q_{1} - 5 (7 q_{2} - 8 q_{3} + 8)) (8 q_{1} + q_{2} - 8 q_{3} + 8) < 0

Q_{1} > 0 \land Q_{2} > 0 \land Q_{3} > 0 \land Q_{1} + 3 Q_{2} + 2 Q_{3} < 1.

Q_{1} > 0 \land Q_{2} > 0 \land Q_{3} > 0 \land Q_{1} + 3 Q_{2} + 2 Q_{3} < 1.

Q_{1} > 0 \land Q_{3} > 0 \land Q_{1} + 3 Q_{2} + 2 Q_{3} < 1 \land Q_{1}^{2} + 3 Q_{2} Q_{1} + (3 Q_{2} + Q_{3})^{2} < 3 Q_{2} + 2 Q_{1} Q_{3} .

Q_{1} > 0 \land Q_{3} > 0 \land Q_{1} + 3 Q_{2} + 2 Q_{3} < 1 \land Q_{1}^{2} + 3 Q_{2} Q_{1} + (3 Q_{2} + Q_{3})^{2} < 3 Q_{2} + 2 Q_{1} Q_{3} .

Q_{1} > 3 Q_{2} + 4 Q_{3},

Q_{1} > 3 Q_{2} + 4 Q_{3},

3 c [0, 0] + c [0, 1] + 2 c [0, 2] + c [1, 1] + 2 c [1, 2] + c [2, 1] + 2 c [2, 2] > 2.

3 c [0, 0] + c [0, 1] + 2 c [0, 2] + c [1, 1] + 2 c [1, 2] + c [2, 1] + 2 c [2, 2] > 2.

W^{(+)} = \frac{1}{6} (i = 0 \sum 2 [2∣ ii ⟩ ⟨ ii ∣ + ∣ i, i - 1 ⟩ ⟨ i, i - 1∣] - 3 P_{+}),

W^{(+)} = \frac{1}{6} (i = 0 \sum 2 [2∣ ii ⟩ ⟨ ii ∣ + ∣ i, i - 1 ⟩ ⟨ i, i - 1∣] - 3 P_{+}),

W^{(+)}=\frac{1}{6}\left(\cdot\begin{array}[]{ *{3}{c} | *{3}{c} | *{3}{c} }1&0&0&0&-1&0&0&0&-1\\ 0&0&0&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0&0\\ \hline\cr 0&0&0&1&0&0&0&0&0\\ -1&0&0&0&1&0&0&0&-1\\ 0&0&0&0&0&0&0&0&0\\ \hline\cr 0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&1&0\\ -1&0&0&0&-1&0&0&0&1\\ \end{array}\right)

W^{(+)}=\frac{1}{6}\left(\cdot\begin{array}[]{ *{3}{c} | *{3}{c} | *{3}{c} }1&0&0&0&-1&0&0&0&-1\\ 0&0&0&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0&0\\ \hline\cr 0&0&0&1&0&0&0&0&0\\ -1&0&0&0&1&0&0&0&-1\\ 0&0&0&0&0&0&0&0&0\\ \hline\cr 0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&1&0\\ -1&0&0&0&-1&0&0&0&1\\ \end{array}\right)

2 Q_{3} + 1 - 2 Q_{1} - 3 Q_{2} < 0,

2 Q_{3} + 1 - 2 Q_{1} - 3 Q_{2} < 0,

1/6 (- c [0, 0] + c [0, 2] + 2 c [1, 0] + c [1, 2] + 2 c [2, 0] + c [2, 2]) < 0.

1/6 (- c [0, 0] + c [0, 2] + 2 c [1, 0] + c [1, 2] + 2 c [2, 0] + c [2, 2]) < 0.

a + b + c = 2, 0 \leq a \leq 1, b, c \geq 0, b c = (1 - a)^{2} .

a + b + c = 2, 0 \leq a \leq 1, b, c \geq 0, b c = (1 - a)^{2} .

a = \frac{2}{3} (cos (\frac{π i}{3}) + 1), b = \frac{2}{3} (- \frac{1}{2} 3 sin (\frac{π i}{3}) - \frac{1}{2} cos (\frac{π i}{3}) + 1),

a = \frac{2}{3} (cos (\frac{π i}{3}) + 1), b = \frac{2}{3} (- \frac{1}{2} 3 sin (\frac{π i}{3}) - \frac{1}{2} cos (\frac{π i}{3}) + 1),

c = \frac{2}{3} (\frac{1}{2} 3 sin (\frac{π i}{3}) - \frac{1}{2} cos (\frac{π i}{3}) + 1) .

c = \frac{2}{3} (\frac{1}{2} 3 sin (\frac{π i}{3}) - \frac{1}{2} cos (\frac{π i}{3}) + 1) .

\frac{- 204 + 7 lo g ( 7 ) + 168 3 cos ^{- 1} ( \frac{11}{14} )}{1134} \approx 0.00325612294236.

\frac{- 204 + 7 lo g ( 7 ) + 168 3 cos ^{- 1} ( \frac{11}{14} )}{1134} \approx 0.00325612294236.

- \frac{( a - 2 ) ^{3}}{9 a ^{2} - 30 a + 27} .

- \frac{( a - 2 ) ^{3}}{9 a ^{2} - 30 a + 27} .

- \frac{A + B}{54 ( 4 - 3 a ) ^{3/2} ( 2 a - 3 )},

- \frac{A + B}{54 ( 4 - 3 a ) ^{3/2} ( 2 a - 3 )},

A = 8 12 - 9 a (6 a^{2} - 17 a + 12) cos^{- 1} (\frac{a ( 3 a - 8 ) + 6}{6 - 4 a})

A = 8 12 - 9 a (6 a^{2} - 17 a + 12) cos^{- 1} (\frac{a ( 3 a - 8 ) + 6}{6 - 4 a})

B = 3 a (2 (9 a^{3} - 57 a^{2} + 108 a - 64) + 3 (3 - 2 a) a lo g (9 - 6 a)) .

B = 3 a (2 (9 a^{3} - 57 a^{2} + 108 a - 64) + 3 (3 - 2 a) a lo g (9 - 6 a)) .

\frac{1}{3}\left(\cdot\begin{array}[]{ *{3}{c} | *{3}{c} | *{3}{c} }4&0&0&0&-1&0&0&0&-1\\ 0&1&0&0&0&2&2&0&0\\ 0&0&1&2&0&0&0&2&0\\ \hline\cr 0&0&2&1&0&0&0&2&0\\ -1&0&0&0&4&0&0&0&-1\\ 0&2&0&0&0&1&2&0&0\\ \hline\cr 0&2&0&0&0&2&1&0&0\\ 0&0&2&2&0&0&0&1&0\\ -1&0&0&0&-1&0&0&0&4\\ \end{array}\right),

\frac{1}{3}\left(\cdot\begin{array}[]{ *{3}{c} | *{3}{c} | *{3}{c} }4&0&0&0&-1&0&0&0&-1\\ 0&1&0&0&0&2&2&0&0\\ 0&0&1&2&0&0&0&2&0\\ \hline\cr 0&0&2&1&0&0&0&2&0\\ -1&0&0&0&4&0&0&0&-1\\ 0&2&0&0&0&1&2&0&0\\ \hline\cr 0&2&0&0&0&2&1&0&0\\ 0&0&2&2&0&0&0&1&0\\ -1&0&0&0&-1&0&0&0&4\\ \end{array}\right),

α (- 3 α Q_{1} + (9 α - 3) Q_{2} + 6 α Q_{3} + Q_{1}) < 0,

α (- 3 α Q_{1} + (9 α - 3) Q_{2} + 6 α Q_{3} + Q_{1}) < 0,

α (3 α - 1) (2 Q_{1} + 3 Q_{2} - 1) > 2 α Q_{3},

α (3 α - 1) (2 Q_{1} + 3 Q_{2} - 1) > 2 α Q_{3},

\frac{1 - 3 α}{2 - 12 α} .

