dorsal/arxiv
View SchemaA priori Probabilities of Separable Quantum States
| Authors | Paul B. Slater |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9810026 |
| URL | https://arxiv.org/abs/quant-ph/9810026 |
| DOI | 10.1088/0305-4470/32/28/306 |
| Journal | J.Phys.A32:5261,1999 |
Abstract
Zyczkowski, Horodecki, Sanpera, and Lewenstein (ZHSL) recently proposed a ``natural measure'' on the N-dimensional quantum systems (quant-ph/9804024), but expressed surprise when it led them to conclude that for N = 2 x 2, disentangled (separable) systems are more probable (0.632) in nature than entangled ones. We contend, however, that ZHSL's (rejected) intuition has, in fact, a sound theoretical basis, and that the a priori probability of disentangled 2 x 2 systems should more properly be viewed as (considerably) less than 0.5. We arrive at this conclusion in two quite distinct ways, the first based on classical and the second, quantum considerations. Both approaches, however, replace (in whole or part) the ZHSL (product) measure by ones based on the volume elements of monotone metrics, which in the classical case amounts to adopting the Jeffreys' prior of Bayesian theory. Only the quantum-theoretic analysis (which yields the smallest probabilities of disentanglement) uses the minimum number of parameters possible, N^2 - 1, as opposed to N^2 + N - 1 (although this "over-parameterization", as recently indicated by Byrd, should be avoidable). However, despite substantial computation, we are not able to obtain precise estimates of these probabilities, and the need for additional (possibly supercomputer) analyses is indicated (particularly so, for higher-dimensional quantum systems, such as the 2 x 3 we also study here).
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"abstract": "Zyczkowski, Horodecki, Sanpera, and Lewenstein (ZHSL) recently proposed a\n``natural measure\u0027\u0027 on the N-dimensional quantum systems (quant-ph/9804024),\nbut expressed surprise when it led them to conclude that for N = 2 x 2,\ndisentangled (separable) systems are more probable (0.632) in nature than\nentangled ones. We contend, however, that ZHSL\u0027s (rejected) intuition has, in\nfact, a sound theoretical basis, and that the a priori probability of\ndisentangled 2 x 2 systems should more properly be viewed as (considerably)\nless than 0.5. We arrive at this conclusion in two quite distinct ways, the\nfirst based on classical and the second, quantum considerations. Both\napproaches, however, replace (in whole or part) the ZHSL (product) measure by\nones based on the volume elements of monotone metrics, which in the classical\ncase amounts to adopting the Jeffreys\u0027 prior of Bayesian theory. Only the\nquantum-theoretic analysis (which yields the smallest probabilities of\ndisentanglement) uses the minimum number of parameters possible, N^2 - 1, as\nopposed to N^2 + N - 1 (although this \"over-parameterization\", as recently\nindicated by Byrd, should be avoidable). However, despite substantial\ncomputation, we are not able to obtain precise estimates of these\nprobabilities, and the need for additional (possibly supercomputer) analyses is\nindicated (particularly so, for higher-dimensional quantum systems, such as the\n2 x 3 we also study here).",
"arxiv_id": "quant-ph/9810026",
"authors": [
"Paul B. Slater"
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"doi": "10.1088/0305-4470/32/28/306",
"journal_ref": "J.Phys.A32:5261,1999",
"title": "A priori Probabilities of Separable Quantum States",
"url": "https://arxiv.org/abs/quant-ph/9810026"
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