On the dimension of subspaces with bounded Schmidt rank
TL;DR: In this paper, the authors consider the problem of finding the largest subspace of a given bipartite quantum system when the subspace contains only highly entangled states and show that for every pair of local dimensions dA and dB, and every r, the largest dimension of a subspace consisting only of entangled states of Schmidt rank r or larger is known.
Abstract: We consider the question of how large a subspace of a given bipartite quantum system can be when the subspace contains only highly entangled states. This is motivated in part by results of Hayden et al., which show that in large d x d--dimensional systems there exist random subspaces of dimension almost d^2, all of whose states have entropy of entanglement at least log d - O(1). It is also related to results due to Parthasarathy on the dimension of completely entangled subspaces, which have connections with the construction of unextendible product bases. Here we take as entanglement measure the Schmidt rank, and determine, for every pair of local dimensions dA and dB, and every r, the largest dimension of a subspace consisting only of entangled states of Schmidt rank r or larger. This exact answer is a significant improvement on the best bounds that can be obtained using random subspace techniques. We also determine the converse: the largest dimension of a subspace with an upper bound on the Schmidt rank. Finally, we discuss the question of subspaces containing only states with Schmidt equal to r.
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TL;DR: In this paper, it was shown that there is a dichotomy in that every UCP map that is not entanglement breaking in the sense of Horodecki-Shor-Ruskai must preserve entanglements, and that entangler-preserving maps of every possible rank exist in abundance.
Abstract: Let M and N be full matrix algebras. A unital completely positive (UCP) map \({\phi:M\to N}\) is said to preserve entanglement if its inflation \({\phi\otimes {\rm id}_N : M\otimes N\to N\otimes N}\) has the following property: for every maximally entangled pure state ρ of \({N\otimes N}\), \({\rho\circ(\phi\otimes {\rm id}_N)}\) is an entangled state of \({M\otimes N}\) . We show that there is a dichotomy in that every UCP map that is not entanglement breaking in the sense of Horodecki–Shor–Ruskai must preserve entanglement, and that entanglement preserving maps of every possible rank exist in abundance. We also show that with probability 1, all UCP maps of relatively small rank preserve entanglement, but that this is not so for UCP maps of maximum rank.
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TL;DR: In this paper , it was shown that a set of bipartite entangled subspaces can be constructed under joining of their adjacent subsystems, and that direct sums of such constructions under certain conditions are genuinely entangled.
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TL;DR: In this paper , the authors investigated the relation between entangled witnesses and its mirrored ones, and presented a conjecture that the mirrored operator obtained from an optimal entangled witness is either a positive operator or a decomposable observer, which implies that the bound entangled states cannot be detected.
Abstract: Entanglement witnesses (EWs) are a versatile tool in the verification of entangled states. The framework of mirrored EW doubles the power of a given EW by introducing its twin-a mirrored EW-whereby two EWs related by mirroring can bound the set of separable states more efficiently. In this work, we investigate the relation between the EWs and its mirrored ones, and present a conjecture which claims that the mirrored operator obtained from an optimal EW is either a positive operator or a decomposable EW, which implies that positive-partial-transpose entangled states, also known as the bound entangled states, cannot be detected. This conjecture is reached by studying numerous known examples of optimal EWs. However, the mirrored EWs obtained from the non-optimal ones can be non-decomposable as well. We also show that mirrored operators obtained from the extremal decomposable witnesses are positive semi-definite. Interestingly, the witnesses that violate the well known conjecture of Structural Physical Approximation, do satisfy our conjecture. The intricate relation between these two conjectures is discussed and it reveals a novel structure of the separability problem.
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Book•
01 Jan 1978
TL;DR: In this paper, a comprehensive, self-contained treatment of complex manifold theory is presented, focusing on results applicable to projective varieties, and including discussion of the theory of Riemann surfaces and algebraic curves, algebraic surfaces and the quadric line complex.
Abstract: A comprehensive, self-contained treatment presenting general results of the theory. Establishes a geometric intuition and a working facility with specific geometric practices. Emphasizes applications through the study of interesting examples and the development of computational tools. Coverage ranges from analytic to geometric. Treats basic techniques and results of complex manifold theory, focusing on results applicable to projective varieties, and includes discussion of the theory of Riemann surfaces and algebraic curves, algebraic surfaces and the quadric line complex as well as special topics in complex manifolds.
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