dorsal/arxiv
View SchemaLocal commutativity versus Bell inequality violation for entangled states and versus non-violation for separable states
| Authors | Michael Seevinck, Jos Uffink |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0703134 |
| URL | https://arxiv.org/abs/quant-ph/0703134 |
| DOI | 10.1103/PhysRevA.76.042105 |
| Journal | Phys. Rev. A 76, 042105 (2007) |
Abstract
By introducing a quantitative `degree of commutativity' in terms of the angle between spin-observables we present two tight quantitative trade-off relations in the case of two qubits: First, for entangled states, between the degree of commutativity of local observables and the maximal amount of violation of the Bell inequality: if both local angles increase from zero to \pi/2 (i.e., the degree of local commutativity decreases), the maximum violation of the Bell inequality increases. Secondly, a converse trade-off relation holds for separable states: if both local angles approach \pi/2 the maximal value obtainable for the correlations in the Bell inequality decreases and thus the non-violation increases. As expected, the extremes of these relations are found in the case of anti-commuting local observables where respectively the bounds of 2\sqrt{2} and \sqrt{2} hold for the expectation of the Bell operator. The trade-off relations show that non-commmutativity gives ``a more than classical result" for entangled states, whereas "a less than classical result" is obtained for separable states. The experimental relevance of the trade-off relation for separable states is that it provides an experimental test for two qubit entanglement. Its advantages are twofold: in comparison to violations of Bell inequalities it is a stronger criterion and in comparison to entanglement witnesses it needs to make less strong assumptions about the observables implemented in the experiment.
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"abstract": "By introducing a quantitative `degree of commutativity\u0027 in terms of the angle\nbetween spin-observables we present two tight quantitative trade-off relations\nin the case of two qubits: First, for entangled states, between the degree of\ncommutativity of local observables and the maximal amount of violation of the\nBell inequality: if both local angles increase from zero to \\pi/2 (i.e., the\ndegree of local commutativity decreases), the maximum violation of the Bell\ninequality increases. Secondly, a converse trade-off relation holds for\nseparable states: if both local angles approach \\pi/2 the maximal value\nobtainable for the correlations in the Bell inequality decreases and thus the\nnon-violation increases. As expected, the extremes of these relations are found\nin the case of anti-commuting local observables where respectively the bounds\nof 2\\sqrt{2} and \\sqrt{2} hold for the expectation of the Bell operator. The\ntrade-off relations show that non-commmutativity gives ``a more than classical\nresult\" for entangled states, whereas \"a less than classical result\" is\nobtained for separable states. The experimental relevance of the trade-off\nrelation for separable states is that it provides an experimental test for two\nqubit entanglement. Its advantages are twofold: in comparison to violations of\nBell inequalities it is a stronger criterion and in comparison to entanglement\nwitnesses it needs to make less strong assumptions about the observables\nimplemented in the experiment.",
"arxiv_id": "quant-ph/0703134",
"authors": [
"Michael Seevinck",
"Jos Uffink"
],
"categories": [
"quant-ph"
],
"doi": "10.1103/PhysRevA.76.042105",
"journal_ref": "Phys. Rev. A 76, 042105 (2007)",
"title": "Local commutativity versus Bell inequality violation for entangled states and versus non-violation for separable states",
"url": "https://arxiv.org/abs/quant-ph/0703134"
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