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Volumn 77, Issue 3, 2008, Pages

Robustness of entangled states that are positive under partial transposition

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PERTURBATION TECHNIQUES;

EID: 40949104829     PISSN: 10502947     EISSN: 10941622     Source Type: Journal    
DOI: 10.1103/PhysRevA.77.032318     Document Type: Article
Times cited : (20)

References (37)
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    • We have Tr (W) = d1 d2 Tr (W I d1 d2) 0, by condition (i). If Tr (W) =0, we then see that the hyperplane in B, orthogonal to the vector W, touches the compact and convex set S at its point I/ (d1 d2). Thus, I/ (d1 d2) turns out to be a point on the boundary of S, and hence, each open neighborhood of I/ (d1 d2) will always contain at least one inseparable state of AB, which is not true, whatever be the suitable metric we choose (see).
    • We have Tr (W) = d1 d2 Tr (W I d1 d2) 0, by condition (i). If Tr (W) =0, we then see that the hyperplane in B, orthogonal to the vector W, touches the compact and convex set S at its point I/ (d1 d2). Thus, I/ (d1 d2) turns out to be a point on the boundary of S, and hence, each open neighborhood of I/ (d1 d2) will always contain at least one inseparable state of AB, which is not true, whatever be the suitable metric we choose (see).
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    • The state σρ S is "nearest" to the state ρ (P-B) in the sense that the distance d (ρ,σ) of ρ from any element σ of S achieves its minimum at the point σρ.
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    • Note that this is not true, in general, for an arbitrary element of S. For example, if |Ψ A Ψ| is an entangled state in D, then the (unnormalized) state ρ= |ψ A ψ| ψ B ψ | +η |Ψ Ψ| is always entangled for all η>0 (where we have assumed that d1 ≤ d2, and for |ψ A = i=1 d1 λi |i A, we have | ψ B = i=1 d1 λi |i B). This is because ρ TB (the partial transposition of ρ with respect to the subsystem B) is not positive semidefinite when η>0. Hence there does not exist λ>0 such that the ball B (|ψ A ψ| ψ B ψ |; λ) contains only separable states.
    • Note that this is not true, in general, for an arbitrary element of S. For example, if |Ψ A Ψ| is an entangled state in D, then the (unnormalized) state ρ= |ψ A ψ| ψ B ψ | +η |Ψ Ψ| is always entangled for all η>0 (where we have assumed that d1 ≤ d2, and for |ψ A = i=1 d1 λi |i A, we have | ψ B = i=1 d1 λi |i B). This is because ρ TB (the partial transposition of ρ with respect to the subsystem B) is not positive semidefinite when η>0. Hence there does not exist λ>0 such that the ball B (|ψ A ψ| ψ B ψ |; λ) contains only separable states.
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* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.