Vladislav Kargin

On the Chernoff bound for efficiency of quantum hypothesis testing

The paper estimates the Chernoff rate for the efficiency of quantum hypothesis testing. For both joint and separate measurements, approximate bounds for the rate are given if both states are mixed, and exact expressions are derived if at least one of the states is pure. The efficiencies of tests with separate and joint measurements are compared. The results are illustrated by a test of quantum entanglement.

On the Largest Lyapunov Exponent for Products of Gaussian Matrices

The paper provides a new integral formula for the largest Lyapunov exponent of Gaussian matrices, which is valid in the real, complex and quaternion-valued cases. This formula is applied to derive asymptotic expressions for the largest Lyapunov exponent when the size of the matrix is large and compare the Lyapunov exponents in models with a spike and no spikes.

On superconvergence of sums of free random variables

This paper derives sufficient conditions for superconvergence of sums of bounded free random variables and provides an estimate for the rate of superconvergence.

An inequality for the distance between densities of free convolutions

This paper contributes to the study of the free additive convolution of probability measures. It shows that under some conditions, if measures μi and νi, i=1,2, are close to each other in terms of the Le vy metric and if the free convolution μ1⊞μ2 is sufficiently smooth, then ν1⊞ν2 is absolutely continuous, and the densities of measures ν1⊞ν2 and μ1⊞μ2 are close to each other. In particular, convergence in distribution μ(n)1→μ1, μ(n)2→μ2 implies that the density of μ(n)1⊞μ(n)2 is defined for all sufficiently large n and converges to the density of μ1⊞μ2. Some applications are provided, including: (i) a new proof of the local version of the free central limit theorem, and (ii) new local limit theorems for sums of free projections, for sums of ⊞-stable random variables and for eigenvalues of a sum of two N-by-N random matrices.

On coordination games with quantum correlations

A necessary condition is derived that helps to determine whether an entangled quantum system can improve coordination in a game with incomplete information.

A large deviation inequality for vector functions on finite reversible Markov Chains

Let

On Eigenvalues of the Sum of Two Random Projections

S_N

On estimation in the reduced-rank regression with a large number of responses and predictors

be the sum of vector-valued functions defined on a finite Markov chain. An analogue of the Bernstein--Hoeffding inequality is derived for the probability of large deviations of

On Pfaffian Random Point Fields

S_N

Continuous-Time Quantum Walk on Integer Lattices and Homogeneous Trees

and relates the probability to the spectral gap of the Markov chain. Examples suggest that this inequality is better than alternative inequalities if the chain has a sufficiently large spectral gap and the function is high-dimensional.