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26 January 2021
 
  » arxiv » quant-ph/0505093

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Dimension-Independent Positive-Partial-Transpose Probability Ratios
Paul B. Slater ;
Date 13 May 2005
Subject quant-ph
AbstractWe 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.
Source arXiv, quant-ph/0505093
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