Entropy Gap Research Articles

Reflected entropy and Markov gap in noninertial frames

Reflected entropy and Markov gap in noninertial frames

Reflected entropy and Markov gap in Lifshitz theories

We study the reflected entropy in (1+1)-dimensional Lifshitz field theory whose groundstate is described by a quantum mechanical model. Starting from tripartite Lifshitz groundstates, both critical and gapped, we derive explicit formulas for the Rényi reflected entropies reduced to two adjacent or disjoint intervals, directly in the continuum. We show that the reflected entropy in Lifshitz theory does not satisfy monotonicity, in contrast to what is observed for free relativistic fields. We analytically compute the full reflected entanglement spectrum for two disjoint intervals, finding a discrete set of eigenvalues which is that of a thermal density matrix. Furthermore, we investigate the Markov gap, defined as the difference between reflected entropy and mutual information, and find it to be universal and nonvanishing, signaling irreducible tripartite entanglement in Lifshitz groundstates. We also obtain analytical results for the reflected entropies and the Markov gap in 2 + 1 dimensions. Finally, as a byproduct of our results on reflected entropy, we provide exact formulas for two other entanglement-related quantities, namely the computable cross-norm negativity and the operator entanglement entropy.

Tower of quantum scars in a partially many-body localized system

Isolated quantum many-body systems are often well-described by the eigenstate thermalization hypothesis. There are, however, mechanisms that cause different behavior: many-body localization and quantum many-body scars. Here, we show how one can find disordered Hamiltonians hosting a tower of scars by adapting a known method for finding parent Hamiltonians. Using this method, we construct a spin-1/2 model which is both partially localized and contains scars. We demonstrate that the model is partially localized by studying numerically the level spacing statistics and bipartite entanglement entropy. As disorder is introduced, the adjacent gap ratio transitions from the Gaussian orthogonal ensemble to the Poisson distribution and the entropy shifts from volume-law to area-law scaling. We investigate the properties of scars in a partially localized background and compare with a thermal background. At strong disorder, states initialized inside or outside the scar subspace display different dynamical behavior but have similar entanglement entropy and Schmidt gap. We demonstrate that localization stabilizes scar revivals of initial states with support both inside and outside the scar subspace. Finally, we show how strong disorder introduces additional approximate towers of eigenstates.

Counting, equidistribution and entropy gaps at infinity with applications to cusped Hitchin representations

Abstract We show that if an eventually positive, non-arithmetic, locally Hölder continuous potential for a topologically mixing countable Markov shift with (BIP) has an entropy gap at infinity, then one may apply the renewal theorem of Kesseböhmer and Kombrink to obtain counting and equidistribution results. We apply these general results to obtain counting and equidistribution results for cusped Hitchin representations, and more generally for cusped Anosov representations of geometrically finite Fuchsian groups.

Poisson boundaries of II1 factors

We introduce Poisson boundaries of II$_1$ factors with respect to density operators that give the traces. The Poisson boundary is a von Neumann algebra that contains the II$_1$ factor and is a particular example of the boundary of a unital completely positive map as introduced by Izumi. Studying the inclusion of the II$_1$ factor into its boundary, we develop a number of notions, such as double ergodicity and entropy, that can be seen as natural analogues of results regarding the Poisson boundaries introduced by Furstenberg. We use the techniques developed to answer a problem of Popa by showing that all finite factors satisfy his MV property. We also extend a result of Nevo by showing that property (T) factors give rise to an entropy gap.

Metric Learning in Freewill EEG Pre-Movement and Movement Intention Classification for Brain Machine Interfaces.

Decoding movement related intentions is a key step to implement BMIs. Decoding EEG has been challenging due to its low spatial resolution and signal to noise ratio. Metric learning allows finding a representation of data in a way that captures a desired notion of similarity between data points. In this study, we investigate how metric learning can help finding a representation of the data to efficiently classify EEG movement and pre-movement intentions. We evaluate the effectiveness of the obtained representation by comparing classification the performance of a Support Vector Machine (SVM) as a classifier when trained on the original representation, called Euclidean, and representations obtained with three different metric learning algorithms, including Conditional Entropy Metric Learning (CEML), Neighborhood Component Analysis (NCA), and the Entropy Gap Metric Learning (EGML) algorithms. We examine different types of features, such as time and frequency components, which input to the metric learning algorithm, and both linear and non-linear SVM are applied to compare the classification accuracies on a publicly available EEG data set for two subjects (Subject B and C). Although metric learning algorithms do not increase the classification accuracies, their interpretability using an importance measure we define here, helps understanding data organization and how much each EEG channel contributes to the classification. In addition, among the metric learning algorithms we investigated, EGML shows the most robust performance due to its ability to compensate for differences in scale and correlations among variables. Furthermore, from the observed variations of the importance maps on the scalp and the classification accuracy, selecting an appropriate feature such as clipping the frequency range has a significant effect on the outcome of metric learning and subsequent classification. In our case, reducing the range of the frequency components to 0–5 Hz shows the best interpretability in both Subject B and C and classification accuracy for Subject C. Our experiments support potential benefits of using metric learning algorithms by providing visual explanation of the data projections that explain the inter class separations, using importance. This visualizes the contribution of features that can be related to brain function.

