Bound Of Entropy Research Articles

Some inequalities for spectral geometric mean with applications

Recently, the spectral geometric mean has been studied by some papers. In this paper, we first estimate the Hölder-type inequality of the spectral geometric mean of positive invertible operators on the Hilbert space for all real order in terms of the generalized Kantorovich constant and show the relation between the weighted geometric mean and the spectral geometric one under the usual operator order. Moreover, we show their operator norm version. Next, in the matrix case, we show the log-majorization for the spectral geometric mean and their applications. Among others, we show the order relation among three quantum Tsallis relative entropies. Finally we give a new lower bound of the Tsallis relative entropy.

Entropy Bounds for Device-Independent Quantum Key Distribution with Local Bell Test.

One of the main challenges in device-independent quantum key distribution (DIQKD) is achieving the required Bell violation over long distances, as the channel losses result in low overall detection efficiencies. Recent works have explored the concept of certifying nonlocal correlations over extended distances through the use of a local Bell test. Here, an additional quantum device is placed in close proximity to one party, using short-distance correlations to verify nonlocal behavior at long distances. However, existing works have either not resolved the question of DIQKD security against active attackers in this setup, or used methods that do not yield tight bounds on the key rates. In this work, we introduce a general formulation of the key rate computation task in this setup that can be combined with recently developed methods for analyzing standard DIQKD. Using this method, we show that if the short-distance devices exhibit sufficiently high detection efficiencies, positive key rates can be achieved in the long-distance branch with lower detection efficiencies as compared to standard DIQKD setups. This highlights the potential for improved performance of DIQKD over extended distances in scenarios where short-distance correlations are leveraged to validate quantum correlations.

Entanglement island and Page curve for one-sided charged black hole

In this paper, we extend the method of calculating the entanglement entropy of Hawking radiation of black holes using the “in” vacuum state, which describes one-sided asymptotically flat neutral black hole formed by gravitational collapse, to dynamic charged black holes. We explore the influence of charge on the position of the boundary of island ∂I and the Page time. Due to their distinct geometric structures, we discuss non-extremal and extremal charged black holes separately. In non-extremal cases, the emergence of island saves the bound of entropy at late times, and the entanglement entropy of Hawking radiation satisfies the Page curve. Moreover, we also find that the position of the boundary of island ∂I depends on the position of the cutoff surface (observers), differing from the behavior in eternal charged black holes. In extremal black holes, when the island exists, the entanglement entropy is approximately equal to the Bekenstein-Hawking entropy, while the entanglement entropy becomes ill-defined when island is absent. Our analysis underscores how different geometric configurations significantly influence the behavior of entropy.

How are Entanglement Entropies Related to Entropy Bounds?

How are Entanglement Entropies Related to Entropy Bounds?

Generalized covariant entropy bound in Einstein gravity with quadratic curvature corrections

We explore the generalized covariant entropy bound in the theory where Einstein gravity is perturbed by quadratic curvature terms, which can be viewed as the first-order quantum correction to Einstein gravity. By replacing the Bekenstein-Hawking entropy with the holographic entanglement entropy of this theory and introducing two reasonable physical assumptions, we demonstrate that the corresponding Generalized Covariant Entropy Bound is satisfied under a first-order approximation of the perturbation from the quadratic curvature terms. Our findings suggest that the entropy bound and the Generalized Second Law of black holes are satisfied in the Einstein gravity under the first-order perturbation from the quadratic curvature corrections, and they also imply that the generalized covariant entropy bound may still hold even after considering the quantum correction of gravity, but in this case, we may need to use holographic entanglement entropy as the formula for gravitational entropy.

