DimensionIndependent PositivePartialTranspose Probability Ratios
Abstract
We conduct quasiMonte Carlo numerical integrations in two very high (80 and 79)dimensional domains  the parameter spaces of rank9 and rank8 qutritqutrit (9 x 9) density matrices. We, then, estimate the ratio of the probability  in terms of the HilbertSchmidt metric  that a generic rank9 density matrix has a positive partial transpose (PPT) to the probability that a generic rank8 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 quantph/0410238) also yielded estimates close to 2 of the comparable ratios for qubitqubit and qubitqutrit 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 HilbertSchmidt (rankNM/rank(NM1)) PPT probability ratios are 2 for all NMdimensional quantum systems.
 Publication:

arXiv eprints
 Pub Date:
 May 2005
 DOI:
 10.48550/arXiv.quantph/0505093
 arXiv:
 arXiv:quantph/0505093
 Bibcode:
 2005quant.ph..5093S
 Keywords:

 Quantum Physics
 EPrint:
 8 pages, 1 figure