dorsal/arxiv
View SchemaEntanglement in SO(3)-invariant bipartite quantum systems
| Authors | Heinz-Peter Breuer |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0503079 |
| URL | https://arxiv.org/abs/quant-ph/0503079 |
| DOI | 10.1103/PhysRevA.71.062330 |
| Journal | Phys. Rev. A71, 062330 (2005) |
Abstract
The structure of the state spaces of bipartite (N tensor N) quantum systems which are invariant under product representations of the group SO(3) of three-dimensional proper rotations is analyzed. The subsystems represent particles of arbitrary spin j which transform according to an irreducible representation of the rotation group. A positive map theta is introduced which describes the time reversal symmetry of the local states and which is unitarily equivalent to the transposition of matrices. It is shown that the partial time reversal transformation theta_2 = (I tensor theta) acting on the composite system can be expressed in terms of the invariant 6-j symbols introduced by Wigner into the quantum theory of angular momentum. This fact enables a complete geometrical construction of the manifold of states with positive partial transposition and of the sets of separable and entangled states of (4 tensor 4) systems. The separable states are shown to form a three-dimensional prism and a three-dimensional manifold of bound entangled states is identified. A positive maps is obtained which yields, together with the time reversal, a necessary and sufficient condition for the separability of states of (4 tensor 4) systems. The relations to the reduction criterion and to the recently proposed cross norm criterion for separability are discussed.
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"abstract": "The structure of the state spaces of bipartite (N tensor N) quantum systems\nwhich are invariant under product representations of the group SO(3) of\nthree-dimensional proper rotations is analyzed. The subsystems represent\nparticles of arbitrary spin j which transform according to an irreducible\nrepresentation of the rotation group. A positive map theta is introduced which\ndescribes the time reversal symmetry of the local states and which is unitarily\nequivalent to the transposition of matrices. It is shown that the partial time\nreversal transformation theta_2 = (I tensor theta) acting on the composite\nsystem can be expressed in terms of the invariant 6-j symbols introduced by\nWigner into the quantum theory of angular momentum. This fact enables a\ncomplete geometrical construction of the manifold of states with positive\npartial transposition and of the sets of separable and entangled states of (4\ntensor 4) systems. The separable states are shown to form a three-dimensional\nprism and a three-dimensional manifold of bound entangled states is identified.\nA positive maps is obtained which yields, together with the time reversal, a\nnecessary and sufficient condition for the separability of states of (4 tensor\n4) systems. The relations to the reduction criterion and to the recently\nproposed cross norm criterion for separability are discussed.",
"arxiv_id": "quant-ph/0503079",
"authors": [
"Heinz-Peter Breuer"
],
"categories": [
"quant-ph"
],
"doi": "10.1103/PhysRevA.71.062330",
"journal_ref": "Phys. Rev. A71, 062330 (2005)",
"title": "Entanglement in SO(3)-invariant bipartite quantum systems",
"url": "https://arxiv.org/abs/quant-ph/0503079"
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