Thermodynamics of the Einstein-Maxwell system

At first glance, thermodynamic properties of gravity with asymptotically AdS conditions and those with box boundary conditions, where the spatial section of the boundary is a sphere of finite radius, appear similar. Both exhibit a similar phase structure and Hawking-Page phase transition. However, when we introduce a U(1) gauge field to the system, discrepancies in thermodynamic properties between the two cases has been reported in [7] (JHEP 11 (2016) 041). In this paper, by accepting the assumption that all Euclidean saddles contribute to the partition function, I found that these discrepancies are resolved due to the contribution from the “bag of gold (BG),” which is the class of Euclidean geometries whose area of bolt is bigger than that of the boundary. As a result, the Hawking-Page phase structure is restored, with the unexpected properties that the upper bound of thermodynamic entropy is always larger than the boundary area divided by 4G when the chemical potential is non-zero, and that such high entropy states are realized at sufficiently high temperature.

The influence of hyperchaoticity, synchronization, and Shannon entropy on the performance of a physical reservoir computer.

In this paper, we analyze the dynamic effect of a reservoir computer (RC) on its performance. Modified Kuramoto's coupled oscillators are used to model the RC, and synchronization, Lyapunov spectrum (and dimension), Shannon entropy, and the upper bound of the Kolmogorov-Sinai entropy are employed to characterize the dynamics of the RC. The performance of the RC is analyzed by reproducing the distribution of random, Gaussian, and quantum jumps series (shelved states) since a replica of the time evolution of a completely random series is not possible to generate. We demonstrate that hyperchaotic motion, moderate Shannon entropy, and a higher degree of synchronization of Kuramoto's oscillators lead to the best performance of the RC. Therefore, an appropriate balance of irregularity and order in the oscillator's dynamics leads to better performances.

Multi-LRA: Multi logical residual architecture for spiking neural networks

Recently, it has been noticed that certain current state-of-the-art (SOTA) spiking neural networks incorporated non-spike data by use of residual connections and count coding. However, these architectures may not be well-suited for the neuromorphic chips optimized primarily for processing binary inputs and floating-point weights. On the other hand, some pure-spike structures, such as SEW-AND and SEW-IAND, have been found to exhibit spike vanishing and limited capacity. To overcome these shortcomings, we analyze the energy consumption and expressive ability of binary and integer inputs on neuromorphic chips, and then propose a new Multi Logical Residual Architecture (Multi-LRA), which eliminates non-spike computation on neuromorphic chips and addresses the issues of spike vanishing and limited capacity in SEW-AND and SEW-IAND. Furthermore, we prove that the upper bound of conditional entropy of Multi-LRA is higher than that of SEW-AND and logical-OR, which means that Multi-LRA has better capacity and may avoid the spike vanishing. Finally, experiments on CIFAR-10 have shown that Multi-LRA can significantly reduce energy consumption because of its pure-spike computation and logical operation, where Multi-LRA-ResNet50 can reach the SOTA accuracy 96.27% in just 4 time steps. In addition, the effectiveness of Multi-LRA has been validated on ImageNet, CIFAR10-DVS and DVS128 Gesture.

Entropy Bounds for Self-Shrinkers with Symmetries

Entropy Bounds for Self-Shrinkers with Symmetries

Are Entropy Bounds Epistemic?

Are Entropy Bounds Epistemic?

Fgh-Convex Functions and Entropy Bounds

In this paper, we introduce an universal definition (fgh-convex) that results in several types of convexity. Particular cases of the fgh-convex are for instance the harmonically convex, geometrically convex, GA-convex, log-convex, and several others. Also, we obtain some useful inequalities such as Jensen, generalization of Jensen Hermite-Hadamard, Mercer inequalities. Moreover, with the use of these inequalities, we obtained bounds for Shannon’s entropy and Kapur’s entropy. Finally, we found an application of the obtained inequalities in means.

Statistical solutions and Kolmogorov entropy for the lattice long-wave–short-wave resonance equations in weighted space

This article studies the lattice long-wave–short-wave resonance equations in weighted spaces. The authors first prove the global well-posedness of the initial value problem and the existence of the pullback attractor for the process generated by the solution mappings in the weighted space. Then they establish that the process possesses a family of invariant Borel probability measures supported by the pullback attractor. Afterwards, they verify that this family of Borel probability measures satisfies the Liouville theorem and is a statistical solution of the lattice long-wave–short-wave resonance equations. Finally, they prove an upper bound of the Kolmogorov entropy of the statistical solution.