\frac{1 - 3 α}{2 - 12 α} .

\frac{1}{162 ( 6 ( 1 - 3 α ) α + 1 ) ^{2}}

\frac{1}{162 ( 6 ( 1 - 3 α ) α + 1 ) ^{2}}

83 π (6 (1 - 3 α) α + 1)^{2} - 18 (3 α - 1) (15 α + 2) (6 α (3 α - 1) - 1)

83 π (6 (1 - 3 α) α + 1)^{2} - 18 (3 α - 1) (15 α + 2) (6 α (3 α - 1) - 1)

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Full text

Jagged Islands of Bound Entanglement and Witness-Parameterized Probabilities

Paul B. Slater

[email protected]

Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106-4030

Abstract

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 Löffler of “magic” simplices of Bell states that were introduced by Baumgartner, Hiesmayr and Narnhofer. The Hilbert-Schmidt probabilities of positive-partial-transpose (PPT) states–within which we search for bound-entangled states–are $\frac{8\pi}{27\sqrt{3}}\approx 0.537422$ ( $d=3$ ) 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 $-\frac{4}{9}+\frac{4\pi}{27\sqrt{3}}+\frac{\log(3)}{6}\approx 0.00736862$ 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 $\frac{24\text{csch}^{-1}\left(\frac{8}{\sqrt{17}}\right)}{17\sqrt{17}}-\frac{91}{544}\approx 0.00218722$ ( $d=4$ ). (For $d=3$ , we also obtain $\frac{2}{81}\left(4\sqrt{3}\pi-21\right)\approx 0.0189035$ based on the realignment criterion. Thus, the total entanglement probability appears to equal $(1-\frac{8\pi}{27\sqrt{3}})+\frac{2}{81}\left(4\sqrt{3}\pi-21\right)=\frac{13}{27}\approx 0.481481$ .) The families, encompassing these results, are parameterized using generalized Choi and Jafarizadeh-Behzadi-Akbari witnesses. In the $d=3$ , analyses, we utilized the mutually unbiased bases (MUB) test of Hiesmayr and Löffler, and also the Choi $W^{(+)}$ test. 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 $\frac{7\pi}{25\sqrt{5}}\approx 0.39338962$ ) and 0.115732 (conjecturally, $\frac{1}{8}+\frac{\log\left(3-\sqrt{5}\right)}{13\sqrt{5}}\approx 0.115737$ ) for the original (8- [two-qutrit] and 15 [two-ququart]-dimensional) magic simplices themselves.

Hilbert-Schmidt measure, PPT-probabilities, bound entanglement

pacs:

Valid PACS 03.67.Mn, 02.50.Cw, 02.40.Ft, 02.10.Yn, 03.65.-w

I Introduction

In their landmark 1998 paper, “Volume of the set of separable states”, Życzkowski, Horodecki, Sanpera and Lewenstein stated: “As it was mentioned in the introduction for $N\geq 8$ there are states which are inseparable but have positive partial transposition. Moreover, it has been recently shown that all states of such type represent ‘bound’ entanglement in the sense that they cannot be distilled to the singlet form. The immediate question that arises is how frequently such peculiar states appear in the set of all the states of a given composite system. This question is related to the role of time reversal in the context of entanglement of mixed states….we provide a qualitative argument that the volume of the set of those states is also nonzero” (Życzkowski et al., 1998, sec. V).

We will here offer some quantitative arguments in this direction (cf. Bae et al. (2009)), where the sets of primary interest are the “magic” simplices of Bell states (sec. II) Baumgartner et al. (2006, 2008); Derkacz and Jakóbczyk (2007); Sentís et al. (2018)–for which it had been noted that the “Hilbert-Schmidt metric defines a natural metric on the space state” Baumgartner et al. (2006)–and generalized Horodecki states (sec. IV) Chruściński and Rutkowski (2011); Jafarizadeh et al. (2009).

II Magic Simplices Analyses

Within the magic simplex setting of Baumgartner, Hiesmayr and Narnhofer Baumgartner et al. (2006, 2008), the case of bound entanglement of two photonic qutrits using the orbital angular momentum degree of freedom of light was investigated in a 2013 paper of B. C. Hiesmayr and W. Löffler Hiesmayr and Löffler (2013). They noted that this was the simplest case of bound entanglement, with “complications, such as those arising in multipartite systems, not occurring”.

Their equation (7) for a density matrix $\rho_{d}$ in the ( $d^{2}-1$ )-dimensional simplices ( $d=3,4$ ) took the form,

[TABLE]

the $P_{i,j}$ ’s being orthonormal Bell states. (No explicit ranges were given for the $q$ ’s, and our initial analyses assumed that they would have to be nonnegative. But, it, then, seemed somewhat puzzling that a main example of Hiesmayr and Löffler employed negative $q$ ’s. Also, we observed that bound entanglement did not seem possible with strictly nonnegative $q$ ’s. Eventually, we arrived at the clearly powerful change-of-variables approach–to be shortly detailed–greatly facilitating the exact integrations we had been attempting.)

“This family also includes for $d=3$ the one-parameter Horodecki–state, the first found bound entangled state. Namely, for $q_{1}=\frac{30-5\lambda}{21},\;q_{2}=-\frac{8\lambda}{21},\;q_{3}=\frac{5-2\lambda}{7}$ with $\lambda\in[0,5]$ . This state is PPT for $\lambda\in[1,4]$ and was shown to be bound entangled for $\lambda\in\{3,4]$ ” Hiesmayr and Löffler (2013). Let us note now–as a prototypical example of our subsequent more demanding three- and four-parameter calculations–that the Hilbert-Schmidt PPT-probability for this one-parameter ( $\lambda$ ) Horodecki-state is $\frac{3}{5}$ , the probability of entanglement is also $\frac{3}{5}$ , and the bound-entangled probability, $\frac{1}{5}$ (cf. Chruściński and Rutkowski (2011), and sec. IV below).

