dorsal/arxiv
View SchemaDimension-Independent Positive-Partial-Transpose Probability Ratios
| Authors | Paul B. Slater |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0505093 |
| URL | https://arxiv.org/abs/quant-ph/0505093 |
Abstract
We conduct quasi-Monte Carlo numerical integrations in two very high (80 and 79)-dimensional domains -- the parameter spaces of rank-9 and rank-8 qutrit-qutrit (9 x 9) density matrices. We, then, estimate the ratio of the probability -- in terms of the Hilbert-Schmidt metric -- that a generic rank-9 density matrix has a positive partial transpose (PPT) to the probability that a generic rank-8 density matrix has a PPT (a precondition to separability/nonentanglement). Close examination of the numerical results generated -- despite certain large fluctuations -- indicates that the true ratio may, in fact, be 2. Our earlier investigation (eprint quant-ph/0410238) also yielded estimates close to 2 of the comparable ratios for qubit-qubit and qubit-qutrit pairs (the only two cases where the PPT condition fully implies separability). Therefore, it merits conjecturing (as Zyczkowski was the first to do) that such Hilbert-Schmidt (rank-NM/rank-(NM-1)) PPT probability ratios are 2 for all NM-dimensional quantum systems.
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"abstract": "We conduct quasi-Monte Carlo numerical integrations in two very high (80 and\n79)-dimensional domains -- the parameter spaces of rank-9 and rank-8\nqutrit-qutrit (9 x 9) density matrices. We, then, estimate the ratio of the\nprobability -- in terms of the Hilbert-Schmidt metric -- that a generic rank-9\ndensity matrix has a positive partial transpose (PPT) to the probability that a\ngeneric rank-8 density matrix has a PPT (a precondition to\nseparability/nonentanglement). Close examination of the numerical results\ngenerated -- despite certain large fluctuations -- indicates that the true\nratio may, in fact, be 2. Our earlier investigation (eprint quant-ph/0410238)\nalso yielded estimates close to 2 of the comparable ratios for qubit-qubit and\nqubit-qutrit pairs (the only two cases where the PPT condition fully implies\nseparability). Therefore, it merits conjecturing (as Zyczkowski was the first\nto do) that such Hilbert-Schmidt (rank-NM/rank-(NM-1)) PPT probability ratios\nare 2 for all NM-dimensional quantum systems.",
"arxiv_id": "quant-ph/0505093",
"authors": [
"Paul B. Slater"
],
"categories": [
"quant-ph"
],
"title": "Dimension-Independent Positive-Partial-Transpose Probability Ratios",
"url": "https://arxiv.org/abs/quant-ph/0505093"
},
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