Entropy Of Subset Research Articles

The holographic map of an evaporating black hole

We construct a holographic map that takes the semi-classical state of an evaporating black hole and its Hawking radiation to a microscopic model that reflects the scrambling dynamics of the black hole. The microscopic model is given by a nested sequence of random unitaries, each one implementing a scrambling time step of the black hole evolution. Differently from other models, energy conservation and the thermal nature of the Hawking radiation are taken into account. We show that the QES formula follows for the entropy of multiple subsets of the radiation and black hole. We further show that a version of entanglement wedge reconstruction can be proved by computing suitable trace norms and quantum fidelities involving the action of a unitary on a subset of Hawking partners. If the Hawking partner is in an island, its unitary can be reconstructed by a unitary on the radiation. We also adopt a similar setup and analyse reconstruction of unitaries acting on an infalling system.

Relative topological conditional entropy and a Ledrappier's type variational principle for it

In this paper we provide a sufficient condition for the existence of invariant measures with maximal relative measure-theoretic entropy, by introducing a new invariant for any factor map between topological dynamical systems, the concept of relative topological conditional entropy. It is proved a Ledrappier's type variational principle concerning relative topological conditional entropy. Consequently, if the factor map has zero relative topological conditional entropy (such a factor map is called asymptotically $ h $-expansive), then there exist invariant measures with maximal relative measure-theoretic entropy (i.e., whose relative measure-theoretic entropy equals exactly relative topological conditional entropy of the factor map). We explore further properties of relative topological conditional entropy, for example, the addition formula for a product factor map, the estimation of it for composition of two factor maps, and so on. We also interpret relative topological conditional entropy, respectively, using relative topological entropy of subsets and Bowen's dimensional entropy.

An entropy formula over the dynamical balls and its applications

The Bowen topological entropy was introduced by Bowen in way which resembles the Hausdorff dimension. It is well-known that the Bowen topological entropy and the topological entropy of subsets are not equal in general. We show in the paper that the supremum of the Bowen topological entropy of the dynamical balls is the same as that of the topological entropy of the dynamical balls, which naturally implies that every positively countably expansive map is also positively entropy expansive. This answers a question posed by Artigue, Carvalho, Cordeiro, and Vieitez [Proc. Amer. Math. Soc. 150 (2022), pp. 3369–3378]. Besides, our main result is applied to find some other interesting phenomena.

Topological Entropy for Arbitrary Subsets of Infinite Product Spaces

In this note, a notion of generalized topological entropy for arbitrary subsets of the space of all sequences in a compact topological space is introduced. It is shown that for a continuous map on a compact space, the generalized topological entropy of the set of all orbits of the map coincides with the classical topological entropy of the map. Some basic properties of this new notion of entropy are considered; among them are the behavior of the entropy with respect to disjoint union, cartesian product, component restriction and dilation, shift mapping, and some continuity properties with respect to Vietoris topology. As an example, it is shown that any self-similar structure of a fractal given by a finite family of contractions gives rise to a notion of intrinsic topological entropy for subsets of the fractal. A generalized notion of Bowen’s entropy associated to any increasing sequence of compatible semimetrics on a topological space is introduced and some of its basic properties are considered. As a special case for $1\leq p\leq \infty $ , the Bowen p-entropy of sets of sequences of any metric space is introduced. It is shown that the notions of generalized topological entropy and Bowen $\infty $ -entropy for compact metric spaces coincide.

Islands in the stream of Hawking radiation

We consider the island formula for the entropy of subsets of the Hawking radiation in the adiabatic limit where the black hole evaporation is very slow. We find a simple concrete ‘on-shell’ formula for the generalized entropy which involves the image of the island out in the stream of radiation, the ‘island in the stream’. The resulting recipe for the entropy allows us to calculate the quantum information properties of the radiation and verify various constraints including the Araki-Lieb inequality and strong subadditivity.