II.1 Transformation between magic simplex parameters, and associated constraints

In the $d=3$ (two-qutrit) framework, we transform between the nine nonnegative parameters ( $c[k,l]\geq 0$ , with $k,l=0,\ldots 2$ ), summing to 1, employed for the full/original magic simplex of Baumgartner, Hiesmayr and Narnhofer (Baumgartner et al., 2008, sec. 4), and the three ( $q_{1},q_{2},q_{3}$ ) of the Hiesmayr-Löfller subset. To do so, we use the three equations,

[TABLE]

and

[TABLE]

for $\{k,l\}={\{0,1\},\{0,2\},\{1,1\},\{1,2\},\{2,1\},\{2,2\}}$ , and

[TABLE]

for $\{k,l\}={\{1,0\},\{2,0\}}$ . (For the indicated Horodecki-state, we have $\{Q_{1},Q_{2},Q_{3}\}=\left\{-\frac{2\lambda}{189}+\frac{2}{9\lambda}+\frac{2}{7},\frac{1}{189}\left(-11\lambda+\frac{42}{\lambda}+45\right),\frac{\lambda^{2}-21}{27\lambda}\right\}.)$

If we, then, employ $Q_{1},Q_{2}$ and $Q_{3}$ as our principal variables, rather than $q_{1},q_{2},q_{3}$ , using the linear transformations,

[TABLE]

our ensuing analyses simplify greatly. For example, the requirement that $\rho_{3}$ is a nonnegative definite density matrix–ensured by requiring that its nine leading nested minors all be nonnegative–is transformed from

[TABLE]

to

[TABLE]

In Fig. 1 we show the convex set of possible $\rho_{3}$ ’s, in terms of this parameterization.

Additionally, the constraint that the partial transpose of $\rho_{3}$ is nonnegative definite becomes

[TABLE]

(We report and employ the $d=4$ [two-ququart] analogues of the results (2)-(7) in sec. II.3.)

II.1.1 MUB test

Further, the Hiesmayr-Löffler mutually-unbiased-bases (MUB) criterion for bound entanglement, $I_{4}=\Sigma_{k=1}^{4}C_{A_{k},B_{k}}>2$ , where $C_{A_{k},B_{k}}$ are correlation functions for observables $A_{k},B_{k}$ (Hiesmayr and Löffler, 2013, Fig. 1) takes the form

[TABLE]

or, in terms of the original magic simplex parameters

[TABLE]

II.1.2 Choi test

Also, Example 2 in Bae et al. (2018) states that the “Choi EW $W^{(+)}$ obtained from the Choi map in $d=3$ …is given by

[TABLE]

where $\mathrm{P}_{+}=|\phi^{+}\rangle\langle\phi^{+}|$ with the Bell state $|\phi^{+}\rangle=(|00\rangle+|11\rangle+|22\rangle)/\sqrt{3}$ .” It was noted there that this witness (zeros are denoted by dots)

[TABLE]

is applicable in the two-qutrit setting. (A related operator $W^{(-)}$ is presented in Bae et al. (2018), but it is decomposable and, thus, not capable of detecting bound entanglement. Further, in our particular analysis here, no entanglement at all was detected with its use.) In the Hiesmayr-Löffler $d=3$ two-qutrit density-matrix setting (1), the entanglement requirement that $\mbox{Tr}[W\rho_{3}]<0$ takes the simple form

[TABLE]

or, in terms of the original magic simplex parameters,

[TABLE]

II.2 Two-qutrit analyses ( $d=3$ )

In Table 1, we report certain computations based on the preceding constraints. Hopefully, these provide insight into the underlying (Hilbert-Schmidt) geometry of the Hiesmayr-Löffler $d=3$ magic simplex model.

To begin, we obtained findings (Fig. 2, cf. kgl ) for the $d=3$ two-qutrit Hiesmayr-Löffler model that the PPT-probability is $\frac{8\pi}{27\sqrt{3}}\approx 0.537422$ , and that the entanglement probabilities revealed by the MUB and Choi tests are both $\frac{1}{6}\approx 0.16667$ .

Further, the MUB and Choi bound-entangled probabilities were each determined to equal $-\frac{4}{9}+\frac{4\pi}{27\sqrt{3}}+\frac{\log(3)}{6}\approx 0.00736862$ . So, the intersection of the PPT-set with either of the two sets accounts for less than one-percent of the total probability. (This smallness appears to be very much in line with that exhibited in Figure 3 in Hiesmayr and Löffler (2013)–also (Baumgartner et al., 2006, Figs. 2, 3).)

If we enforce both the MUB and Choi tests for entanglement, but not the PPT constraint, we obtain a probability of $\frac{1}{9}\approx 0.11111$ . This doubles to $\frac{2}{9}\approx 0.22222$ , if only one of the two tests needs to be met, again without PPT necessarily holding. (“An entanglement witness is an observable detecting entanglement for a subset of states. We present a framework that makes an entanglement witness twice [emphasis added] as powerful due to the general existence of a second (lower) bound, in addition to the (upper) bound of the very definition” Bae et al. (2018).) Continuing, then, the probability for either of the tests to be met, but not the other, is $\frac{1}{18}\approx 0.05555$ .

If, on the other hand, the PPT-constraint is satified, but both of the two entanglement tests are failed, the associated probability is $\frac{1}{9}(8-3\log(3))\approx 0.5226847927$ .

The intersection is void between those (bound-entangled) states satisfying both PPT and MUB criteria and those (bound-entangled) states satisfying both PPT and Choi criteria. (In their two-qutrit analysis, Gabuldin and Mandilara concluded that the bound-entangled states had “negligible volume and that these form tiny ‘islands’ sporadically distributed over the surface of the polytope of separable states” Gabdulin and Mandilara (2019). In a continuous variable study DiGuglielmo et al. (2011), “the tiny regions in parameter space where bound entanglement does exist” were noted” )

In Fig. 3 we show these non-intersecting ragged/craggy/jagged islands of bound-entangled states. (The notion of fractal-type behavior comes to mind.) The bound-entangled probabilities of $-\frac{4}{9}+\frac{4\pi}{27\sqrt{3}}+\frac{\log(3)}{6}\approx 0.00736862$ are equal to 36 times the depicted volumes.

In Fig. 4 we show the effect upon the calculated probability of allowing the right-hand side (zero) of the constraint (11) to instead vary. When the bound is zero–as in (11)–the indicated bound-entangled probability of $-\frac{4}{9}+\frac{4\pi}{27\sqrt{3}}+\frac{\log(3)}{6}\approx 0.00736862$ is obtained. If the bound is further loosened to as high as $\frac{43}{32}\approx 1.34375$ , where the associated constraint loses all effect, the curve attains the PPT-probability of $\frac{8\pi}{27\sqrt{3}}\approx 0.537422$ . At the other extreme, for a bound as strong as $-\frac{3}{16}\approx-0.1875$ , the probability yielded is simply zero.

In Fig. 5 we show the points lying at the boundary between the separable and bound-entanglement states based on the MUB witness, obtained by setting $Q_{2}=\frac{1}{3}(Q_{1}-4Q_{3})$ , so that the inequality constraint (8) becomes an equality.

