Composition of PPT Maps

Kennedy, Matthew; Manor, Nicholas; Paulsen, Vern I.

doi:10.48550/arxiv.1710.08475

Cited by 1 publication

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“…Due to the Choi-Jamiołkowski isomorphism [11,24] between quantum states and quant um channels, there is an equivalent 'channel' form of the PPT square conjecture given by Bäuml [7, lemma 14] and Christandl [29]: if Φ and Ψ are PPT quantum channels, then their composition Φ • Ψ must be entanglement breaking. Recently, Kennedy et al showed that the PPT square conjecture holds asymptotically; namely, they proved that the distance between the iteration of any PPT channel and the set of all entanglement breaking channels goes to zero [25]. This result has been improved by Rahaman et al [28], where they showed that every unital PPT channel becomes entanglement breaking after a finite number of iterations.…”

Section: Introductionmentioning

confidence: 99%

The PPT square conjecture holds generically for some classes of independent states

Collins¹,

Yin²,

Zhong³

2018

J. Phys. A: Math. Theor.

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Let |ψ ψ| be a random pure state on C d 2 ⊗ C s , where ψ is a random unit vector uniformly distributed on the sphere in C d 2 ⊗ C s . Let ρ 1 be random induced states ρ 1 = Tr C s (|ψ ψ|) whose distribution is µ d 2 ,s ; and let ρ 2 be random induced states following the same distribution µ d 2 ,s independent from ρ 1 . Let ρ be a random state induced by the entanglement swapping of ρ 1 and ρ 2 . We show that the empirical spectrum of ρ − 1l/d 2 converges almost surely to the Marcenko-Pastur law with parameter c 2 as d → ∞ and s/d → c.As an application, we prove that the state ρ is separable generically if ρ 1 , ρ 2 are PPT entangled.

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