Table 1 Summary of various features of uniformly sampled n-partite pure states of local dimension d according to the Haar measure.

From: Entanglement transitivity problems

(n,d)

Nsample(×103)

NPT (%)

PPT (%)

NPT ⇒ NPT (%)

PPT ⇒ NPT (%)

NPT + PPT ⇒ NPT (%)

\({{{\rm{Max}}}}(1-{{{\mathcal{F}}}})\)

\(1-{{{\mathcal{F}}}} < \epsilon ( \% )\)

\(1-{{{\mathcal{F}}}} < 1{0}^{-6}\) among ⇒ NPT (%)

(3,2)

1000

100

0

100

0 (-)

0 (-)

1.21 × 10−9

100; 100; 100

100

(3,3)

100

100

0

100

0 (-)

0 (-)

1.73 × 10−6

25.51; 69.55; 99.99

99.99

(3,4)

10

100

0

100

0 (-)

0 (-)

1.33 × 10−6

72.77; 85.47; 99.84

99.84

(3,5)

10

100

0

100

0 (-)

0 (-)

1.29 × 10−6

83.64; 94.31; 99.79

99.79

(4,2)

100

46.74

2.64

7.32 (15.66)

0.02 (0.87)

3.36 (6.64)

1†

0.29; 1.50; 3.20

26.18

(4,3)

10

99.93

0

0 (0)

0 (-)

0 (0)

1†

0.00; 0.00; 0.00

—

(5,2)

10

0.35

45.75

0 (0)

0 (0)

0 (0)

1†

0.00; 0.00; 0.00

—

(5,3)

1.030

0.10

54.85

0 (0)

0 (0)

0 (0)

1†

0.00; 0.00; 0.00

—

  1. The second column gives the number of pure states sampled Nsample in each scenario (n, d). The next two columns list the fraction of states giving (n−1) neighboring two-body marginals that are, respectively, all NPT (i.e., none of which being PPT) and all PPT. The next three columns summarize how generic the phenomenon of (meta)transivitiy is among such states when the target system T lie at the two ends of an n-body chain. We give from left to right, respectively, the fraction among all sampled states exhibiting transitivity (i.e., with only entangled marginals), metatransitivity with only separable marginals, and metatransitivity with mixed marginals. Enclosed in each bracket is the corresponding fraction among samples having the associated kind of marginals. The next two columns summarize the extent to which the (n−1) two-body marginals lead to a unique global pure state. These are expressed in terms of the largest value of the infidelity \(1-{{{\mathcal{F}}}}\), where \({{{\mathcal{F}}}}=\mathop{\min }\nolimits_{{\rho }_{{{{\rm{S}}}}}}\left\langle \psi \right|{\rho }_{{{{\rm{S}}}}}\left|\psi \right\rangle\) and \(\left|\psi \right\rangle\) is the sampled pure state; the three numbers listed in the second last column are, respectively, for ϵ = 10−8, 10−7, and 10−6. The final column shows the fraction of (meta)transitivity examples having a unique global state (with an infidelity threshold set to 10−6). Throughout, we use 1† to represent a number that differs from 1 by less than 10−8.
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