II.2.1 Extension of Choi witness $W^{(+)}$ to $W^{(+)}(a)$

From the papers Chruściński et al. (2018); Ha and Kye (2011) we obtained a general (initially three-parameter ( $[a,b,c]$ ) set of entanglement witnesses $W^{(+)}(a)$ (with the Choi witness already employed corresponding to $a=1$ ). It was generated by replacing (making use of cyclical permutations of $a,b,c$ ) the diagonal entries of the witness (10) by $\{a,b,c,c,a,b,b,c,a\}$ . The conditions imposed on the three parameters are (Ha and Kye, 2011, eq. (1))

[TABLE]

To start our associated analyses, for $a=\frac{1}{4}\left(3-\sqrt{5}\right)$ , we found an entanglement probability of $\frac{5}{132}\left(5+\sqrt{5}\right)\approx 0.274093$ and a bound-entangled probability of 0.00149772192.

Further, we note that $a=\frac{1}{3}$ is the particular case $i=4$ of (Ha and Kye, 2011, eq. (1))

[TABLE]

and

[TABLE]

For $1\leq i\leq 5$ the corresponding witness is optimal Ha and Kye (2011).

Now, for $a=\frac{1}{3}$ , we have an entanglement probability of $\frac{125}{486}=\frac{5^{3}}{2\cdot 3^{5}}\approx 0.257202$ , with a bound-entangled probability of

[TABLE]

The case $a=1$ yields back our initial Choi witness $W^{(+)}$ analysis, for which a bound-entangled probability of $-\frac{4}{9}+\frac{4\pi}{27\sqrt{3}}+\frac{\log(3)}{6}\approx 0.00736862$ was obtained.

More, generally still, the entanglement probabilities for this class of witnesses $W^{(+)}(a)$ , are given by (Fig. 6)

[TABLE]

First one-parameter family of bound-entangled probabilities.

Further, the bound-entangled probability obtained by application of the family of generalized Choi witnesses $W^{(+)}(a)$ to the Hiesmayr-Löffler $d=3$ two-qutrit model is given by the expression (Fig. 7)

[TABLE]

where

[TABLE]

and

[TABLE]

In regard to this formula (17), C. Dunkl remarked: “I have one intuitive observation: the mixture of special functions (including trig) in the answers appears to imply that the boundaries of the sets you are measuring are complicated and have pieces of various properties (e.g. flat, curved …)”. As an expansion upon this remark (cf. Fig. 3)–suggestive of the jagged-island phenomenon–let us note that the bound-entangled probability function (17) was obtained by the integration of the value 36, firstly of $Q_{2}$ over the interval $[\frac{1}{6}\left(-Q_{1}-2Q_{3}+1\right)-\frac{1}{6}\sqrt{-3Q_{1}^{2}+12Q_{3}Q_{1}-2Q_{1}-4Q_{3}+1},\frac{1}{6}\left(-Q_{1}-2Q_{3}+1\right)-\frac{1}{6}\sqrt{-\frac{\left((a-2)\left(3Q_{1}-1\right)+6aQ_{3}\right){}^{2}}{a(3a-4)}}]$ , secondly of $Q_{3}$ over the interval $[0,\frac{a\left(\sqrt{\frac{(3a-4)\left(3Q_{1}-1\right)\left(3(3a-4)Q_{1}+4\right)}{a^{2}}}+3\right)+(15-9a)Q_{1}-5}{9a}]$ , and thirdly of $Q_{1}$ over $[\frac{-a^{2}+2a-1}{2a-3},\frac{1}{3}]$ .

In Fig. 8, we display the ratio of the bound-entangled probability (17) to the entanglement probability (16) .

II.2.2 Entanglement witnesses from mutually unbiased bases

Moving on to further forms of witnesses, we observed that for the entanglement witness given in (Chruściński et al., 2018, eq. (32)),

[TABLE]

there was no entanglement at all detected for the $d=3$ Hiesmayr-Löffler model under study here.

This last witness is the particular case $\phi_{1}=\phi_{2}=\pi,\phi_{3}=\phi_{4}=0$ of a family parameterized by an $L$ -dimensional torus ( $L=2,3,4$ , being the number of MUBs’ used in the construction).

As an exercise, we randomly generated values of $\phi_{1},\phi_{2},\phi_{3}$ and $\phi_{4}$ $\in[0,2\pi]$ –and evaluated their (bound and non-bound/free) entanglement probabilities with respect to the $d=3$ Hiesmayr-Löffler model. Neither of the probabilities generated exceeded the maximum ones indicated ( $\frac{8}{27}\approx 0.296296$ and $\approx 0.0082381739$ in Figs. 6 and 7).

II.2.3 Jafarizadeh-Behzadi-Akbari witnesses

In Jafarizadeh et al. (2009), M. A. Jafarizadeh, N. Behzadi, and Y. Akbari (JBA), constructed “[o]n the basis of linear programming, new sets of entanglement witnesses (EWs) for $3\otimes 3$ and $4\otimes 4$ systems”. In the $3\otimes 3$ case, their two witnesses $\mathcal{W}_{\alpha}$ and $\mathcal{W^{\prime}}_{\alpha}$ were defined over $\alpha\in[\frac{1}{3},\frac{2}{3}]$ (Jafarizadeh et al., 2009, eqs. (17), (18)).

The associated entanglement constraints for the $d=3$ Hiesmayr-Löffler model took the forms

[TABLE]

and

[TABLE]

respectively.

The entanglement probabilities for the two witnesses were identical (Fig. 9), being given by

[TABLE]

Second one-parameter family of bound-entangled probabilities.

The bound-entangled probability (Fig. 10) based on either of these two JBA witnesses is given by the product of

[TABLE]

and

[TABLE]

In Fig. 11, we jointly plot a rescaled version of this bound-entangled probability figure based on the JBA witnesses along with that earlier-derived one (Fig. 7) using the generalized Choi witnesses.

In Figs. 12 and 13, we show the probabilities of the intersections and unions of the bound-entangled regions revealed through use of the one-parameter families of generalized JBA and Choi witnesses.

In Fig. 14 we show a pair of jagged islands for $\alpha=\frac{3}{5}$ .

It was commented upon in Jafarizadeh et al. (2009) that the expectation value of $\mathcal{W}_{\alpha}$ “with respect to the all separable states is positive hence it can be an EW for $\alpha\in[\frac{1}{3},\frac{2}{3}]$ ”. The formula (22) was derived assuming these restrictions–but we have observed that if we take the limit $\alpha\rightarrow\infty$ , it attains the value

[TABLE]

Possibly this is not an achievable (or near-achievable) bound-entanglement probability measurement. For $\alpha=\frac{2}{3}$ , the formula (22) has the considerably lesser value $\frac{1}{162}\left(8\sqrt{3}\pi+9(\log(27)-8)\right)\approx 0.00736862$ .

II.3 Two-ququart analyses ( $d=4$ )

Similarly to our $d=3$ analyses (sec. II.1), in the $d=4$ (two-ququart) framework, to transform between the sixteen nonnegative parameters ( $c[k,l]\geq 0$ , with $k,l=0,\ldots 3$ ), summing to 1, for the full/original magic simplex (Baumgartner et al., 2008, sec. 4) and the four ( $q_{1},q_{2},q_{3},q_{4}$ ) of the Hiesmayr-Löfller subset, we must take

[TABLE]

and

[TABLE]

for $\{k,l\}={\{0,1\},\{1,1\},\{2,1\}},\{3,1\}$ , and

[TABLE]

for $\{k,l\}={\{0,2\},\{1,2\},\{2,2\},\{3,2\}}$ , and

[TABLE]

for $\{k,l\}={\{1,0\},\{2,0\},\{3,0\}}$ .

