Symmetric Projections of the Entropy Region

Entropy inequalities play a central role in proving converse coding theorems for network information theoretic problems. This thesis studies two new aspects of entropy inequalities. First, inequalities relating average joint entropies rather than entropies over individual subsets are studied. It is shown that the closures of the average entropy regions where the averages are over all subsets of the same size and all sliding windows of the same size respectively are identical, implying that averaging over sliding windows always suffices as far as unconstrained entropy inequalities are concerned. Second, the existence of non-Shannon type inequalities under partial symmetry is studied using the concepts of Shannon and non-Shannon groups. A complete classification of all permutation groups over four elements is established. With five random variables, it is shown that there are no non-Shannon type inequalities under cyclic symmetry

Towards measuring Entanglement Entropies in Many Body Systems

We explore the relation between entanglement entropy of quantum many body systems and the distribution of corresponding, properly selected, observables. Such a relation is necessary to actually measure the entanglement entropy. We show that in general, the Shannon entropy of the probability distribution of certain symmetry observables gives a lower bound to the entropy. In some cases this bound is saturated and directly gives the entropy. We also show other cases in which the probability distribution contains enough information to extract the entropy: we show how this is done in several examples including BEC wave functions, the Dicke model, XY spin chain and chains with strong randomness

Two-parameter deformations of logarithm, exponential, and entropy: A consistent framework for generalized statistical mechanics

A consistent generalization of statistical mechanics is obtained by applying the maximum entropy principle to a trace-form entropy and by requiring that physically motivated mathematical properties are preserved. The emerging differential-functional equation yields a two-parameter class of generalized logarithms, from which entropies and power-law distributions follow: these distributions could be relevant in many anomalous systems. Within the specified range of parameters, these entropies possess positivity, continuity, symmetry, expansibility, decisivity, maximality, concavity, and are Lesche stable. The Boltzmann-Shannon entropy and some one parameter generalized entropies already known belong to this class. These entropies and their distribution functions are compared, and the corresponding deformed algebras are discussed.Comment: Version to appear in PRE: about 20% shorter, references updated, 13 PRE pages, 3 figure

Nonequilibrium quantum-impurities: from entropy production to information theory

Nonequilibrium steady-state currents, unlike their equilibrium counterparts, continuously dissipate energy into their physical surroundings leading to entropy production and time-reversal symmetry breaking. This letter discusses these issues in the context of quantum impurity models driven out of equilibrium by attaching the impurity to leads at different chemical potentials and temperatures. We start by pointing out that entropy production is often hidden in traditional treatments of quantum-impurity models. We then use simple thermodynamic arguments to define the rate of entropy production. Using the scattering framework recently developed by the authors we show that the rate of entropy production has a simple information theoretic interpretation in terms of the Shannon entropy and Kullback-Leibler divergence of nonequilibrium distribution function. This allows us to show that the entropy production is strictly positive for any nonequilibrium steady-state. We conclude by applying these ideas to the Resonance Level Model and the Kondo model.Comment: 5 pages, 1 figure new version with minor clarification

Finite-size scaling of the Shannon-R\'enyi entropy in two-dimensional systems with spontaneously broken continuous symmetry

We study the scaling of the (basis dependent) Shannon entropy for two-dimensional quantum antiferromagnets with N\'eel long-range order. We use a massless free-field description of the gapless spin wave modes and phase space arguments to treat the fact that the finite-size ground state is rotationally symmetric, while there are degenerate physical ground states which break the symmetry. Our results show that the Shannon entropy (and its R\'enyi generalizations) possesses some universal logarithmic term proportional to the number $N_\text{NG}$ of Nambu-Goldstone modes. In the case of a torus, we show that $S_{n>1} \simeq {\rm const.} N+ \frac{N_\text{NG}}{4}\frac{n}{n-1} \ln{N}$ and $S_1 \simeq {\rm const.} N - \frac{N_\text{NG}}{4} \ln{N}$, where $N$ is the total number of sites and $n$ the R\'enyi index. The result for $n>1$ is in reasonable agreement with the quantum Monte Carlo results of Luitz et al. [Phys. Rev. Lett. 112, 057203 (2014)], and qualitatively similar to those obtained previously for the entanglement entropy. The Shannon entropy of a line subsystem (embedded in the two-dimensional system) is also considered. Finally, we present some density-matrix renormalization group (DMRG) calculations for a spin$\frac{1}{2}$ XY model on the square lattice in a cylinder geometry. These numerical data confirm our findings for logarithmic terms in the $n=\infty$ R\'enyi entropy (also called $-\ln{p_{\rm max}}$). They also reveal some universal dependence on the cylinder aspect ratio, in good agreement with the fact that, in that case, $p_{\rm max}$ is related to a non-compact free-boson partition function in dimension 1+1.Comment: 15 pages, 3 figures, v2: published versio

Quantum simulation of bosonic-fermionic non-interacting particles in disordered systems via quantum walk

We report on the theoretical analysis of bosonic and fermionic non-interacting systems in a discrete two-particle quantum walk affected by different kinds of disorder. We considered up to 100-step QWs with a spatial, temporal and space-temporal disorder observing how the randomness and the wavefunction symmetry non-trivially affect the final spatial probability distribution, the transport properties and the Shannon entropy of the walkers.Comment: 13 pages, 10 figures. arXiv admin note: text overlap with arXiv:1101.2638 by other author