dorsal/arxiv
View SchemaState space structure and entanglement of rotationally invariant spin systems
| Authors | Heinz-Peter Breuer |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0506224 |
| URL | https://arxiv.org/abs/quant-ph/0506224 |
| DOI | 10.1088/0305-4470/38/41/013 |
| Journal | J. Phys. A: Math. Gen. 38 (2005) 9019-9037 |
Abstract
We investigate the structure of SO(3)-invariant quantum systems which are composed of two particles with spins j_1 and j_2. The states of the composite spin system are represented by means of two complete sets of rotationally invariant operators, namely by the projections P_J onto the eigenspaces of the total angular momentum J, and by certain invariant operators Q_K which are built out of spherical tensor operators of rank K. It is shown that these representations are connected by an orthogonal matrix whose elements are expressible in terms of Wigner's 6-j symbols. The operation of the partial time reversal of the combined spin system is demonstrated to be diagonal in the Q_K-representation. These results are employed to obtain a complete characterization of spin systems with j_1 = 1 and arbitrary j_2 > 1. We prove that the Peres-Horodecki criterion of positive partial transposition (PPT) is necessary and sufficient for separability if j_2 is an integer, while for half-integer spins j_2 there always exist entangled PPT states (bound entanglement). We construct an optimal entanglement witness for the case of half-integer spins and design a protocol for the detection of entangled PPT states through measurements of the total angular momentum.
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"abstract": "We investigate the structure of SO(3)-invariant quantum systems which are\ncomposed of two particles with spins j_1 and j_2. The states of the composite\nspin system are represented by means of two complete sets of rotationally\ninvariant operators, namely by the projections P_J onto the eigenspaces of the\ntotal angular momentum J, and by certain invariant operators Q_K which are\nbuilt out of spherical tensor operators of rank K. It is shown that these\nrepresentations are connected by an orthogonal matrix whose elements are\nexpressible in terms of Wigner\u0027s 6-j symbols. The operation of the partial time\nreversal of the combined spin system is demonstrated to be diagonal in the\nQ_K-representation. These results are employed to obtain a complete\ncharacterization of spin systems with j_1 = 1 and arbitrary j_2 \u003e 1. We prove\nthat the Peres-Horodecki criterion of positive partial transposition (PPT) is\nnecessary and sufficient for separability if j_2 is an integer, while for\nhalf-integer spins j_2 there always exist entangled PPT states (bound\nentanglement). We construct an optimal entanglement witness for the case of\nhalf-integer spins and design a protocol for the detection of entangled PPT\nstates through measurements of the total angular momentum.",
"arxiv_id": "quant-ph/0506224",
"authors": [
"Heinz-Peter Breuer"
],
"categories": [
"quant-ph"
],
"doi": "10.1088/0305-4470/38/41/013",
"journal_ref": "J. Phys. A: Math. Gen. 38 (2005) 9019-9037",
"title": "State space structure and entanglement of rotationally invariant spin systems",
"url": "https://arxiv.org/abs/quant-ph/0506224"
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