Bounds for The Reduced Relative Entropies

Two reverse inequalities associated with Tsallis relative operator entropy via generalized Kantorovich constant and their applications

We analyze the relative entropy of certain KMS states for scalar self-interacting quantum field theories over Minkowski backgrounds that have been recently constructed by Fredenhagen and Lindner in [FL14] in the framework of perturbative algebraic quantum field theory. The definition we are using is a generalization of the Araki relative entropy to the case of field theories. In particular, we shall see that the analyzed relative entropy is positive in the sense of perturbation theory, hence, even if the relative modular operator is not at disposal in this context, the proposed extension is compatible with perturbation theory. In the second part of the paper we analyze the adiabatic limits of these states showing that also the density of relative entropy obtained dividing the relative entropy by the spatial volume of the region where interaction takes place is positive and finite. In the last part of the paper we discuss the entropy production for states obtained by an ergodic mean (time average) of perturbed KMS states evolved with the free evolution recently constructed by the authors of the present paper. We show that their entropy production vanishes even if return to equilibrium [Ro73, HKT74] does not hold. This means that states constructed in this way are thermodynamically simple, namely they are not so far from equilibrium states.

Natural sounds have substantial acoustic structure (predictability, nonrandomness) in their spectral and temporal compositions. Listeners are expected to exploit this structure to distinguish simultaneous sound sources; however, previous studies confounded acoustic structure and listening experience. Here, sensitivity to acoustic structure in novel sounds was measured in discrimination and identification tasks. Complementary signal-processing strategies independently varied relative acoustic entropy (the inverse of acoustic structure) across frequency or time. In one condition, instantaneous frequency of low-pass-filtered 300-ms random noise was rescaled to 5 kHz bandwidth and resynthesized. In another condition, the instantaneous frequency of a short gated 5-kHz noise was resampled up to 300 ms. In both cases, entropy relative to full bandwidth or full duration was a fraction of that in 300-ms noise sampled at 10 kHz. Discrimination of sounds improved with less relative entropy. Listeners identified a probe sound as a target sound (1%, 3.2%, or 10% relative entropy) that repeated amidst distractor sounds (1%, 10%, or 100% relative entropy) at 0 dB SNR. Performance depended on differences in relative entropy between targets and background. Lower-relative-entropy targets were better identified against higher-relative-entropy distractors than lower-relative-entropy distractors; higher-relative-entropy targets were better identified amidst lower-relative-entropy distractors. Results were consistent across signal-processing strategies.

We propose an information-theoretical measure, the relative cluster entropy \mathcal{D_C}[P\|Q]𝒟𝒞[P∥Q], to discriminate among cluster partitions characterised by probability distribution functions P and Q. The measure is illustrated with the clusters generated by pairs of fractional Brownian motions with Hurst exponents H_1H1 and H_2H2 respectively. For subdiffusive, normal and superdiffusive sequences, the relative entropy sensibly depends on the difference between H_1H1 and H_2H2. By using the minimum relative entropy principle, cluster sequences characterized by different correlation degrees are distinguished and the optimal Hurst exponent is selected. As a case study, real-world cluster partitions of market price series are compared to those obtained from fully uncorrelated sequences (simple Browniam motions) assumed as a model. The minimum relative cluster entropy yields optimal Hurst exponents H_1H1=0.55, H_1H1=0.57, and H_1H1=0.63 respectively for the prices of DJIA, S and P500, NASDAQ: a clear indication of non-markovianity. Finally, we derive the analytical expression of the relative cluster entropy and the outcomes are discussed for arbitrary pairs of power-laws probability distribution functions of continuous random variables.

In this paper, we obtain new inequalities for relative operator entropy [Formula: see text] in terms of Tsallis’ relative entropy [Formula: see text] [Formula: see text] in the case of positive invertible operators [Formula: see text] [Formula: see text]. Further bounds for [Formula: see text] are also provided.

Reverse inequalities involving two relative operator entropies and two relative entropies

The second law of thermodynamics is discussed and reformulated from a quantum information theoretic perspective for open quantum systems using relative entropy. Specifically, the relative entropy of a quantum state with respect to equilibrium states is considered and its monotonicity property with respect to an open quantum system evolution is used to obtain second law-like inequalities. We discuss this first for generic quantum systems in contact with a thermal bath and subsequently turn to a formulation suitable for the description of local dynamics in a relativistic quantum field theory. A local version of the second law similar to the one used in relativistic fluid dynamics can be formulated with relative entropy or even relative entanglement entropy in a space-time region bounded by two light cones. We also give an outlook towards isolated quantum field theories and discuss the role of entanglement for relativistic fluid dynamics.

