Orbital approach to microstate free entropy

Motivated by Voiculescu's liberation theory, we introduce the orbital free entropy $\chi_orb$ for non-commutative self-adjoint random variables (also for "hyperfinite random multi-variables"). Besides its basic properties the relation of $\chi_orb$ with the usual free entropy $\chi$ is shown. Moreover, the dimension counterpart $\delta_{0,orb}$ of $\chi_orb$ is discussed, and we obtain the relation of $\delta_{0,orb}$ with the original free entropy dimension $\delta_0$ with applications to $\delta_0$ itself.Comment: 38 pages; Section 5 was largely improved and Section 6 was adde

Stationary quantum source coding

In this paper the quantum source coding theorem is obtained for a completely ergodic source. This results extends Shannon's classical theorem as well as Schumacher's quantum noiseless coding theorem for memoryless sources. The control of the memory effects requires earlier results of Hiai and Petz on high probability subspaces.Comment: 8 page

Large deviations for functions of two random projection matrices

In this paper two independent and unitarily invariant projection matrices P(N) and Q(N) are considered and the large deviation is proven for the eigenvalue density of all polynomials of them as the matrix size $N$ converges to infinity. The result is formulated on the tracial state space $TS({\cal A})$ of the universal $C^*$-algebra ${\cal A}$ generated by two selfadjoint projections. The random pair $(P(N),Q(N))$ determines a random tracial state $\tau_N \in TS({\cal A})$ and $\tau_N$ satisfies the large deviation. The rate function is in close connection with Voiculescu's free entropy defined for pairs of projections.Comment: 22 page

On the quantum Renyi relative entropies and related capacity formulas

We show that the quantum $\alpha$-relative entropies with parameter $\alpha\in (0,1)$ can be represented as generalized cutoff rates in the sense of [I. Csiszar, IEEE Trans. Inf. Theory 41, 26-34, (1995)], which provides a direct operational interpretation to the quantum $\alpha$-relative entropies. We also show that various generalizations of the Holevo capacity, defined in terms of the $\alpha$-relative entropies, coincide for the parameter range $\alpha\in (0,2]$, and show an upper bound on the one-shot epsilon-capacity of a classical-quantum channel in terms of these capacities.Comment: v4: Cutoff rates are treated for correlated hypotheses, some proofs are given in greater detai

Generalized Log-Majorization and Multivariate Trace Inequalities

© 2017, Springer International Publishing. We show that recent multivariate generalizations of the Araki–Lieb–Thirring inequality and the Golden–Thompson inequality (Sutter et al. in Commun Math Phys, 2016. doi:10.1007/s00220-016-2778-5) for Schatten norms hold more generally for all unitarily invariant norms and certain variations thereof. The main technical contribution is a generalization of the concept of log-majorization which allows us to treat majorization with regard to logarithmic integral averages of vectors of singular values

A nonlinear model dynamics for closed-system, constrained, maximal-entropy-generation relaxation by energy redistribution

We discuss a nonlinear model for the relaxation by energy redistribution within an isolated, closed system composed of non-interacting identical particles with energy levels e_i with i=1,2,...,N. The time-dependent occupation probabilities p_i(t) are assumed to obey the nonlinear rate equations tau dp_i/dt=-p_i ln p_i+ alpha(t)p_i-beta(t)e_ip_i where alpha(t) and beta(t) are functionals of the p_i(t)'s that maintain invariant the mean energy E=sum_i e_ip_i(t) and the normalization condition 1=sum_i p_i(t). The entropy S(t)=-k sum_i p_i(t) ln p_i(t) is a non-decreasing function of time until the initially nonzero occupation probabilities reach a Boltzmann-like canonical distribution over the occupied energy eigenstates. Initially zero occupation probabilities, instead, remain zero at all times. The solutions p_i(t) of the rate equations are unique and well-defined for arbitrary initial conditions p_i(0) and for all times. Existence and uniqueness both forward and backward in time allows the reconstruction of the primordial lowest entropy state. The time evolution is at all times along the local direction of steepest entropy ascent or, equivalently, of maximal entropy generation. These rate equations have the same mathematical structure and basic features of the nonlinear dynamical equation proposed in a series of papers ended with G.P.Beretta, Found.Phys., 17, 365 (1987) and recently rediscovered in S. Gheorghiu-Svirschevski, Phys.Rev.A, 63, 022105 and 054102 (2001). Numerical results illustrate the features of the dynamics and the differences with the rate equations recently considered for the same problem in M.Lemanska and Z.Jaeger, Physica D, 170, 72 (2002).Comment: 11 pages, 7 eps figures (psfrag use removed), uses subeqn, minor revisions, accepted for Physical Review