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1 Concurrence of Lorentzpositive maps
, 2008
"... Let H(d) be the space of complex hermitian matrices of size d ×d and let H+(d) ⊂ H(d) be the cone of positive semidefinite matrices. A linear operator Φ: H(d1) → H(d2) is said to be positive if Φ[H+(d1)] ⊂ H+(d2). The concurrence C(Φ; ·) of a positive operator Φ: H(d1) → H(d2) is a realvalued q ..."
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Let H(d) be the space of complex hermitian matrices of size d ×d and let H+(d) ⊂ H(d) be the cone of positive semidefinite matrices. A linear operator Φ: H(d1) → H(d2) is said to be positive if Φ[H+(d1)] ⊂ H+(d2). The concurrence C(Φ; ·) of a positive operator Φ: H(d1) → H(d2) is a realvalued q function on the cone H+(d1), defined as the largest convex function which coincides with 2 σ d2 2 (Φ(ξξ∗)) on all rank 1 matrices ξξ ∗ ∈ H+(d1). Here σ d 2: H(d) → R denotes the second symmetric function, defined by σ d 2(A) = P i<j µiµj, where µ1,..., µd are the eigenvalues of A. The concurrence of a bipartite density matrix X is defined as the concurrence C(Φ; X) with Φ being the partial trace. A analogous concept can be considered for Lorentzpositive maps. Let Ln ⊂ R n be the ndimensional Lorentz cone. Then a linear map Υ: R m → R n is called Lorentzpositive if Υ[Lm] ⊂ Ln. For this class of maps we are able to compute the concurrence explicitly. This allows us to obtain formulae for the concurrence of positive operators having H(2) as input space and consequently of bipartite density matrices of rank 2. Namely, let Φ: H(2) → H(d2) be a positive operator, and let λ1,..., λ4 be the generalized eigenvalues of the pencil σ d2 2 (Φ(X)) − q λ det X, in decreasing order. Then the concurrence is given by the expression C(Φ; X) = 2 σ d2 2 (Φ(X)) − λ2 detX. As an application, we compute the concurrences of the density matrices of all graphs with 2 edges. Similar results apply for a function which we call Ifidelity, with the second largest generalized eigenvalue λ2 replaced by the smallest generalized eigenvalue λ4. 1
PARTIAL TRANSPOSE OF PERMUTATION MATRICES
, 2007
"... The main purpose of this paper is to look at the notion of partial transpose from the combinatorial side. In this perspective, we solve some basic enumeration problems involving partial transpose of permutation matrices. Specifically, we count the number of permutations matrices which are invarian ..."
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The main purpose of this paper is to look at the notion of partial transpose from the combinatorial side. In this perspective, we solve some basic enumeration problems involving partial transpose of permutation matrices. Specifically, we count the number of permutations matrices which are invariant under partial transpose. We count the number of permutation matrices which are still permutation matrices after partial transpose. We solve this problem also for transpositions. In this case, there is little evidence to justify a link between some permutations, partial transpose, and certain domino tilings.
Center for Combinatorics, Nankai University,
, 2008
"... The partial transpose of a block matrix M is the matrix obtained by transposing the blocks of M independently. We approach the notion of partial transpose from a combinatorial point of view. In this perspective, we solve some basic enumeration problems concerning the partial transpose of permutation ..."
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The partial transpose of a block matrix M is the matrix obtained by transposing the blocks of M independently. We approach the notion of partial transpose from a combinatorial point of view. In this perspective, we solve some basic enumeration problems concerning the partial transpose of permutation matrices. More specifically, we count the number of permutations matrices which are equal to their partial transpose and the number of permutation matrices whose partial transpose is still a permutation. We solve these problems also when restricted to symmetric permutation matrices only.