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arXiv:quant-ph/0110082

Manifolds of interconvertible pure states

Magdalena M. Sinołęcka, Karol Życzkowski, Marek Kuś

Published Oct 12, 2001• 3 citations
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Abstract

Local orbits of a pure state of an N x N bi-partite quantum system are analyzed. We compute their dimensions which depends on the degeneracy of the vector of coefficients arising by the Schmidt decomposition. In particular, the generic orbit has 2N^2 -N-1 dimensions, the set of separable states is 4(N-1) dimensional, while the manifold of maximally entangled states has N^2-1 dimensions.

DOI
10.48550/arxiv.quant-ph/0110082

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Local orbits of a pure state of an N x N bi-partite quantum system are analyzed. We compute their dimensions which depends on the degeneracy of the vector of coefficients arising by the Schmidt decomposition. In particular, the generic orbit has 2N^2 -N-1 dimensions, the set of separable states is 4(N-1) dimensional, while the manifold of maximally entangled states has N^2-1 dimensions.
Summary. A simple derivation of finite Schmidt decomposition of pure states describing finite dimensional systems interacting with the infinite dimensional ones is presented. In particular, maximally entangled pure states in such systems are being characterized. The concept of mixed-pure states has been introduced and some criterions for checking separability and entanglement of them are presented .The notion of LOCC equivalence and LOCC semi-order on the space of pure states of systems analyzed has been adopted suitably. In particular a Nielsen -like theorem has been extended to the pure (d, infinity )- states case. The notion of spin-orbit entanglement in the context of atomic physics is being discussed from a mathematical perspective. Keywords: quantum entanglement, Schmidt decomposition, mixed-pure states, LOCC semi-order, spin-orbit entanglement
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Geometric properties of the set of quantum entangled states are investigated. We propose an explicit method to compute the dimension of local orbits for any mixed state of the general $K\ifmmode\times\else\texttimes\fi{}M$ problem and characterize the set of effectively different states (which cannot be related by local transformations). Thus, we generalize earlier results obtained for the simplest $2\ifmmode\times\else\texttimes\fi{}2$ system, which lead to a stratification of the six-dimensional set of N=4N=4N=4 pure states. We define the concept of absolutely separable states, for which all globally equivalent states are separable.