Renyi Relative Entropy Research Articles

Characterization of Distributive Mixing in Polymer Processing Equipment using Renyi Entropies

Abstract A new method for characterization of distributive mixing in processing equipment, based on Renyi entropies, was developed. This method was applied to a twin-flight single screw extruder, in which tracer positions were determined through computer simulations of the flow field. The various entropies were calculated using particle concentrations in equal area domains of the mixer. Renyi entropies, which are function of a parameter β, were calculated for extruders of different lengths. We discuss the merit of using Renyi entropies for different values of β by pointing to the different mixing characteristics they probe. The relative Renyi entropy varies between 0 and 1 and represents a measure of distributive mixing quality, with 1 corresponding to perfect mixing and 0 corresponding to poorest mixing. We compare this new method of distributive mixing characterization to traditional ones based on the concepts of Scale and Intensity of Segregation, and the calculations based on Pairwise Correlations and Correlation Sums. The results show good agreement between the relative Renyi entropy and the traditional methods. Other advantages of the Renyi entropy such as reduced calculation time and geometric independence are discussed. For the case of a twin-flight single screw extruder, it is shown that a longer extruder is not necessarily more beneficial to distributive mixing.

Closest separable state when measured by a quasi-relative entropy

It is well known that for pure states the relative entropy of entanglement is equal to the reduced entropy, and the closest separable state is explicitly known as well. The same holds for Renyi relative entropy per recent results. We ask the same question for a quasi-relative entropy of entanglement, which is an entanglement measure defined as the minimum distance to the set of separable state, when the distance is measured by the quasi-relative entropy. First, we consider a maximally entangled state, and show that the closest separable state is the same for any quasi-relative entropy as for the relative entropy of entanglement. Then, we show that this also holds for a certain class of functions and any pure state. And at last, we consider any pure state on two qubit systems and a large class of operator convex function. For these, we find the closest separable state, which may not be the same one as for the relative entropy of entanglement.

Quantum Renyi relative entropies on a spin chain with interface defects

We compute the quantum Renyi relative entropies in an infinite spinless fermionic chain with a defect. Doing a numerical analysis we will show that the resulting quantity depends non trivially on the effective central charge of the theory. Moreover, we will see that an explicit analytic expression can be written for all of them and from that one can read the quantum fidelity and the relative entropy.

Renyi relative entropies and renormalization group flows

Quantum Renyi relative entropies provide a one-parameter family of distances between density matrices, which generalizes the relative entropy and the fidelity. We study these measures for renormalization group flows in quantum field theory. We derive explicit expressions in free field theory based on the real time approach. Using monotonicity properties, we obtain new inequalities that need to be satisfied by consistent renormalization group trajectories in field theory. These inequalities play the role of a second law of thermodynamics, in the context of renormalization group flows. Finally, we apply these results to a tractable Kondo model, where we evaluate the Renyi relative entropies explicitly. An outcome of this is that Anderson’s orthogonality catastrophe can be avoided by working on a Cauchy surface that approaches the light-cone.

Strong Converse for the Feedback-Assisted Classical Capacity of Entanglement-Breaking Channels

Quantum entanglement can be used in a communication scheme to establish a correlation between successive channel inputs that is impossible by classical means. It is known that the classical capacity of quantum channels can be enhanced by such entangled encoding schemes, but this is not always the case. In this paper, we prove that a strong converse theorem holds for the classical capacity of an entanglement-breaking channel even when it is assisted by a classical feedback link from the receiver to the transmitter. In doing so, we identify a bound on the strong converse exponent, which determines the exponentially decaying rate at which the success probability tends to zero, for a sequence of codes with communication rate exceeding capacity. Proving a strong converse, along with an achievability theorem, shows that the classical capacity is a sharp boundary between reliable and unreliable communication regimes. One of the main tools in our proof is the sandwiched Renyi relative entropy. The same method of proof is used to derive an exponential bound on the success probability when communicating over an arbitrary quantum channel assisted by classical feedback, provided that the transmitter does not use entangled encoding schemes.