Polynomial Entropy of Subsets for Free Semigroup Actions

Polynomial Entropy of Subsets for Free Semigroup Actions

Decimation and interleaving operations in one-sided symbolic dynamics

This paper studies subsets of one-sided shift spaces on a finite alphabet. Such subsets arise in symbolic dynamics, in fractal constructions, and in number theory. We study a family of decimation operations, which extract subsequences of symbol sequences in infinite arithmetic progressions, and show these operations are closed under composition. We also study a family of n-ary interleaving operations, one for each n≥1. Given subsets X0,X1,...,Xn−1 of such a shift space, the n-ary interleaving operation produces a set whose elements combine individual elements xi, one from each Xi, by interleaving their symbol sequences in arithmetic progressions (modn). We determine algebraic relations between decimation and interleaving operations and the shift map. We study set-theoretic n-fold closure operations X↦X[n], which interleave decimations of X of modulus level n. A set is n-factorizable if X=X[n]. The n-fold interleaving operations are closed under composition and are idempotent. To each X we assign the set N(X) of all values n≥1 for which X=X[n]. We characterize the possible sets N(X) as nonempty sets of positive integers that form a distributive lattice under the divisibility partial order and are downward closed under divisibility. We show that all sets of this type occur. We introduce a class of weakly shift-stable sets and show that this class is closed under all decimation, interleaving, and shift operations. We study two notions of entropy for subsets of the full one-sided shift and show that they coincide for weakly shift-stable X, but can be different in general. We give a formula for entropy of interleavings of weakly shift-stable sets in terms of individual entropies.

Hawking radiation correlations of evaporating black holes in JT gravity

We consider the Hawking radiation emitted by an evaporating black hole in JT gravity and compute the entropy of arbitrary subsets of the radiation in the slow evaporation limit, and find a zoo of possible island saddles. The Hawking radiation is shown to have long range correlations. We compute the mutual information between early and late modes and bound from below their squashed entanglement. A small subset of late modes are shown to be correlated with modes in a suitably large subset of the radiation previously emitted as well as later modes. We show how there is a breakdown of the semi-classical approximation in the form of a violation of the Araki–Lieb triangle entropy inequality, if the interior of the black hole and the radiation are considered to be separate systems. Finally, we consider how much of the radiation must be collected, and how early, to recover information thrown into the black hole as it evaporates.

The Explicit Coding Rate Region of Symmetric Multilevel Diversity Coding

It is well known that superposition coding, namely separately encoding the independent sources, is optimal for symmetric multilevel diversity coding (SMDC) (Yeung-Zhang 1999) for any L ≥ 2, where L is the number of levels of the coding system. However, the characterization of the coding rate region therein involves uncountably many linear inequalities and the constant term (i.e., the lower bound) in each inequality is given in terms of the solution of a linear optimization problem. Thus this implicit characterization of the coding rate region does not enable the determination of the achievability of a given rate tuple. In principle, the achievability of a given rate tuple can be determined by direct computation, but the complexity is prohibitive even for L = 5. In this paper, for any fixed L, we obtain in closed form a finite set of linear inequalities for characterizing the coding rate region. We further show by the symmetry of the problem that only a much smaller subset of this finite set of inequalities needs to be verified in determining the achievability of a given rate tuple. Yet, the cardinality of this smaller set grows at least exponentially fast with L. We also present a subset entropy inequality, which together with our explicit characterization of the coding rate region, is sufficient for proving the optimality of superposition coding.

Bowen topological entropy of subsets for amenable group actions

We extend the definition of Bowen topological entropy of subsets to continuous action of amenable groups on a compact metrizable space. We investigate how Bowen topological entropy behaves for restricted actions of finite-index subgroups and verify the validity of the subgroup formula. We also show that the Bowen topological entropy of subsets for amenable group actions can be determined via the local entropies of measures. As applications, we compute the Bowen topological entropy of a subset for Bernoulli shift over amenable groups.

