dorsal/arxiv
View SchemaBetter bound on the exponent of the radius of the multipartite separable ball
| Authors | Leonid Gurvits, Howard Barnum |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0409095 |
| URL | https://arxiv.org/abs/quant-ph/0409095 |
| DOI | 10.1103/PhysRevA.72.032322 |
Abstract
We show that for an m-qubit quantum system, there is a ball of radius asymptotically approaching kappa 2^{-gamma m} in Frobenius norm, centered at the identity matrix, of separable (unentangled) positive semidefinite matrices, for an exponent gamma = (1/2)((ln 3/ln 2) - 1), roughly .29248125. This is much smaller in magnitude than the best previously known exponent, from our earlier work, of 1/2. For normalized m-qubit states, we get a separable ball of radius sqrt(3^(m+1)/(3^m+3)) * 2^{-(1 + \gamma)m}, i.e. sqrt{3^{m+1}/(3^m+3)}\times 6^{-m/2} (note that \kappa = \sqrt{3}), compared to the previous 2 * 2^{-3m/2}. This implies that with parameters realistic for current experiments, NMR with standard pseudopure-state preparation techniques can access only unentangled states if 36 qubits or fewer are used (compared to 23 qubits via our earlier results). We also obtain an improved exponent for m-partite systems of fixed local dimension d_0, although approaching our earlier exponent as d_0 approaches infinity.
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"abstract": "We show that for an m-qubit quantum system, there is a ball of radius\nasymptotically approaching kappa 2^{-gamma m} in Frobenius norm, centered at\nthe identity matrix, of separable (unentangled) positive semidefinite matrices,\nfor an exponent gamma = (1/2)((ln 3/ln 2) - 1), roughly .29248125. This is much\nsmaller in magnitude than the best previously known exponent, from our earlier\nwork, of 1/2. For normalized m-qubit states, we get a separable ball of radius\nsqrt(3^(m+1)/(3^m+3)) * 2^{-(1 + \\gamma)m}, i.e. sqrt{3^{m+1}/(3^m+3)}\\times\n6^{-m/2} (note that \\kappa = \\sqrt{3}), compared to the previous 2 * 2^{-3m/2}.\nThis implies that with parameters realistic for current experiments, NMR with\nstandard pseudopure-state preparation techniques can access only unentangled\nstates if 36 qubits or fewer are used (compared to 23 qubits via our earlier\nresults). We also obtain an improved exponent for m-partite systems of fixed\nlocal dimension d_0, although approaching our earlier exponent as d_0\napproaches infinity.",
"arxiv_id": "quant-ph/0409095",
"authors": [
"Leonid Gurvits",
"Howard Barnum"
],
"categories": [
"quant-ph"
],
"doi": "10.1103/PhysRevA.72.032322",
"title": "Better bound on the exponent of the radius of the multipartite separable ball",
"url": "https://arxiv.org/abs/quant-ph/0409095"
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