dorsal/arxiv
View SchemaQuantum entanglement and geometry of determinantal varieties
| Authors | Hao Chen |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0110103 |
| URL | https://arxiv.org/abs/quant-ph/0110103 |
Abstract
Quantum entanglement was first recognized as a feature of quantum mechanics in the famous paper of Einstein, Podolsky and Rosen [18]. Recently it has been realized that quantum entanglement is a key ingredient in quantum computation, quantum communication and quantum cryptography ([16],[17],[6]). 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 in 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. In this way algebraic geometry and complex differential geometry of these algebraic sets turn to be powerful tools for the understanding of quantum enatanglement. Our construction has applications in the following important topics in quantum information theory: 1) separability criterion, it is proved the algebraic sets have to be the sum of the linear subspaces if the mixed states are separable; 2) lower bound of Schmidt numbers, that is, generic low rank bipartite mixed states are entangled in many degrees of freedom; 3) simulation of Hamiltonians, it is proved the simulation of semi-positive Hamiltonians of the same rank implies the projective isomorphisms of the corresponding algebraic sets; 4) construction of bound enatanglement, examples of the entangled mixed states which are invariant under partial transpositions (thus PPT bound entanglement) are constructed systematically from our new separability criterion. On the other hand many examples of entangled mixed states with rich algebraic-geometric structure in their associated determinantal varieties are constructed and studied from this point of view.
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"abstract": "Quantum entanglement was first recognized as a feature of quantum mechanics\nin the famous paper of Einstein, Podolsky and Rosen [18]. Recently it has been\nrealized that quantum entanglement is a key ingredient in quantum computation,\nquantum communication and quantum cryptography ([16],[17],[6]). In this paper,\nwe introduce algebraic sets, which are determinantal varieties in the complex\nprojective spaces or the products of complex projective spaces, for the mixed\nstates in bipartite or multipartite quantum systems as their invariants under\nlocal unitary transformations. These invariants are naturally arised from the\nphysical consideration of measuring mixed states by separable pure states. In\nthis way algebraic geometry and complex differential geometry of these\nalgebraic sets turn to be powerful tools for the understanding of quantum\nenatanglement. Our construction has applications in the following important\ntopics in quantum information theory: 1) separability criterion, it is proved\nthe algebraic sets have to be the sum of the linear subspaces if the mixed\nstates are separable; 2) lower bound of Schmidt numbers, that is, generic low\nrank bipartite mixed states are entangled in many degrees of freedom; 3)\nsimulation of Hamiltonians, it is proved the simulation of semi-positive\nHamiltonians of the same rank implies the projective isomorphisms of the\ncorresponding algebraic sets; 4) construction of bound enatanglement, examples\nof the entangled mixed states which are invariant under partial transpositions\n(thus PPT bound entanglement) are constructed systematically from our new\nseparability criterion. On the other hand many examples of entangled mixed\nstates with rich algebraic-geometric structure in their associated\ndeterminantal varieties are constructed and studied from this point of view.",
"arxiv_id": "quant-ph/0110103",
"authors": [
"Hao Chen"
],
"categories": [
"quant-ph",
"cs.IT",
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"title": "Quantum entanglement and geometry of determinantal varieties",
"url": "https://arxiv.org/abs/quant-ph/0110103"
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