A Billingsley-type theorem for the pressure of an action of an amenable group

This paper extends the definition of Bowen topological entropy of subsets to Pesin-Pitskel topological pressure for the continuous action of amenable groups on a compact metric space. We introduce the local measure theoretic pressure of subsets and investigate the relation between local measure theoretic pressure of Borel probability measures and Pesin-Pitskel topological pressure on an arbitrary subset of a compact metric space.

Subset Source Coding

This paper studies the fundamental limits of storage for structured data, where statistics and structure are both critical to the application. Accordingly, a framework is proposed for optimal lossless and lossy compression of subsets of the possible realizations of a discrete memoryless source (DMS). For the lossless subset-compression problem, it turns out that the optimal source code may not index the conventional source-typical sequences, but rather index certain subset-typical sequences consistent with the subset of interest. Building upon an achievability and a strong converse, an analytic expression is given, based on the Shannon entropy, relative entropy, and subset entropy, which identifies such subset-typical sequences for a broad class of subsets of a DMS. Intuitively, subset-typical sequences belong to those typical sets which highly intersect the subset of interest but are still closest to the source distribution in the sense of relative entropy. For the lossy subset-compression problem, an upper bound is derived on the subset rate-distortion function in terms of the subset mutual information optimized over the set of conditional distributions that satisfy the expected distortion constraint with respect to the subset-typical distribution and over a set of certain auxiliary subsets. By proving a strong converse result, this upper bound is shown to be tight for a class of symmetric subsets. As shown in our numerical examples, more often than not, one achieves a gain in the fundamental limits, in that the optimal compression rate for the subset in both the lossless and lossy settings can be strictly smaller than the source entropy and the source rate-distortion function, respectively, although exceptions are also possible.

Harnessing Bursty Interference in Multicarrier Systems With Output Feedback

We study parallel two-user interference channels when the interference is bursty and feedback is available from the respective receivers. Presence of interference in each subcarrier is modeled as a memoryless Bernoulli random state. The states across subcarriers are drawn from an arbitrary joint distribution with the same marginal probability for each subcarrier and instantiated independent and identically distributed (i.i.d.) over time. For the linear deterministic setup with symmetric interference in each subcarrier, we give a complete characterization of the capacity region. For the analogous setup with Gaussian noise, we give outer bounds and a tight generalized degrees of freedom characterization. We propose a novel helping mechanism, which enables subcarriers in very strong interference regime to help in recovering interfered signals for subcarriers in strong and weak interference regimes. Depending on the interference and burstiness regime, the inner bounds either employ the proposed helping mechanism to code across subcarriers or treat the subcarriers separately. The outer bounds demonstrate a connection to a subset entropy inequality by Madiman and Tetali.

Bowen entropy for actions of amenable groups

Bowen introduced a definition of topological entropy of subset inspired by Hausdorff dimension in 1973 [1]. In this paper we consider the Bowen entropy for amenable group action dynamical systems and show that, under the tempered condition, the Bowen entropy of the whole compact space for a given Folner sequence equals the topological entropy. For the proof of this result, we establish a variational principle related to the Bowen entropy and the Brin–Katok local entropy formula for dynamical systems with amenable group actions.

Tri-level Thresholding using Invasive Weed Optimization based on Nonextensive Fuzzy Entropy

This study presents a tri-level thresholding method for image segmentation with invasive weed optimization (IWO) algorithm. The objective of the proposed approach is to handle the nonextensivity and vagueness of image in segmentation, in the meanwhile to reduce the computation time. In this study, the histogram of image is converted to fuzzy domain by membership function firstly. Then the thresholding method is constructed through maximizing the sum of nonextensive entropy of subsets of the each part of fuzzy histogram. The IWO algorithm is used to search the optimal thresholds to reduce the computation time in the new method. Experiments on synthetic and real-world images are given to demonstrate the effectiveness of the proposed approach compared with the other methods.

