Silver mean conjectures for 15-d volumes and 14-d hyperareas of the separable two-qubit systems

Extensive numerical integration results lead us to conjecture that the silver mean, that is, s = \sqrt{2}-1 = .414214 plays a fundamental role in certain geometries (those given by monotone metrics) imposable on the 15-dimensional convex set of two-qubit systems. For example, we hypothesize that the volume of separable two-qubit states, as measured in terms of (four times) the minimal monotone or Bures metric is s/3, and 10s in terms of (four times) the Kubo-Mori monotone metric. Also, we conjecture, in terms of (four times) the Bures metric, that that part of the 14-dimensional boundary of separable states consisting generically of rank-four 4 x 4 density matrices has volume (``hyperarea'') 55s/39 and that part composed of rank-three density matrices, 43s/39, so the total boundary hyperarea would be 98s/39. While the Bures probability of separability (0.07334) dominates that (0.050339) based on the Wigner-Yanase metric (and all other monotone metrics) for rank-four states, the Wigner-Yanase (0.18228) strongly dominates the Bures (0.03982) for the rank-three states.Comment: 30 pages, 6 tables, 17 figures; nine new figures and one new table in new section VII.B pertaining to 14-dimensional hyperareas associated with various monotone metric

Sequential Quasi-Monte Carlo

We derive and study SQMC (Sequential Quasi-Monte Carlo), a class of algorithms obtained by introducing QMC point sets in particle filtering. SQMC is related to, and may be seen as an extension of, the array-RQMC algorithm of L'Ecuyer et al. (2006). The complexity of SQMC is $O(N \log N)$, where $N$ is the number of simulations at each iteration, and its error rate is smaller than the Monte Carlo rate $O_P(N^{-1/2})$. The only requirement to implement SQMC is the ability to write the simulation of particle $x_t^n$ given $x_{t-1}^n$ as a deterministic function of $x_{t-1}^n$ and a fixed number of uniform variates. We show that SQMC is amenable to the same extensions as standard SMC, such as forward smoothing, backward smoothing, unbiased likelihood evaluation, and so on. In particular, SQMC may replace SMC within a PMCMC (particle Markov chain Monte Carlo) algorithm. We establish several convergence results. We provide numerical evidence that SQMC may significantly outperform SMC in practical scenarios.Comment: 55 pages, 10 figures (final version