If we employ $Q_{1},Q_{2}$ $Q_{3},Q_{4}$ as our four principal variables, rather than $q_{1},q_{2},q_{3},q_{4}$ in the Hiesmayr-Löffler parameterizaion (1), using the linear transformations,

[TABLE]

and

[TABLE]

our ensuing analyses simplify greatly. The requirement that $\rho_{4}$ is a nonnegative definite density matrix–or, equivalently, that its sixteen leading nested minors are nonnegative–takes the form

[TABLE]

The constraint that the partial transpose of $\rho_{4}$ is nonnegative definite is

[TABLE]

With these formulas, we are able to establish that the corresponding PPT-probability is $\frac{1}{2}+\frac{\log\left(2-\sqrt{3}\right)}{8\sqrt{3}}\approx 0.404957$ (again, quite elegant, but seemingly of a different analytic form than the $d=3$ counterpart of $\frac{8\pi}{27\sqrt{3}}$ ).

We were not able originally to compute bound-entangled probabilities in this two-ququart framework, not being successful in attempting to extend the Hiesmayr-Löffler and Choi witnesses to that setting. (“The general case (even for $d=4$ ) is much more involved and the general structure of circulant entanglement witnesses is not known” Chruściński and Wudarski (2011).)

II.3.1 Chruściński witnesses

However, Dariuz Chruściński subsequently provided the particular entanglement witness.

[TABLE]

The constraint required for bound entanglement that $\mbox{Tr}[W_{2ququarts}\rho_{4}]<0$ , then, takes the form

[TABLE]

The Hilbert-Schmidt entanglement probability that the Hiesmayr-Løffler density matrix $\rho_{4}$ –given by (1)–satisfies this constraint is simply $\frac{2}{9}$ .

Joining the constraints (29), (30) and (32), we attempted the corresponding exact four-dimensional integration for the bound-entangled probability. Mathematica was able to reduce it to a clearly challenging one-dimensional integration. However, we were apparently able to obviate this formidable task by doing a numerical integration using a WorkingPrecision $\rightarrow 24$ option. Inputting the result obtained to the WolframAlpha website, an exact value of

[TABLE]

was suggested, which matched the numerical output to considerably more than twenty decimal places. (As a matter of, at least, initial curiosity, the WolframAlpha site also suggested–to equally high-precision–an exact value of $\frac{1}{864}(-64b_{4}(2)-177)$ , where “ $b_{4}(2)$ is the Madelung constant $b_{4}(2)$ ”. Equating the two exact suggested values implies that $b_{4}(2)=-4\log(2)$ . We could not immediately confirm this identity, but the mathworld.wolfram.com Madelung Constants page informs us that, seemingly relatedly, $b_{2}(2)=-\pi\log(2)$ .) At a later point, though, N. Tessore was able to fully confirm the validity of (33) through strictly symbolic integration .

D. Chruściński also indicated that a modification of $W_{2ququarts}$ could be achieved by replacing the sixteen diagonal entries of (31) with the sequence $\{2,1,0,0,0,2,1,0,0,0,2,1,1,0,0,2\}$ . The associated entanglement constraint is, then,

[TABLE]

Doing so, leads to a reduced (from $\frac{2}{9}$ ) entanglement (without PPT required) probability of $\frac{1}{8}$ , but a substantially increased bound-entangled probability of Vas

[TABLE]

This is 4.24019 times greater than the bound-entangled probability, $\frac{8\log(2)}{27}-\frac{59}{288}$ , obtained with the initially used two-ququart witness (31). (For several efforts to graphically represent the results of the immediately preceding set of two-ququart analyses, along the lines of Fig. 2, see kgl –in particular, the animation there–obtained by variation of the $Q$ ’s–of MichaelE2.)

The joint imposition of the two last constraints (32) and (34) gives a (lesser) bound-entanglement probability of

[TABLE]

It would be of interest to embed this last pair of two-ququart entanglement witnesses provided by Chruściński into a parameterized family, similarly to the case of the two-qutrit entanglement witness $W^{(+)}(a)$ analyzed in sec. II.2.1. But to do so, however, seemed quite involved (Chruściński and Wudarski, 2012a, sec. 3).

But, in Chruściński and Wudarski (2012b), we noted the presentation of a set of (nd-)optimal entanglement witnesses for $a,b,c,d\geq 0,a+b+c+d=3$ (cf. (Chruściński, 2014, Example 5.2)),

[TABLE]

Two classes of parameter constraints were considered,

[TABLE]

and

[TABLE]

For neither of these classes, did we detect any bound entanglement with respect to the $d=4$ Hiesmayr-Löffler system. The entanglement probability (cf.(16)) in class I does take the form (Fig. 15),

[TABLE]

and in class II (Fig. 16),

[TABLE]

These two functions are both equal to $\frac{625}{4992}\approx 0.1252$ at $a=\frac{1}{2}$ . The intersection of the two entanglement tests at that point, then, yields a probability of $\frac{512}{4125}\approx 0.124121212$ , while their union gives $\frac{512}{3135}\approx 0.163317384$ .

At $a=1$ , the intersection of the two tests yields a probability of $\frac{2}{16335}\approx 0.000122436$ , while their union gives $\frac{32}{165}\approx 0.193939$ . At that same point, the case I entanglement probability is $\frac{2}{27}\approx 0.0740741$ , and the case II entanglement probability is $\frac{1}{9}\approx 0.11111$ .

In Figs. 17 and 18, we show the entanglement probabilities arising from the intersection and union of classes I and II, while in Fig. 19, we show the ratio of the union curve to the intersection curve. (We were able to obtain a formula for the former curve–but quite large in nature.)

To conclude this section, let us note that for the witness $W_{4}$ specified by (Chruściński and Sarbicki, 2014, eq. (8.4)), we obtained an entanglement (without PPT-requirement) probability of $\frac{8}{25}\approx 0.32$ for the $d=4$ Hiesmayr-Löffler states, but no (with PPT-requirement) bound-entanglement. Further, for the witness $W_{4}^{spin}$ specified by (Chruściński and Sarbicki, 2014, eq. (8.26)), no entanglement at all was detected.

II.3.2 JBA witnesses

Analogously to our analyses in sec. II.2.3, we constructed for the $d=4$ Hiesmayr-Löffler model, the entanglement constraints for the pair of Jafarizadeh-Behzadi-Akbari witnesses $\mathcal{W}_{\alpha}$ and $\mathcal{W^{\prime}}_{\alpha}$ given in (Jafarizadeh et al., 2009, eqs.(38) and (39)). These took the forms,

[TABLE]

and

[TABLE]

respectively.

The entanglement probability based on either $\mathcal{W}_{\alpha}$ or $\mathcal{W^{\prime}}_{\alpha}$ took the form

[TABLE]

In Fig. 20, we show this probability curve.