An Image Segmentation Method Based on Renyi Relative Entropy and Gaussian Distribution

Background: Image segmentation is a necessary prerequisite for many higher level computer vision tasks, such as object recognition, image understanding, and image retrieval. However, the segmentation problem is inherently ill-posed due to the large number of possible partitionings for any single image. So, image segmentation remains one of the major challenges in image analysis. There are also many patents on these problems. This paper introduces a new image segmentation method to readers, and the effectiveness of the new method is demonstrated by experiments. Keywords: Image segmentation, histogram thresholding, Renyi relative entropy, Gaussian distribution, parametric statistical model.

On variational expressions for quantum relative entropies

Distance measures between quantum states like the trace distance and the fidelity can naturally be defined by optimizing a classical distance measure over all measurement statistics that can be obtained from the respective quantum states. In contrast, Petz showed that the measured relative entropy, defined as a maximization of the Kullback-Leibler divergence over projective measurement statistics, is strictly smaller than Umegaki's quantum relative entropy whenever the states do not commute. We extend this result in two ways. First, we show that Petz' conclusion remains true if we allow general positive operator valued measures. Second, we extend the result to Renyi relative entropies and show that for non-commuting states the sandwiched Renyi relative entropy is strictly larger than the measured Renyi relative entropy for $\alpha \in (\frac12, \infty)$, and strictly smaller for $\alpha \in [0,\frac12)$. The latter statement provides counterexamples for the data-processing inequality of the sandwiched Renyi relative entropy for $\alpha < \frac12$. Our main tool is a new variational expression for the measured Renyi relative entropy, which we further exploit to show that certain lower bounds on quantum conditional mutual information are superadditive.

Anomeric effect revisited: Perspective from information-theoretic approach in density functional reactivity theory

In this work, we apply the quantities from the information-theoretic approach in density functional reactivity theory to examine the origin and nature of the anomeric effect, which is the preference of heteroatomic substituents within the cyclohexane ring to be in the axial orientation rather than the sterically less hindered equatorial position. Using α-d-glucopyranose as an example, we confirmed its complex stereoelectronic origin. With Shannon entropy, Fisher information, GBP entropy, information gain, Onicescu information energy, and relative Renyi entropy, we are able to accurately account for the axial-equatorial energy difference in general, with the anomeric effect included as a special case.

Macrostate equivalence of two general ensembles and specific relative entropies.

The two criteria of ensemble equivalence, i.e., macrostate equivalence and measure equivalence, are investigated for a general pair of states. Macrostate equivalence implies the two ensembles are indistinguishable by the measurement of macroscopic quantities obeying the large-deviation principle, and measure equivalence means that the specific relative entropy of these two states vanishes in the thermodynamic limit. It is shown that measure equivalence implies a macrostate equivalence for a general pair of states by deriving an inequality connecting the large-deviation rate functions to the specific relative Renyi entropies. The result is applicable to both quantum and classical systems. As applications, a sufficient condition for thermalization, the time scale of quantum dynamics of macrovariables, and the second law with strict irreversibility in a quantum quench are discussed.

Strong Converse Exponents for a Quantum Channel Discrimination Problem and Quantum-Feedback-Assisted Communication

This paper studies the difficulty of discriminating between an arbitrary quantum channel and a "replacer" channel that discards its input and replaces it with a fixed state. We show that, in this particular setting, the most general adaptive discrimination strategies provide no asymptotic advantage over non-adaptive tensor-power strategies. This conclusion follows by proving a quantum Stein's lemma for this channel discrimination setting, showing that a constant bound on the Type I error leads to the Type II error decreasing to zero exponentially quickly at a rate determined by the maximum relative entropy registered between the channels. The strong converse part of the lemma states that any attempt to make the Type II error decay to zero at a rate faster than the channel relative entropy implies that the Type I error necessarily converges to one. We then refine this latter result by identifying the optimal strong converse exponent for this task. As a consequence of these results, we can establish a strong converse theorem for the quantum-feedback-assisted capacity of a channel, sharpening a result due to Bowen. Furthermore, our channel discrimination result demonstrates the asymptotic optimality of a non-adaptive tensor-power strategy in the setting of quantum illumination, as was used in prior work on the topic. The sandwiched Renyi relative entropy is a key tool in our analysis. Finally, by combining our results with recent results of Hayashi and Tomamichel, we find a novel operational interpretation of the mutual information of a quantum channel N as the optimal type II error exponent when discriminating between a large number of independent instances of N and an arbitrary "worst-case" replacer channel chosen from the set of all replacer channels.

