R\'enyi Entropy Power Inequalities via Normal Transport and Rotation

Following a recent proof of Shannon's entropy power inequality (EPI), a comprehensive framework for deriving various EPIs for the R\'enyi entropy is presented that uses transport arguments from normal densities and a change of variable by rotation. Simple arguments are given to recover the previously known R\'enyi EPIs and derive new ones, by unifying a multiplicative form with constant c and a modification with exponent {\alpha} of previous works. In particular, for log-concave densities, we obtain a simple transportation proof of a sharp varentropy bound.Comment: 17 page. Entropy Journal, to appea

Majorization uncertainty relations for mixed quantum states

Majorization uncertainty relations are generalized for an arbitrary mixed quantum state $\rho$ of a finite size $N$. In particular, a lower bound for the sum of two entropies characterizing probability distributions corresponding to measurements with respect to arbitrary two orthogonal bases is derived in terms of the spectrum of $\rho$ and the entries of a unitary matrix $U$ relating both bases. The obtained results can also be formulated for two measurements performed on a single subsystem of a bipartite system described by a pure state, and consequently expressed as uncertainty relation for the sum of conditional entropies.Comment: 13 pages, 7 figure

Universal proofs of entropic continuity bounds via majorization flow

We introduce a notion of majorization flow, and demonstrate it to be a powerful tool for deriving simple and universal proofs of continuity bounds for entropic functions relevant in information theory. In particular, for the case of the alpha-R\'enyi entropy, whose connections to thermodynamics are discussed in this article, majorization flow yields a Lipschitz continuity bound for the case alpha > 1, thus resolving an open problem and providing a substantial improvement over previously known bounds.Comment: 29 pages; v2: added Cor. 3.2, Section 7, shortened some proofs, minor fixes; v3: added Section 6.2, minor fixe

On Generalized Stam Inequalities and Fisher–Rényi Complexity Measures

Information-theoretic inequalities play a fundamental role in numerous scientific and technological areas (e.g., estimation and communication theories, signal and information processing, quantum physics, …) as they generally express the impossibility to have a complete description of a system via a finite number of information measures. In particular, they gave rise to the design of various quantifiers (statistical complexity measures) of the internal complexity of a (quantum) system. In this paper, we introduce a three-parametric Fisher–Rényi complexity, named ( p , β , λ ) -Fisher–Rényi complexity, based on both a two-parametic extension of the Fisher information and the Rényi entropies of a probability density function ρ characteristic of the system. This complexity measure quantifies the combined balance of the spreading and the gradient contents of ρ , and has the three main properties of a statistical complexity: the invariance under translation and scaling transformations, and a universal bounding from below. The latter is proved by generalizing the Stam inequality, which lowerbounds the product of the Shannon entropy power and the Fisher information of a probability density function. An extension of this inequality was already proposed by Bercher and Lutwak, a particular case of the general one, where the three parameters are linked, allowing to determine the sharp lower bound and the associated probability density with minimal complexity. Using the notion of differential-escort deformation, we are able to determine the sharp bound of the complexity measure even when the three parameters are decoupled (in a certain range). We determine as well the distribution that saturates the inequality: the ( p , β , λ ) -Gaussian distribution, which involves an inverse incomplete beta function. Finally, the complexity measure is calculated for various quantum-mechanical states of the harmonic and hydrogenic systems, which are the two main prototypes of physical systems subject to a central potential.The authors are very grateful to the CNRS (Steeve Zozor) and the Junta de Andalucía and the MINECO–FEDER under the grants FIS2014–54497 and FIS2014–59311P (Jesús Sánchez-Dehesa) for partial financial support

Heisenberg and Entropic Uncertainty Measures for Large-Dimensional Harmonic Systems

The D-dimensional harmonic system (i.e., a particle moving under the action of a quadratic potential) is, together with the hydrogenic system, the main prototype of the physics of multidimensional quantum systems. In this work, we rigorously determine the leading term of the Heisenberg-like and entropy-like uncertainty measures of this system as given by the radial expectation values and the Rényi entropies, respectively, at the limit of large D. The associated multidimensional position-momentum uncertainty relations are discussed, showing that they saturate the corresponding general ones. A conjecture about the Shannon-like uncertainty relation is given, and an interesting phenomenon is observed: the Heisenberg-like and Rényi-entropy-based equality-type uncertainty relations for all of the D-dimensional harmonic oscillator states in the pseudoclassical ( D → ∞ ) limit are the same as the corresponding ones for the hydrogenic systems, despite the so different character of the oscillator and Coulomb potentials.This work has been partially supported by the projects FQM-7276 and FQM-207 of the Junta de Andalucía and the MINECO (Ministerio de Economia y Competitividad)-FEDER (European Regional Development Fund) Grants FIS2014- 54497P and FIS2014-59311-P. Irene V. Toranzo acknowledges the support of MEunder the program FPU

GAIT: A Geometric Approach to Information Theory

We advocate the use of a notion of entropy that reflects the relative abundances of the symbols in an alphabet, as well as the similarities between them. This concept was originally introduced in theoretical ecology to study the diversity of ecosystems. Based on this notion of entropy, we introduce geometry-aware counterparts for several concepts and theorems in information theory. Notably, our proposed divergence exhibits performance on par with state-of-the-art methods based on the Wasserstein distance, but enjoys a closed-form expression that can be computed efficiently. We demonstrate the versatility of our method via experiments on a broad range of domains: training generative models, computing image barycenters, approximating empirical measures and counting modes.Comment: Replaces the previous version named "GEAR: Geometry-Aware R\'enyi Information

Exact asymptotics of long-range quantum correlations in a nonequilibrium steady state

Out-of-equilibrium states of many-body systems tend to evade a description by standard statistical mechanics, and their uniqueness is epitomized by the possibility of certain long-range correlations that cannot occur in equilibrium. In quantum many-body systems, coherent correlations of this sort may lead to the emergence of remarkable entanglement structures. In this work, we analytically study the asymptotic scaling of quantum correlation measures -- the mutual information and the fermionic negativity -- within the zero-temperature steady state of voltage-biased free fermions on a one-dimensional lattice containing a noninteracting impurity. Previously, we have shown that two subsystems on opposite sides of the impurity exhibit volume-law entanglement, which is independent of the absolute distances of the subsystems from the impurity. Here we go beyond this result and derive the exact form of the subleading logarithmic corrections to the extensive terms of correlation measures, in excellent agreement with numerical calculations. In particular, the logarithmic term of the mutual information asymptotics can be encapsulated in a concise formula, depending only on simple four-point ratios of subsystem length-scales and on the impurity scattering probabilities at the Fermi energies. This echoes the case of equilibrium states, where such logarithmic terms may convey universal information about the physical system. To compute these exact results, we devise a hybrid method that relies on Toeplitz determinant asymptotics for correlation matrices in both real space and momentum space, successfully circumventing the inhomogeneity of the system. This method can potentially find wider use for analytical calculations of entanglement measures in similar scenarios.Comment: 21+5 pages, 4 figures. The main results appearing here were originally reported as part of arXiv:2205.12991, and are removed from there to allow a more elaborate discussio

Equality in the Matrix Entropy-Power Inequality and Blind Separation of Real and Complex sources

The matrix version of the entropy-power inequality for real or complex coefficients and variables is proved using a transportation argument that easily settles the equality case. An application to blind source extraction is given.Comment: 5 pages, in Proc. 2019 IEEE International Symposium on Information Theory (ISIT 2019), Paris, France, July 7-12, 201