The (intersection) probability that both entanglement constraints are satisfied is $\frac{1-4\alpha}{3(8\alpha-1)}$ , while the (union) probability that at least one of the two is satisfied is $\frac{2-8\alpha}{3-24\alpha}$ .

Third one-parameter family of bound-entangled probabilities.

Now, in Fig. 21, we show the bound-entangled probability based on either the two-ququart JBA witnesses $\mathcal{W}_{\alpha}$ or $\mathcal{W^{\prime}}_{\alpha}$ . This is given by

[TABLE]

where

[TABLE]

and

[TABLE]

At $\alpha=\frac{3}{8}\approx 0.375$ (outside $[\frac{1}{4},\frac{1}{3}]$ ), the value is identical to the bound-entangled probability $\frac{24\text{csch}^{-1}\left(\frac{8}{\sqrt{17}}\right)}{17\sqrt{17}}-\frac{91}{544}\approx 0.00218722$ , reported above in eq. (35), with respect to the indicated modification of the entanglement witness $W_{2ququarts}$ (31). For given $\alpha\in[\frac{1}{4},\frac{1}{3}]$ , the two witnesses $\mathcal{W}_{\alpha}$ and $\mathcal{W^{\prime}}_{\alpha}$ appear to form disjoint bound entanglement islands (cf. Fig. 3).

In Fig. 22 we show such a pair of islands for $\alpha=\frac{7}{24}$ and $Q_{4}=\frac{1}{384}$ ,

while in Fig. 23 we show such a pair of islands for $\alpha=\frac{1}{3}$ and $Q_{1}=\frac{1}{6}$ .

The formula (45) was derived assuming $\alpha\in[\frac{1}{4},\frac{1}{3}]$ –as seemed suggested in Jafarizadeh et al. (2009) (the authors claiming that the witnesses are non-decomposable [nd] there). Nevertheless, it appears to hold–as for $\alpha=\frac{3}{8}$ –as well for $\alpha>\frac{1}{3}$ . In the limit $\alpha\rightarrow\infty$ , it attains the value

[TABLE]

We are not aware of whether or not this is an achievable (or near-achievable) bound-entanglement probability measurement. But, for $\alpha=\frac{1}{3}$ , the formula (45) has the considerably lesser value $\frac{-92575+\frac{7798329\cosh^{-1}\left(\frac{12}{11}\right)}{\sqrt{23}}-\frac{13484361\cosh^{-1}\left(\sqrt{\frac{23}{22}}\right)}{\sqrt{23}}}{389344}\approx 0.00162026$ .

III Realignment analyses of Hiesmayr-Löffler magic simplices

III.1 Two-qutrit ( $d=3$ ) case

Application of the realignment (CCNR) test for entanglement Chen and Wu (2002); Shang et al. (2018) yielded an entanglement probability of $\frac{1}{81}\left(27+\sqrt{3}\log\left(97+56\sqrt{3}\right)\right)\approx 0.445977$ and an exact bound-entangled probability of $\frac{2}{81}\left(4\sqrt{3}\pi-21\right)\approx 0.0189035$ , considerably larger than the 0.00736862 and 0.00325613 reported above. The realignment constraint that, if satisfied, ensures entanglement is

[TABLE]

The bound-entanglement islands obtained by enforcing this constraint are displayed in Fig. 24. (It is interesting to note that the region of free entanglement also, in fact, separates into two–not as severely jagged–islands of its own.) These realignment islands completely contain the corresponding Choi and MUB islands, with an additional probability of $\frac{1}{27}(10-9\log(3))\approx 0.00416627$ .

This (highly fragmented) domain of additional bound-entangled probability found through use of the realignment criterion, but by neither the Choi nor the MUB witnesses is shown in Fig. 25.

We also implemented the entanglement detection criterion (ESIC) based on symmetric informationally complete positive operator-valued measures (SIC POVMs) Shang et al. (2018). We found it to be equivalent to the realignment (CCNR) test.

Motivated by the recent preprint, “Bound entangled states fit for robust experimental verification” Sentís et al. (2018), we found that the CCNR reached a maximum of $\frac{7}{6}$ at $Q_{1}=\frac{1}{4},Q_{2}=\frac{1}{24}(3-\sqrt{5}),Q_{3}=0$ . (In the original Hiesmayr-Löffler coordinates, this converts to $q_{1}=\frac{5}{24}\left(\sqrt{5}-3\right),q_{2}=-1-\frac{\sqrt{5}}{3},q_{3}=-\frac{\sqrt{5}}{4}$ .) It would be interesting to further analyze this result in the framework of Sentís et al. (2018), to ascertain how robust–in terms of their criteria–this state is for experimental verification. The (rank-7) density matrix in question takes the form (zeros being represented by dots)

[TABLE]

However, undermining the initially presumed robustness of this state is the observation that three of the eigenvalues of its partial transpose are zero, so any perturbation of the state might lead to negative eigenvalues of the partial transpose, and thus its departure from the PPT domain.

III.2 Two-ququart ( $d=4$ ) case

Here, we observed free entanglement and bound-entangled probability CCNR-based estimates of 0.4509440211445637 and 0.01265489845176, respectively.

IV Generalized Horodecki state analyses

At the conclusion of their paper Jafarizadeh et al. (2009), Jafarizadeh, Behzadi, and Akbari consider generalized Horodecki states of the form $\rho_{i}=\Sigma_{i=1}^{n}a_{i}O_{i}$ , where the $O_{i}$ ’s are unit-trace orthogonal operators. These were used as a basis for constructing EWs, in particular, the pair of witnesses $\mathcal{W}_{\alpha}$ and $\mathcal{W^{\prime}}_{\alpha}$ we have employed above in certain two-qutrit (sec. II.2.3) and two-ququart (sec. II.3.2) analyses. (Let us also note that Chruściński and Rutkowski have provided a multi-parameter family of 2-qudit PPT entangled states which generalize the celebrated Horodecki-state Chruściński and Rutkowski (2011).)

IV.1 Two-qutrit case

Following the suggested approach of JBA for obtaining generalized Horodecki states, in the two-qutrit instance, we have found a PPT-probability of $\frac{1}{27}\left(4\sqrt{3}\pi-9\right)\approx 0.4278$ . The entanglement probability with either of the two witnesses is $\frac{1}{2}$ . The bound-entangled probability for either witness is $\frac{1}{27}\left(2\sqrt{3}\pi-9\right)\approx 0.0697322$ for $\alpha\in[\frac{1}{3},\frac{2}{3}]$ .

IV.2 Two-ququart case

In the two-ququart generalized Horodecki case, the PPT-probability is $\frac{1}{3}\approx 0.3333$ . The entanglement probability with either witness is $\frac{1}{2}$ –again, independently of $\alpha$ . The bound-entangled probability with either witness is $\frac{1}{24}\approx 0.0416667$ for $\alpha\in[\frac{1}{4},\frac{1}{3}]$ .