Symmetrical Multilevel Diversity Coding and Subset Entropy Inequalities

Symmetrical multilevel diversity coding (SMDC) is a classical model for coding over distributed storage. In this setting, a simple separate encoding strategy known as superposition coding was shown to be optimal in terms of achieving the minimum sum rate (Roche, Yeung, and Hau, 1997) and the entire admissible rate region (Yeung and Zhang, 1999) of the problem. The proofs utilized carefully constructed induction arguments, for which the classical subset entropy inequality of Han (1978) played a key role. This paper includes two parts. In the first part the existing optimality proofs for classical SMDC are revisited, with a focus on their connections to subset entropy inequalities. First, a new sliding-window subset entropy inequality is introduced and then used to establish the optimality of superposition coding for achieving the minimum sum rate under a weaker source-reconstruction requirement. Second, a subset entropy inequality recently proved by Madiman and Tetali (2010) is used to develop a new structural understanding to the proof of Yeung and Zhang on the optimality of superposition coding for achieving the entire admissible rate region. Building on the connections between classical SMDC and the subset entropy inequalities developed in the first part, in the second part the optimality of superposition coding is further extended to the cases where there is either an additional all-access encoder (SMDC-A) or an additional secrecy constraint (S-SMDC).

Cell and tissue deformation measurements: Texture correlation with third-order approximation of displacement gradients

Cells remarkably are capable of large deformations during motility and when subjected to mechanical force. Measurement of mechanical deformation (i.e. displacements, strain) is critical to understand functional changes in cells and biological tissues following disease, and to elucidate basic relationships between applied force and cellular biosynthesis. Microscopy-based imaging modalities provide the ability to noninvasively visualize small cell or tissue structures and track their motion over time, often using two-dimensional (2D) digital image (texture) correlation algorithms. For the measurement of complex and nonlinear motion in cells and tissues, implementation of texture correlation algorithms with high order approximations of displacement mapping terms are needed to minimize error. Here, we extend a texture correlation algorithm with up to third-order approximation of displacement mapping terms for the measurement of cell and tissue deformation. We additionally investigate relationships between measurement error and image texture, defined by subset entropy. Displacement measurement error is significantly reduced when the order of displacement mapping terms in the texture correlation algorithm matches or exceeds the order of the deformation observed. Displacement measurement error is also inversely proportional to subset entropy, with well-defined cell and tissue structures leading to high entropy and low error. For cell and tissue studies where complex or nonlinear displacements are expected, texture correlation algorithms with high order terms are required to best characterize the observed deformation.

Bowen’s equation in the non-uniform setting

AbstractWe show that Bowen’s equation, which characterizes the Hausdorff dimension of certain sets in terms of the topological pressure of an expanding conformal map, applies in greater generality than has been heretofore established. In particular, we consider an arbitrary subset Z of a compact metric space and require only that the lower Lyapunov exponents be positive on Z, together with a tempered contraction condition. Among other things, this allows us to compute the dimension spectrum for Lyapunov exponents for maps with parabolic periodic points, and to relate the Hausdorff dimension to the topological entropy for arbitrary subsets of symbolic space with the appropriate metric.

Study of optimal subset size in digital image correlation of speckle pattern images

This paper investigates the effect of subset size, associated with image pattern quality and subset displacement functions, on the accuracy of deformation measurements by digital image correlation(DIC). A concept of subset entropy is introduced in this work to quantify the subset image pattern quality for DIC analysis and its effectiveness was demonstrated by experimental studies. By employing white-light images with almost uniform subset entropy and first-order displacement functions, the effect of subset size on DIC analysis was investigated for the deformation cases of translation, uniform deformation, and simulated quadratic deformation, respectively. The results show that the chosen subset size must be large enough for precise displacement measurements when subset displacement functions match underlying actual deformation. On the other hand, optimal subset size in DIC for nonhomgeneous deformation measurements appears as a result of a tradeoff between the influence of random errors and systematic errors.

Metric Entropy of Subsets of Absolutely Convergent Fourier Series

We estimate the metric entropy of compact subsets of the algebra A of absolutely convergent Fourier series