Swiveled Rényi entropies

This paper introduces "swiveled Renyi entropies" as an alternative to the Renyi entropic quantities put forward in [Berta et al., Phys. Rev. A 91, 022333 (2015)]. What distinguishes the swiveled Renyi entropies from the prior proposal of Berta et al. is that there is an extra degree of freedom: an optimization over unitary rotations with respect to particular fixed bases (swivels). A consequence of this extra degree of freedom is that the swiveled Renyi entropies are ordered, which is an important property of the Renyi family of entropies. The swiveled Renyi entropies are however generally discontinuous at $\alpha=1$ and do not converge to the von Neumann entropy-based measures in the limit as $\alpha\rightarrow1$, instead bounding them from above and below. Particular variants reduce to known Renyi entropies, such as the Renyi relative entropy or the sandwiched Renyi relative entropy, but also lead to ordered Renyi conditional mutual informations and ordered Renyi generalizations of a relative entropy difference. Refinements of entropy inequalities such as monotonicity of quantum relative entropy and strong subadditivity follow as a consequence of the aforementioned properties of the swiveled Renyi entropies. Due to the lack of convergence at $\alpha=1$, it is unclear whether the swiveled Renyi entropies would be useful in one-shot information theory, so that the present contribution represents partial progress toward this goal.

Generalized ‘Useful’ Relative Information Measures of Type (α, β)

In this paper we define some new generalized measures of ‘useful’ relative information and study their particular cases. We also obtain their relations with common entropy measures. Moreover, we define the quantum generalized Tsallis entropy and show that projective measurement does not decrease the quantum generalized Tsallis entropy of a quantum state. Finally, we define generalized quantum Tsallis and Renyi relative entropies and discuss their particular cases.

Quantum Hypothesis Testing and the Operational Interpretation of the Quantum Rényi Relative Entropies

We show that the new quantum extension of Renyi's \alpha-relative entropies, introduced recently by Muller-Lennert, Dupuis, Szehr, Fehr and Tomamichel, J. Math. Phys. 54, 122203, (2013), and Wilde, Winter, Yang, Commun. Math. Phys. 331, (2014), have an operational interpretation in the strong converse problem of quantum hypothesis testing. Together with related results for the direct part of quantum hypothesis testing, known as the quantum Hoeffding bound, our result suggests that the operationally relevant definition of the quantum Renyi relative entropies depends on the parameter \alpha: for \alpha<1, the right choice seems to be the traditional definition, whereas for \alpha>1 the right choice is the newly introduced version. As a sideresult, we show that the new Renyi \alpha-relative entropies are asymptotically attainable by measurements for \alpha>1, and give a new simple proof for their monotonicity under completely positive trace-preserving maps.

Relative entropies in conformal field theory.

Relative entropy is a measure of distinguishability for quantum states, and it plays a central role in quantum information theory. The family of Renyi entropies generalizes to Renyi relative entropies that include, as special cases, most entropy measures used in quantum information theory. We construct a Euclidean path-integral approach to Renyi relative entropies in conformal field theory, then compute the fidelity and the relative entropy of states in one spatial dimension at zero and finite temperature using a replica trick. In contrast to the entanglement entropy, the relative entropy is free of ultraviolet divergences, and is obtained as a limit of certain correlation functions. The relative entropy of two states provides an upper bound on their trace distance.

