Power-Law distributions and Fisher's information measure

We show that thermodynamic uncertainties (TU) it preserve their form in passing from Boltzmann-Gibbs' statistics to Tsallis' one provided that we express these TU in terms of the appropriate variable conjugate to the temperature in a nonextensive context.Comment: accepted for publication in Physica

Security of quantum bit string commitment depends on the information measure

Unconditionally secure non-relativistic bit commitment is known to be impossible in both the classical and the quantum world. However, when committing to a string of n bits at once, how far can we stretch the quantum limits? In this letter, we introduce a framework of quantum schemes where Alice commits a string of n bits to Bob, in such a way that she can only cheat on a bits and Bob can learn at most b bits of information before the reveal phase. Our results are two-fold: we show by an explicit construction that in the traditional approach, where the reveal and guess probabilities form the security criteria, no good schemes can exist: a+b is at least n. If, however, we use a more liberal criterion of security, the accessible information, we construct schemes where a=4 log n+O(1) and b=4, which is impossible classically. Our findings significantly extend known no-go results for quantum bit commitment.Comment: To appear in PRL. Short version of quant-ph/0504078, long version to appear separately. Improved security definition and result, one new lemma that may be of independent interest. v2: added funding reference, no other change

Information measure for financial time series: quantifying short-term market heterogeneity

A well-interpretable measure of information has been recently proposed based on a partition obtained by intersecting a random sequence with its moving average. The partition yields disjoint sets of the sequence, which are then ranked according to their size to form a probability distribution function and finally fed in the expression of the Shannon entropy. In this work, such entropy measure is implemented on the time series of prices and volatilities of six financial markets. The analysis has been performed, on tick-by-tick data sampled every minute for six years of data from 1999 to 2004, for a broad range of moving average windows and volatility horizons. The study shows that the entropy of the volatility series depends on the individual market, while the entropy of the price series is practically a market-invariant for the six markets. Finally, a cumulative information measure - the `Market Heterogeneity Index'- is derived from the integral of the proposed entropy measure. The values of the Market Heterogeneity Index are discussed as possible tools for optimal portfolio construction and compared with those obtained by using the Sharpe ratio a traditional risk diversity measure

Net Fisher information measure versus ionization potential and dipole polarizability in atoms

The net Fisher information measure, defined as the product of position and momentum Fisher information measures and derived from the non-relativistic Hartree-Fock wave functions for atoms with Z=1-102, is found to correlate well with the inverse of the experimental ionization potential. Strong direct correlations of the net Fisher information are also reported for the static dipole polarizability of atoms with Z=1-88. The complexity measure, defined as the ratio of the net Onicescu information measure and net Fisher information, exhibits clearly marked regions corresponding to the periodicity of the atomic shell structure. The reported correlations highlight the need for using the net information measures in addition to either the position or momentum space analogues. With reference to the correlation of the experimental properties considered here, the net Fisher information measure is found to be superior than the net Shannon information entropy.Comment: 16 pages, 6 figure

Unique additive information measures - Boltzmann-Gibbs-Shannon, Fisher and beyond

It is proved that the only additive and isotropic information measure that can depend on the probability distribution and also on its first derivative is a linear combination of the Boltzmann-Gibbs-Shannon and Fisher information measures. Power law equilibrium distributions are found as a result of the interaction of the two terms. The case of second order derivative dependence is investigated and a corresponding additive information measure is given.Comment: 10 pages, 1 figures, shortene