arXiv subjects
Publications and source records attributed to L. A Markovich.
The q-deformed entropies of quantum and classical systems are discussed. Standard and q-deformed entropic inequalities for X - states of the two-qubit and a state of single qudit with $j=3/2$ are presented.
In some applications it is necessary to estimate derivatives of probability densities defined on the positive semi-axis. The quality of nonparametric estimates of the probability densities and their derivatives are strongly influenced by smoothing parameters (bandwidths). In this paper an expression for the optimal smoothing parameter of the gamma kernel estimate of the density derivative is obtained. For this parameter data-driven estimates based on methods called "rule of thumb" and "cross-validation" are constructed. The quality of the estimates is verified and demonstrated on examples of density derivatives generated by Maxwell and Weibull distributions.
We consider nonparametric estimation of the derivative of a probability density function with the bounded support on $[0,\infty)$. Estimates are looked up in the class of estimates with asymmetric gamma kernel functions. The use of gamma kernels is due to the fact they are nonnegative, change their shape depending on the position on the semi-axis and possess other good properties. We found analytical expressions for bias, variance, mean integrated squared error (MISE) of the derivative estimate. An optimal bandwidth, the optimal MISE, and rate of mean square convergence of the estimates for density derivative have also been found.
Entropic inequalities related to the quantum mutual information for bipartite system and tomographic mutual information is studied for Werner state of two qubits. Quantum correlations corresponding to entanglement properties of the qubits in Werner state are discussed.