From the quantum relative Tsallis entropy to its conditional form: Separability criterion beyond local and global spectra

The quantum relative Renyi entropy of two density matrices was recently extended when the two do not commute, from which a conditional entropy is identified. This is here extended to the corresponding Tsallis relative entropy and to its conditional form. This new expression of Tsallis conditional entropy is shown to witness entanglement beyond the method based on global and local spectra of composite quantum states. When the reduced density matrix happens to be a maximally mixed state, this conditional entropy coincides with the expression in terms of Tsallis entropies derived earlier by Abe and Rajagopal (Physica A 289, 157 (2001)). Separability range in one parameter family of W and GHZ states with 3 and 4 qubits is explored here and it is shown that the results inferred from negative Tsallis conditional entropy matches with that obtained through Peres partial transpose criteria for one-parameter family of W states, in one of its partitions. The criteria is shown to be non-spectral through its usefulness in identifying entanglement in isospectral density matrices.

Large deviation theory for non-regular location shift family

We apply non-regular extensions of the large deviation theory to non-regular location shift families. Our calculation contains the location shift families generated by Beta distribution, Weibull distribution, and Gamma distribution. We point out the optimal estimator depends on the choice of our criterion in the non-regular case. The limits of relative Renyi entropies play an important role in our derivation.

Identifying set-wise differential co-expression in gene expression microarray data

BackgroundPrevious differential coexpression analyses focused on identification of differentially coexpressed gene pairs, revealing many insightful biological hypotheses. However, this method could not detect coexpression relationships between pairs of gene sets. Considering the success of many set-wise analysis methods for microarray data, a coexpression analysis based on gene sets may elucidate underlying biological processes provoked by the conditional changes. Here, we propose a differentially coexpressed gene sets (dCoxS) algorithm that identifies the differentially coexpressed gene set pairs between conditions.ResultsdCoxS is a two-step analysis method. In each condition, dCoxS measures the interaction score (IS), which represents the expression similarity between two gene sets using Renyi relative entropy. When estimating the relative entropy, multivariate kernel density estimation was used to model gene-gene correlation structure. Statistical tests for the conditional difference between the ISs determined the significance of differential coexpression of the gene set pair. Simulation studies supported that the IS is a representative measure of similarity between gene expression matrices. Single gene coexpression analysis of two publicly available microarray datasets detected no significant results. However, the dCoxS analysis of the datasets revealed differentially coexpressed gene set pairs related to the biological conditions of the datasets.ConclusiondCoxS identified differentially coexpressed gene set pairs not found by single gene analysis. The results indicate that set-wise differential coexpression analysis is useful for understanding biological processes induced by conditional changes.

Entropic characterization of distributive mixing in polymer processing equipment

AbstractMixing is an integral component of most polymer processing operations as material properties are highly influenced by the quality of mixing. The degree of distributive mixing (system homogeneity) is assessed by calculating the evolution of Renyi relative entropies for the minor component along a continuous processing equipment. The Renyi entropy involves a β parameter, which represents weighting given to the concentration of the minor component in small, localized regions. Different aspects of mixing can thus be analyzed, from the amount of void spaces to the concentration of the region where mixing is the worst in terms of the minor component. This method provides a unified, rigorous and flexible way of characterizing distributive mixing. Specifically, distributive mixing is analyzed in a numerical simulation of a twin‐flight single screw extruder by using particle tracking as a method of describing the mixing process dynamics. Renyi entropies are used to examine three different processing conditions in a twin‐flight, single screw extruder. By changing the throttle ratio, the optimal extruder length varied. The relationship between the optimal length and average residence time was good.

Characterization of Distributive Mixing in Polymer Processing Equipment using Renyi Entropies

Abstract A new method for characterization of distributive mixing in processing equipment, based on Renyi entropies, was developed. This method was applied to a twin-flight single screw extruder, in which tracer positions were determined through computer simulations of the flow field. The various entropies were calculated using particle concentrations in equal area domains of the mixer. Renyi entropies, which are function of a parameter β were calculated for extruders of different lengths. We discuss the merit of using Renyi entropies for different values of β by pointing to the different mixing characteristics they probe. The relative Renyi entropy varies between 0 and 1 and represents a measure of distributive mixing quality, with 1 corresponding to perfect mixing and 0 corresponding to poorest mixing. We compare this new method of distributive mixing characterization to traditional ones based on the concepts of Scale and Intensity of Segregation, and the calculations based on Pairwise Correlations and ...