A formula for the relative entropy \(S(\rho \vert \vert \sigma )=\textrm{Tr}\,\rho \,(\log \,\rho -\log \,\sigma )\) of two gaussian states \(\rho \), \(\sigma \) in the boson Fock space \(\varGamma (\mathbb {C}^n)\) is presented. It is shown that the relative entropy has a classical and a quantum part: The classical part consists of a weighted linear combination of relative Shannon entropies of n pairs of Bernouli trials arising from the thermal state composition of the gaussian states \(\rho \) and \(\sigma \). The quantum part has a sum of n terms, that are functions of the annihilation means and the covariance matrices of 1-mode marginals of the gaussian state \(\rho '\), which is equivalent to \(\rho \) under a disentangling unitary gaussian symmetry operation of the state \(\sigma \). A generalized formula for the Petz-Rényi relative entropy \(S_\alpha (\rho \vert \vert \sigma )=-\frac{1}{\alpha -1}\log \textrm{Tr}\, \rho ^{\alpha }\sigma ^{1-\alpha },\ 0<\alpha <1\) for gaussian states \(\rho \), \(\sigma \) is also presented. Furthermore it is shown that \(S_{\alpha }(\rho \vert \vert \sigma )\) converges to the limit \(S(\rho \vert \vert \sigma )\) as \(\alpha \) increases to 1.KeywordsBoson Fock spaceGaussian stateWeyl operatorSymplectic groupRelative entropy

Generalizations of operator Shannon inequality based on Tsallis and Rényi relative entropies

The sufficiency and weak sufficiency in * -algebras are discussed. Some properties are studied concerning the relative entropy and the sufficiency for invariant states and KMS states in W* and C*-dynami- cal systems. Introduction. The concept of sufficiency is very important in mathematical statistics. The abstract measure theoretic investigation of sufficient statistics was initiated by Halmos and Savage (13). Kullback and Leibler (19) gave the characterization of sufficiency in terms of the information (i.e., the classical relative entropy). Umegaki (33,34) studied the sufficiency and the relative entropy in the noncommutative case of semi-finite von Neumann algebras. Araki (4,5) extended the relative entropy to the case for normal positive linear functionals of general von Neumann algebras and showed its several properties. Furthermore Uhlmann (32) showed the general WYDL concavity using a quadratic inteφolation theory and defined the relative entropy of positive linear functionals of arbitrary *-algebras. In the previous paper (14), we discussed the sufficiency and the relative entropy in von Neumann algebras and gave the characterizations of invariant states and KMS states with respect to the modular automor- phism group of a faithful normal state. In this paper, we further develop the sufficiency and the relative entropy in * -algebras. In §1, we introduce besides the sufficiency another notion of weak sufficiency and establish the relation between them. In §2, we deal with the weak sufficiency of positive linear maps between ^alge- bras. In §3, we mention the Araki's and Uhlmann's relative entropies which are equal in the von Neumann algebra case. We further give a formula of relative entropy for states of C*-algebras. In §4, we establish some properties of invariant states and KMS states in WΓ*-dynamical systems and C*-dynamical systems through the relative entropy and the sufficiency. The theorems there improve or extend the results obtained in (14). Finally we give an application to the Gibbs states of quantum lattice systems.

A new concept of the relative metric entropy is introduced that makes possible quantitative evaluation of the chaotic mixing in a dynamical system under the action of an external noise. It is shown that, in the absence of noise, the relative metric entropy represents an estimate from below for the Kolmogorov entropy of the given dynamical system.

The recently introduced resource theory of imaginarity facilitates a systematic investigation into the role of complex numbers in quantum mechanics and quantum information theory. In this work, well‐defined measures of imaginarity are proposed using various distance metrics, drawing inspiration from recent advancements in quantum entanglement and coherence. Specifically, quantitatively evaluating imaginarity is focused through measures such as Tsallis relative ‐entropy, Sandwiched Rényi relative entropy, and Tsallis relative operator entropy. Additionally, the decay rates of these measures are analyzed. These findings reveal that the Tsallis relative ‐entropy of imaginarity exhibits higher decay rate under quantum channels compared to other measures. Finally, the ordering of single‐qubit states are examined under these imaginarity measures, demonstrating that the order remains invariant under the bit‐flip channel for specific parameter ranges. This study enhances the understanding of imaginarity as a quantum resource and its potential applications in quantum information theory.

We introduce the notion of relative volume entropy for two spacetimes with preferred compact spacelike foliations. This is accomplished by applying the notion of Kullback-Leibler divergence to the volume elements induced on spacelike slices. The resulting quantity gives a lower bound on the number of bits which are necessary to describe one metric given the other. For illustration, we study some examples, in particular gravitational waves, and conclude that the relative volume entropy is a suitable device for quantitative comparison of the inhomogeneity of two spacetimes.

We study the relative entanglement entropies of one interval between excited states of a 1+1 dimensional conformal field theory (CFT). To compute the relative entropy S(ρ1∥ρ0) between two given reduced density matrices ρ1 and ρ0 of a quantum field theory, we employ the replica trick which relies on the path integral representation of Tr(ρ1ρ0n − 1) and define a set of Rényi relative entropies Sn(ρ1∥ρ0). We compute these quantities for integer values of the parameter n and derive via the replica limit the relative entropy between excited states generated by primary fields of a free massless bosonic field. In particular, we provide the relative entanglement entropy of the state described by the primary operator i∂ϕ, both with respect to the ground state and to the state generated by chiral vertex operators. These predictions are tested against exact numerical calculations in the XX spin-chain finding perfect agreement.

The Bregman operator divergence is introduced for density matrices by differentiation of the matrix-valued function x ↦ x log x. This quantity is compared with the relative operator entropy of Fujii and Kamei. It turns out that the trace is the usual Umegaki’s relative entropy which is the only intersection of the classes of quasi-entropies and Bregman divergences.