In Fig. 26 we show a pair of jagged bound-entanglement islands. Six times the displayed volume for each island gives the bound-entanglement probability of $\frac{1}{24}$ . (These plots are independent of $\alpha$ .)

IV.3 Two-ququint case

For the $5\times 5$ two-ququint ( $25\times 25$ density matrices) scenario, the entanglement probability is simply again $\frac{1}{2}$ , independently of $\alpha$ , for the two witnesses. (This case is not explicitly discussed in Jafarizadeh et al. (2009), and we have followed their discussions of the previous lower-dimensional instances.) The PPT-probability is approximately 0.33734924124312192527. The bound-entangled probability is $\approx 0.0370662$ for $\alpha>\frac{1}{5}$ .

It would be of interest to investigate the relations between the JBA generalized Horodecki states we have analyzed and those put forth by Chruściński and Rutkowski Chruściński and Rutkowski (2011).

V Full-dimensional magic simplices numerical analyses

V.1 Two-qutrits

Moving on from the $d=3$ Hiesmayr-Löffler model, we examined the original 8-dimensional “magical” simplex of Bell states of bipartite qutrits studied by Baumgartner, Hiesmayr and Narnhofer Baumgartner et al. (2006, 2008).

In Fig. 27, we plot–using the interesting (golden-ratio-related) “quasi-random” procedure (we have been recently employing Slater (2019)) of Martin Roberts Rob (a, b)–two sets of estimates of the associated Hilbert-Schmidt PPT-probability. One (symmetrically) fixes the Roberts parameter $\alpha_{0}$ at $\frac{1}{2}$ (which may lead to superior estimates), and the other at zero. Large numbers of realizations–5,820,000,000 realizations in the former case, and 6,260,000,000 in the latter–were used. The two estimates obtained were 0.39338785 and 0.39339143, respectively. A well-fitting conjecture for the underlying exact value–of a seemingly similar nature to the $d=3$ Hiesmayr-Löffler counterpart of $\frac{8\pi}{27\sqrt{3}}$ –is $\frac{7\pi}{25\sqrt{5}}\approx 0.3933896249$ . (In particular, the average of the two estimates appears to very strongly converge in this direction–with the last recorded average agreeing with the conjecture to six decimal places.)

In a supplementary analysis to these two, now using a Roberts parameter of $\alpha_{0}=\frac{1}{3}$ , together with 280,000,000 million realizations, we obtained an estimate of 0.393381, plus, additionally, an estimate of 0.00011335 for the associated bound-entangled probability based on the Hiesmayr-Löffler mutually-unbiased-bases criterion (9) of

[TABLE]

Let us here recall that for the Hiesmayr-Löffler $d=3$ counterpart, the bound-entangled probability was found to be considerably greater, that is, $-\frac{4}{9}+\frac{4\pi}{27\sqrt{3}}+\frac{\log(3)}{6}\approx 0.00736862$ , than 0.000113354.

V.2 Two-ququarts

For the 15-dimensional magic simplex of Bell states of bipartite ququarts, two Hilbert-Schmidt PPT-probability quasi-random-based estimates, employing approximately one hundred seventy million realizations each, were 0.115717 ( $\alpha_{0}=\frac{1}{2}$ ) and 0.115778 ( $\alpha_{0}=0$ ). The Hiesmayr-Löffler counterpart is $\frac{1}{2}+\frac{\log\left(2-\sqrt{3}\right)}{8\sqrt{3}}\approx 0.404957$ .

At this point–due to certain technical issues–we undertook a parallel set of analyses (however, shifting the Roberts parameters from 0 and $\frac{1}{2}$ to $\frac{1}{4}$ and $\frac{3}{4}$ ). Additionally, now we recorded whether or not the realignment test Chen and Wu (2002) for entanglement was passed. In Fig. 28, we plot the two sets of estimates of the associated Hilbert-Schmidt PPT-probability. An interesting candidate value is $\frac{1}{8}+\frac{\log\left(3-\sqrt{5}\right)}{13\sqrt{5}}\approx 0.115737$ .

Further, in Fig. 29, we plot the two sets of estimates of the associated Hilbert-Schmidt entanglement probability based on the realignment criterion.

Also, in Fig. 30, we display the pairs of estimates of the Hilbert-Schmidt bound-entangled probability based on the realignment criterion. A conjecture of $\frac{1}{750}\approx 0.00133333$ can be advanced, in this regard.

VI Further discussion

It has been established that for the two-rebit, rebit-retrit and two-retrit $X$ -states (the density matrices for which, by definition, have their only nonzero entries along their diagonal and anti-diagonal Rau (2018)), the Hilbert-Schmidt separability/PPT probabilities are all equal to $\frac{16}{3\pi^{2}}\approx 0.54038$ for the two-rebit, rebit-retrit and two-retrit $X$ -states (cf. Dunkl and Slater (2015)). Numerical and exact analyses of ours strongly indicated that among the ( $9\times 9$ ) two-retrit PPT-states, none is bound entangled in terms of the Hiesmayr-Löffler MUB $I_{d}>2$ criterion. But the question of whether there are bound-entangled $X$ -states should be addressed more thoroughly than so far has been done.

There certainly are many more directions in which efforts to determine probabilities of bound-entangled states can be directed (cf. (Życzkowski, 1999, sec.IV.C) for a $2\times 4$ density matrix analysis, and Zhou et al. (2018) for multipartitite issues).

Much research has been devoted to the determination of Hilbert-Schmidt (and other–Bures, monotone) separability and PPT-probabilities Slater (2018a, b, 2019) (and references therein), but considerably less so, it would seem, as we have attempted here, to the bound-entangled situation. (Perhaps we can regard the Horodecki-state bound-entangled probability of $\frac{1}{5}$ noted above, as the initial result in this area of research.) But of interest in these respects, is the paper Bae et al. (2009), in which there was derived “an explicit analytic estimate for the entanglement of a large class of bipartite quantum states, which extends into bound entanglement regions”.

As to the full 35-dimensional set of two-qutrit states itself, evidence has been presented indicating that the associated Hilbert-Schmidt PPT-probability–on the order of 0.0001027 (Slater, 2019, Fig. 9)–is constant over the Casimir invariants of their qutrit subsystems (Slater, 2016a, sec.III.A). (Also, a “repulsion” effect for the Casimir invariants has been observed in the two-qubit case, where the invariants are the Bloch radii of the individual qubits Slater (2016b). The two-qutrit case was also examined there.)

In an auxiliary full 63-dimensional $2\times 4$ qubit-qudit analysis, based on 1,200 million iterations, use of the realignment criterion Chen and Wu (2002) yielded an estimate of 0.00023410917 for the bound-entangled probability and 0.94234319 (conjecturally, $\frac{589}{625}=\frac{17\cdot 31}{5^{4}}\approx 0.94234616$ ) for for the entanglement probability, in general. In that (quasirandom Hilbert-Schmidt) analysis, we were not able to detect any finite probability at all of genuinely tripartite entanglement using the Greenberger-Horne-Zeilinger test set out in Example 3 in Bae et al. (2018). (However, in a parallel full 80-dimensional two-qutrit study, the realignment test for entanglement was not passed by any randomly generated states (cf.Gabdulin and Mandilara (2019)).)