Topological entropy bounds multiscale entropy

In this paper, we introduce the notion of topological entropy of a discrete dynamical system given by a continuous map f:X→X, where X is metric and compact, with respect to observations of its orbits given by a continuous map φ:X→Rk. We apply our definitions to obtain an upper bound of multiscale entropy, which has been used in time series analysis.

Privacy Amplification Strategies in Sequential Secret Key Distillation Protocols Based on Machine Learning

It is well known that Renyi’s entropy of order 2 determines the maximum possible length of the distilled secret keys in sequential secret key distillation protocols so that no information is leaked to the eavesdropper. There have been no attempts to estimate this key quantity based on information available to the legitimate parties to this protocol in the literature. We propose a new machine learning system, which estimates the lower bound of conditional Renyi entropy with high accuracy, based on 13 characteristics locally measured on the side of legitimate participants. The system is based on a prediction intervals deep neural network, trained for a given source of common randomness. We experimentally evaluated this result for two different sources, namely 14 and 6-dimensional EEG signals, of 50 participants, with varying advantage distillation and information reconciliation strategies with and without additional lossless compression block. Across all proposed systems and analyzed sources on average, the best machine learning strategy, called the hybrid strategy, increases the quantity of generated keys 2.77 times compared to the classical strategy. By introducing the Huffman lossless coder before the PA block, the loss of potential source randomness was reduced from 68.48% to a negligible 0.75%, while the leakage rate per one bit remains in the order of magnitude 10−4.

Constraints on the Duration of Inflation from Entanglement Entropy Bounds

Using the fact that we only observe those modes that exit the Hubble horizon during inflation, one can calculate the entanglement entropy of such long-wavelength perturbations by tracing out the unobservable sub-Hubble fluctuations they are coupled with. On requiring that this perturbative entanglement entropy, which increases with time, obey the covariant entropy bound for an accelerating background, we find an upper bound on the duration of inflation. This presents a new perspective on the (meta-)stability of de Sitter spacetime and an associated lifetime for it.

IR/UV mixing, towers of species and swampland conjectures

By applying the Covariant Entropy Bound (CEB) to an EFT in a box of size 1/ΛIR one obtains that the UV and IR cut-offs of the EFT are necessarily correlated. We argue that in a theory of Quantum Gravity (QG) one should identify the UV cutoff with the ‘species scale’, and give a general algorithm to calculate it in the case of multiple towers becoming light. One then obtains an upper bound on the characteristic mass scale of the tower in terms of the IR cut-off, given by Mtower ≲ {left({Lambda}_{mathrm{IR}}right)}^{2{alpha}_D} in Planck units, with αD = (D − 2 + p)/2p(D − 1), where p depends on the density of states. Identifying the IR cut-off with a (non-vanishing) curvature in AdS one reproduces the statement of the AdS Distance Conjecture (ADC), also giving an explicit lower bound for the α exponent. In particular, we find that the CEB implies α ≥ 1/2 in any dimension if there is a single KK tower, both in AdS and dS vacua. However values α < 1/2 are allowed if the particle tower is multiple or has a string component. We also consider the CKN constraint coming from avoiding gravitational collapse which further requires in general α ≥ 1/D for the lightest tower. We analyse the case of the DGKT-CFI class of Type IIA orientifold models and show it has both particle and string towers below the species scale, so that a careful analysis of how the ADC is defined is needed. We find that this class of models obey but do not saturate the CEB. The UV/IR constraints found apply to both AdS and dS vacua. We comment on possible applications of these ideas to the dS Swampland conjecture as well as to the observed dS phase of the universe.

