464 research outputs found
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
From van der Corput to modern constructions of sequences for quasi-Monte Carlo rules
In 1935 J.G. van der Corput introduced a sequence which has excellent uniform
distribution properties modulo 1. This sequence is based on a very simple
digital construction scheme with respect to the binary digit expansion.
Nowadays the van der Corput sequence, as it was named later, is the prototype
of many uniformly distributed sequences, also in the multi-dimensional case.
Such sequences are required as sample nodes in quasi-Monte Carlo algorithms,
which are deterministic variants of Monte Carlo rules for numerical
integration. Since its introduction many people have studied the van der Corput
sequence and generalizations thereof. This led to a huge number of results.
On the occasion of the 125th birthday of J.G. van der Corput we survey many
interesting results on van der Corput sequences and their generalizations. In
this way we move from van der Corput's ideas to the most modern constructions
of sequences for quasi-Monte Carlo rules, such as, e.g., generalized Halton
sequences or Niederreiter's -sequences
On the use of a Modified Latin Hypercube Sampling (MLHS) approach in the estimation of a Mixed Logit model for vehicle choice
Quasi-random number sequences have been used extensively for many years in the simulation of integrals that do not have a closed-form expression, such as Mixed Logit and Multinomial Probit choice probabilities. Halton sequences are one example of such quasi-random number sequences, and various types of Halton sequences, including standard, scrambled, and shuffled versions, have been proposed and tested in the context of travel demand modeling. In this paper, we propose an alternative to Halton sequences, based on an adapted version of Latin Hypercube Sampling. These alternative sequences, like scrambled and shuffled Halton sequences, avoid the undesirable correlation patterns that arise in standard Halton sequences. However, they are easier to create than scrambled or shuffled Halton sequences. They also provide more uniform coverage in each dimension than any of the Halton sequences. A detailed analysis, using a 16-dimensional Mixed Logit model for choice between alternative-fuelled vehicles in California, was conducted to compare the performance of the different types of draws. The analysis shows that, in this application, the Modified Latin Hypercube Sampling (MLHS) outperforms each type of Halton sequence. This greater accuracy combined with the greater simplicity make the MLHS method an appealing approach for simulation of travel demand models and simulation-based models in general
Qubit-Qutrit Separability-Probability Ratios
Paralleling our recent computationally-intensive (quasi-Monte Carlo) work for
the case N=4 (quant-ph/0308037), we undertake the task for N=6 of computing to
high numerical accuracy, the formulas of Sommers and Zyczkowski
(quant-ph/0304041) for the (N^2-1)-dimensional volume and (N^2-2)-dimensional
hyperarea of the (separable and nonseparable) N x N density matrices, based on
the Bures (minimal monotone) metric -- and also their analogous formulas
(quant-ph/0302197) for the (non-monotone) Hilbert-Schmidt metric. With the same
seven billion well-distributed (``low-discrepancy'') sample points, we estimate
the unknown volumes and hyperareas based on five additional (monotone) metrics
of interest, including the Kubo-Mori and Wigner-Yanase. Further, we estimate
all of these seven volume and seven hyperarea (unknown) quantities when
restricted to the separable density matrices. The ratios of separable volumes
(hyperareas) to separable plus nonseparable volumes (hyperareas) yield
estimates of the separability probabilities of generically rank-six (rank-five)
density matrices. The (rank-six) separability probabilities obtained based on
the 35-dimensional volumes appear to be -- independently of the metric (each of
the seven inducing Haar measure) employed -- twice as large as those (rank-five
ones) based on the 34-dimensional hyperareas. Accepting such a relationship, we
fit exact formulas to the estimates of the Bures and Hilbert-Schmidt separable
volumes and hyperareas.(An additional estimate -- 33.9982 -- of the ratio of
the rank-6 Hilbert-Schmidt separability probability to the rank-4 one is quite
clearly close to integral too.) The doubling relationship also appears to hold
for the N=4 case for the Hilbert-Schmidt metric, but not the others. We fit
exact formulas for the Hilbert-Schmidt separable volumes and hyperareas.Comment: 36 pages, 15 figures, 11 tables, final PRA version, new last
paragraph presenting qubit-qutrit probability ratios disaggregated by the two
distinct forms of partial transpositio
- …