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arXiv:quant-ph/0108034

Schmidt number of pure states in bipartite quantum systems as an algebraic-geometric invariant

Hao Chen

Published Aug 8, 2001• 0 citations
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Abstract

Our previous work about algebraic-geometric invariants of the mixed states are extended and a stronger separability criterion is given. We also show that the Schmidt number of pure states in bipartite quantum systems, a classical concept, is actually an algebraic-geometric invariant.
DOI
10.48550/arxiv.quant-ph/0108034

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Quantum entanglement was first recognized as a feature of quantum mechanics in the famous paper of Einstein, Podolsky, and Rosen. Recently it has been realized that quantum entanglement is a key ingredient in quantum computation, quantum communication, and quantum cryptography. In this paper, we introduce algebraic sets, which are determinantal varieties in the complex projective spaces or the products of complex projective spaces, for the mixed states on bipartite or multipartite quantum systems as their invariants under local unitary transformations. These invariants are naturally arised from the physical consideration of measuring mixed states by separable pure states. Our construction has applications in the following important topics in quantum information theory: (1) separability criterion, it is proved that the algebraic sets must be a union of the linear subspaces if the mixed states are separable; (2) simulation of Hamiltonians, it is proved that the simulation of semipositive Hamiltonians of the same rank implies the projective isomorphisms of the corresponding algebraic sets; (3) construction of bound entangled mixed states, examples of the entangled mixed states which are invariant under partial transpositions (thus PPT bound entanglement) are constructed systematically from our new separability criterion.