Relative entropy of k-order edge capacity for nodes similarity analysis

The solutions to many problems on complex networks depend on the calculation of the similarity between nodes. The existing methods face the problems of the lack of hierarchical information richness or large computational requirements. In order to flexibly analyze the similarity of nodes on an optional multi-order scale as needed, we propose a novel method for calculating the similarity based on the relative entropy of [Formula: see text]-order edge capacity in this paper. The distribution of edges affects the network heterogeneity, information propagation, node centrality and so on. Entropy of [Formula: see text]-order edge capacity can represent the edge distribution feature in the range of [Formula: see text]-order of node. It increases as [Formula: see text] increases and converges at the eccentricity of the node. Relative entropy of [Formula: see text]-order edge capacity can be used to compare the similarity of edge distribution between nodes within [Formula: see text]-order. As order [Formula: see text] increases, upper bound of the relative entropy possibly increases. Relative entropy gets the maximum when nodes compared with isolated nodes. By quantifying the effect difference of the most similar nodes on the network structure and information propagation, we compared relative entropy of [Formula: see text]-order edge capacity with some major similarity methods in the experiments, combined with visual analysis. The results show the rationality and effectiveness of the proposed method.

Information content and minimum-length metric: A drop of light

In the vast amount of results linking gravity with thermodynamics, statistics, information, a path is described which tries to explore this connection from the point of view of (non)locality of the gravitational field. First the emphasis is put on that well-known thermodynamic results related to null hypersurfaces (i.e. to lightsheets and to generalized covariant entropy bound) can be interpreted as implying an irreducible intrinsic nonlocality of gravity. This nonlocality even if possibly concealed at ordinary scales(depending on which matter is source of the gravitational field, and which matter we use to probe the latter) unavoidably shows up at the smallest scales, read the Planck length $l_p$, whichever are the circumstances we are considering. Some consequences are then explored of this nonlocality when embodied in the fabric itself of spacetime by endowing the latter with a minimum length $L$, in particular the well-known and intriguing fact that this brings to get the field equations, and all of gravity with it, as a statistical-mechanical result. This is done here probing the neighborhood of a would-be (in ordinary spacetime) generic event through lightsheets (instead of spacelike or timelike geodesic congruences as in other accounts) from it. The tools for these derivations are nonlocal quantities, and among them the minimum-length Ricci scalar stands out both for providing micro degrees of freedom for gravity in the statistical account and for the fact that intriguingly the ordinary, or `classical', Ricci scalar can not be recovered from it in the $L\to 0$ limit. Emphasis is put on that classical gravity is generically obtained this way for $\hbar\ne 0$, but not in the $\hbar\to 0$ limit (the statistically derived field equations become singular in this limit), adding to previous results in this sense. (truncated Abstract; see the paper for full Abstract)

Modular Computation of Restoration Entropy for Networks of Systems: A Dissipativity Approach

The problem of state estimation based on information received over a finite bit rate channel gives rise to the study of minimal bit rate above which state can be estimated with any desired accuracy. In the past few years, researchers have studied the minimal average bit rate which is sufficient enough for state estimation such that the estimation error stays within a given factor of its initial value. The notion of restoration entropy characterizes this type of bit rate. Recent results proposed numerical schemes to estimate restoration entropy by the computation of singular values of the linearized systems. Such schemes are either complex to implement or suffer severely from computational complexity and the size of the state dimension. In this paper, we describe a modular approach to compute an upper bound of the restoration entropy of a large network by decomposing the network to an interconnection of smaller subsystems. Then, we formulate a distributed optimization problem which is solved for each subsystem separately and then their optimization results are composed to get an upper bound of the restoration entropy for the overall network. We illustrate the effectiveness of our results by two examples.

Inflationary Implications of the Covariant Entropy Bound and the Swampland de Sitter Conjectures

We present a proposal to relate the de Sitter conjecture (dSC) with the time dependence of fluxes via the covariant entropy bound (CEB). By assuming an early phase of accelerated expansion where the CEB is satisfied, we take into account a contribution from time-dependent flux compactification to the four-dimensional entropy which establishes a bound on the usual slow-roll parameters ηH and ϵH. We also show an explicit calculation of entropy from a toroidal flux compactification, from a transition amplitude of time-dependent fluxes which allows us to determine the conditions on which the bounds on the slow-roll parameters are in agreement to the dSC.