Optimal estimation of SU(d) using exact and approximate 2-designs

We consider the problem of estimating an SU(d) quantum operation when n copies of it are available at the same time. It is well known that, if one uses a separable state as the input for the unitaries, the optimal mean square error will decrease as 1/n. However it is shown here that, if a proper entangled state is used, the optimal mean square error will decrease at a 1/n^2 rate. It is also shown that spherical 2-designs (e.g. complete sets of mutually unbiased bases and symmetric informationally complete positive operator valued measures) can be used to design optimal input states. Although 2-designs are believed to exist for every dimension, this has not yet been proven. Therefore, we give an alternative input state based on approximate 2-designs which can be made arbitrarily close to optimal. It is shown that measurement strategies which are based on local operations and classical communication between the ancilla and the rest of the system can be optimal.Comment: 6 pages. v2: Complete rewrite, new results 11 page

Synchronization in Complex Networks: a Comment on two recent PRL papers

I show that the conclusions of [Hwang, Chavez, Amann, & Boccaletti, PRL 94, 138701 (2005); Chavez, Hwang, Amann, Hentschel, & Boccaletti, PRL 94, 218701 (2005)] are closely related to those of previous publications.Comment: 2 page

Intra-industry trade and labor costs: The smooth adjustment hypothesis

We study a situation in which, owing to the exhaustion of non-renewable energy sources, conventional motor vehicles will turn out of use. We consider two scenarios: recycling or dismantling these motor vehicles. M|G|∞ queue system is used to study the process. Through it, we conclude that if the rate of dismantling and recycling of motor vehicles is greater than the rate at which they become idle, the system will tend to get balanced. The model allows also performing a brief study about the recycling or dismantling economic interest.

A consistent test of significance

This paper presents a test of significance consistent under nonparametric alternatives. Under the null hypothesis, a regressor has no effect on the regression model. Our statistic does not require to estimate the model on the alternative hypothesis, which is left unspecified. Hence, no smoothing techniques are required. The statistic is a weighted empirical process which resembles the Cram~r-von Mises. The asymptotic test is consistent under Pitman's alternatives converging to the null at arate n-1/2. A Monte-Cario experiment illustrates the performance ofthe test in small samples. We also inelude two applications involving biomedical and acid rain data

Entanglement is not very useful for estimating multiple phases

The problem of the estimation of multiple phases (or of commuting unitaries) is considered. This is a sub-model of the estimation of a completely unknown unitary operation where it has been shown in recent works that there are considerable improvements by using entangled input states and entangled measurements. Here it is shown that when estimating commuting unitaries, there is practically no advantage in using entangled input states or entangled measurements.Comment: v2. New title, improved Fig.3, other minor changes, Accepted in PR