As a final remark, let us conjecture that the ”nonsmooth/jagged” nature of the boundaries of regions of bound entanglement reported above (Figs 3, 14, 22, 23 and 26), will, in some sense–remaining to be made precise–diminishes with increasing dimensions of, say, bipartite systems (cf. Beigi and Shor (2010)).

VII Compliance with Ethical Standards

The author asserts that he has no conflicts of interest or potential such conflicts. The research reported did not involve human participants and/or animals.

Appendix A Generalized Horodecki-Werner States

A.1 Two-qutrit case

Let us consider the states that are composed of equally-weighted two-qutrit generalized Horodecki ones (in the sense discussed [sec. IV.1]) and the fully mixed two-qutrit state.

Then the PPT-probability for this set is

[TABLE]

The entanglement probability for either Jafarizadeh-Behzadi-Akbari (JBA) witness $\mathcal{W}_{\alpha}$ or $\mathcal{W^{\prime}}_{\alpha}$ is given by

[TABLE]

The bound-entangled probability for either witness is

[TABLE]

for $\alpha\in[\frac{1}{309}\left(71-6\sqrt{17}\right)\approx 0.149713,\frac{1}{6}]$ and

[TABLE]

for $\alpha\in[\frac{1}{6},\frac{11}{39}]$ . (In these last two intervals, all the entanglement probability is bound–as can be seen from (51).)

A.2 Two-ququart case

Now, for the class of equally-weighted generalized Horodecki two-ququart states (in the above JBA sense again [sec. IV.2]) and the fully mixed two-ququart state, the PPT-probability is

[TABLE]

The entanglement probability for either JBA witness is

[TABLE]

In Fig. 31, we show this function over the interval $\alpha\in\frac{1}{8},\frac{19}{48}]$ along with the entanglement probabilities based on the union and intersection of application of the two witnesses.

In a somewhat indirect fashion, We have been able to establish–through examination of numerous specific values of $\alpha$ and certain auxiliary analyses–that the bound-entangled probability function for either witness takes the form (Fig. 32)

[TABLE]

over $\alpha\in[1/872(161-18\sqrt{7})\approx 0.130019,\frac{11}{56}\approx 0.196429]$ . The nature of the function outside this interval remains under examination.

A.3 Two-ququints

The entanglement probability for this still higher-dimensional equally-weighted case for either $\mathcal{W}_{\alpha}$ or $\mathcal{W^{\prime}}_{\alpha}$ is

[TABLE]

The PPT-probability is $\approx 0.758301$ . At $\frac{1}{6}$ , the bound-entangled probability is, interestingly, $\frac{6561}{781250}=\frac{3^{8}}{2\cdot 5^{8}}\approx 0.00839808$ . It jumps at $\alpha=\frac{1}{7}$ to 0.1646314041874492= 0.0119320240085435+ $\frac{8}{625}+\frac{2008781-157760\sqrt{145}}{1953125}+\frac{72\left(696\sqrt{145}-7925\right)}{390625}$ .

In Fig. 33, we show a numerically-derived plot of this function over $\alpha\in[0,\frac{1}{6}]$ .

An important question that remains is whether outside the intervals employed by Jafarizadeh, Behzadi, and Akbari, $\mathcal{W}_{\alpha}$ and $\mathcal{W^{\prime}}_{\alpha}$ can serve as entanglement witnesses–and, thus, whether a number of probabilities presented above in this section that were termed “entanglement” and “bound-entangled” are. in fact, so.

Acknowledgements.

This research was supported by the National Science Foundation under Grant No. NSF PHY-1748958. Nicolas Tessore greatly assisted in the carrying out of many of the calculations reported above. I also strongly thank Dariuz Chruściński for sharing his entanglement-witness expertise.

Bibliography35

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1Życzkowski et al. (1998) K. Życzkowski, P. Horodecki, A. Sanpera, and M. Lewenstein, Physical Review A 58 , 883 (1998).
2Bae et al. (2009) J. Bae, M. Tiersch, S. Sauer, F. de Melo, F. Mintert, B. Hiesmayr, and A. Buchleitner, Physical Review A 80 , 022317 (2009).
3Baumgartner et al. (2006) B. Baumgartner, B. C. Hiesmayr, and H. Narnhofer, Physical Review A 74 , 032327 (2006).
4Baumgartner et al. (2008) B. Baumgartner, B. C. Hiesmayr, and H. Narnhofer, Physics Letters A 372 , 2190 (2008).
5Derkacz and Jakóbczyk (2007) L. Derkacz and L. Jakóbczyk, ar Xiv preprint ar Xiv:0707.1575 (2007).
6Sentís et al. (2018) G. Sentís, J. N. Greiner, J. Shang, J. Siewert, and M. Kleinmann, ar Xiv preprint ar Xiv:1804.07562 (2018).
7Chruściński and Rutkowski (2011) D. Chruściński and A. Rutkowski, Physics Letters A 375 , 2793 (2011).
8Jafarizadeh et al. (2009) M. Jafarizadeh, N. Behzadi, and Y. Akbari, The European Physical Journal D 55 , 197 (2009).

TL;DR

Contribution

Findings

Abstract

Peer Reviews

Videos

Jagged Islands of Bound Entanglement and Witness-Parameterized Probabilities

Abstract

pacs:

I Introduction

II Magic Simplices Analyses

II.1 Transformation between magic simplex parameters, and associated constraints

II.1.1 MUB test

II.1.2 Choi test

II.2 Two-qutrit analyses (d=3d=3d=3)

II.2.1 Extension of Choi witness W(+)W^{(+)}W(+) to W(+)(a)W^{(+)}(a)W(+)(a)

First one-parameter family of bound-entangled probabilities.

II.2.2 Entanglement witnesses from mutually unbiased bases

II.2.3 Jafarizadeh-Behzadi-Akbari witnesses

Second one-parameter family of bound-entangled probabilities.

II.3 Two-ququart analyses (d=4d=4d=4)

II.3.1 Chruściński witnesses

II.3.2 JBA witnesses

Third one-parameter family of bound-entangled probabilities.

III Realignment analyses of Hiesmayr-Löffler magic simplices

III.1 Two-qutrit (d=3d=3d=3) case

III.2 Two-ququart (d=4d=4d=4) case

IV Generalized Horodecki state analyses

IV.1 Two-qutrit case

IV.2 Two-ququart case

IV.3 Two-ququint case

V Full-dimensional magic simplices numerical analyses

V.1 Two-qutrits

V.2 Two-ququarts

VI Further discussion

VII Compliance with Ethical Standards

Appendix A Generalized Horodecki-Werner States

A.1 Two-qutrit case

A.2 Two-ququart case

A.3 Two-ququints

Acknowledgements.

II.2 Two-qutrit analyses ( $d=3$ )

II.2.1 Extension of Choi witness $W^{(+)}$ to $W^{(+)}(a)$

II.3 Two-ququart analyses ( $d=4$ )

III.1 Two-qutrit ( $d=3$ ) case

III.2 Two-ququart ( $d=4$ ) case