On the Similarity Between von Neumann Graph Entropy and Structural Information: Interpretation, Computation, and Applications

The von Neumann graph entropy is a measure of graph complexity based on the Laplacian spectrum. It has recently found applications in various learning tasks driven by the networked data. However, it is computationally demanding and hard to interpret using simple structural patterns. Due to the close relation between the Laplacian spectrum and the degree sequence, we conjecture that the structural information, defined as the Shannon entropy of the normalized degree sequence, might be a good approximation of the von Neumann graph entropy that is both scalable and interpretable. In this work, we thereby study the difference between the structural information and the von Neumann graph entropy named as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">entropy gap</i> . Based on the knowledge that the degree sequence is majorized by the Laplacian spectrum, we for the first time prove that the entropy gap is between 0 and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\log _{2} e$ </tex-math></inline-formula> in any undirected unweighted graphs. Consequently we certify that the structural information is a good approximation of the von Neumann graph entropy that achieves provable accuracy, scalability, and interpretability simultaneously. This approximation is further applied to two entropy-related tasks: network design and graph similarity measure, where a novel graph similarity measure and the corresponding fast algorithms are proposed. Meanwhile, we show empirically and theoretically that maximizing the von Neumann graph entropy can effectively hide the community structure, and then propose an alternative metric called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">spectral polarization</i> to guide the community obfuscation. Our experimental results on graphs of various scales and types show that the very small entropy gap readily applies to a wide range of simple/weighted graphs. As an approximation of the von Neumann graph entropy, the structural information is the only one that achieves both high efficiency and high accuracy among the prominent methods. It is at least two orders of magnitude faster than SLaQ (Tsitsulin <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> , 2020) with comparable accuracy. Our structural information based methods also exhibit superior performance in downstream tasks such as entropy-driven network design, graph comparison, and community obfuscation.

Is A2261 a Fossil Galaxy Cluster in a Transitional Dynamical State?

Abstract Fossil cluster A2261 is well studied, but previous studies give contradictory results on its dynamical states, such as its X-ray central entropy and magnitude gap. To improve our understanding of its dynamical state, we conduct multiobject spectroscopic observations with Hectospec on the MMT, covering an area out to 5 virial radii from the cluster center, and get improved completeness and membership. Using these new data, we calculate multiple dynamical indicators, including Gaussianity, distance offset, and velocity offset. These indicators suggest that A2261 is moderately relaxed. However, a Dressler–Shectman test reveals a group candidate to the south at a projected distance that is near the virial radius and overlaps with an X-ray tail-like feature. One of the galaxies associated with that group would be sufficiently bright to reduce the fossil magnitude gap. This raises the possibility that A2261 could have recently transited in fossil status, if the group had previously crossed the cluster and is only now found outside. In the cluster outskirts, we see an extended feature of galaxies located on the opposite side of the cluster from the group candidate. On even larger scales, we find that this feature connects, both on the sky and in velocity space, with a long (4.4 Mpc) filamentary structure in the Sloan Digital Sky Survey data. This could support the idea that a group was fed into the cluster through the filament, temporarily breaking the fossil status and resulting in a minor merger that weakly disturbed the intracluster medium of the cluster.

Putting the European Economic Metabolism into Perspective - an Entropy Gap Analysis Based on a Unified Sankey Illustration

Putting the European Economic Metabolism into Perspective - an Entropy Gap Analysis Based on a Unified Sankey Illustration

Modelling Data Aided Sensing With UAVs for Efficient Data Collection

With the increasing adoption of fifth-generation (5G) networks and wide bandwidth availability, unmanned aerial vehicles (UAVs) should now be practicable to stream and operate at different altitudes and upload incredibly large quantities of data from sensors. We develop a novel intelligent sensing framework (Choi, 2020) so that sensing and communication occur on demand, according to a given query. Our query-based Data Aided Sensing (DAS) approach differs from existing methods as we start with selecting relevant nodes based on the query and then allocating UAVs for collecting measurements. We develop an economical and efficient approach to collect measurements from massive passive sensors, which adaptively minimize the entropy gap while scheduling UAVs to sensors. At the heart of our solution is a mathematical approach for estimating entropy, cost and energy consumption, thereby deploying dynamically optimal UAV-sensor association (e.g., optimizing entropy and employing bipartite graph). Our system has been evaluated with extensive simulations, providing insightful observations and findings.

On the Entropy Loss and Gap of Condensers

Many algorithms are proven to work under the assumption that they have access to a source of random, uniformly distributed bits. However, in practice, sources of randomness are often imperfect, giving n random bits that have only k &lt; n min-entropy. The value n − k is called the entropy gap of the source. Randomness condensers are hash functions that hash any such source to a shorter source with reduced entropy gap g . The goal is to lose as little entropy as possible in this process. Condensers also have an error parameter ε and use a small seed of uniformly distributed bits whose length we desire to minimize as well. In this work, we study the exact dependencies between the different parameters of seeded randomness condensers. We obtain a non-explicit upper bound, showing the existence of condensers with entropy loss log (1+log 1/ε / g ) + O (1) and seed length log ( n − k / ε g ) + O (1). In particular, this implies the existence of condensers with O (log 1 / ε) entropy gap and constant entropy loss. This extends (with slightly improved parameters) the non-explicit upper bound for condensers presented in the work of Dodis et al. (2014), which gives condensers with entropy loss at least log log 1 / ε. We also give a non-explicit upper bound for lossless condensers, which have entropy gap g ≥ log 1 / ε / ε + O (1) and seed length log ( n − k / ε 2 g ) + O (1). Furthermore, we address an open question raised in (Dodis et al. 2014), where Dodis et al. showed an explicit construction of condensers with constant gap and O (log log 1/ ε) loss, using seed length O ( n log 1 / ε). In the same article they improve the seed length to O ( k log k ) and ask whether it can be further improved. In this work, we reduce the seed length of their construction to O (log ( n / ε)log ( k / &amp;epsiv)) by a simple concatenation. In the analysis, we use and prove a tight equivalence between condensers and extractors with multiplicative error. We note that a similar, but non-tight, equivalence was already proven by Dodis et al. (Dodis et al. 2014) using a weaker variant of extractors called unpredictability extractors . We also remark that this equivalence underlies the work of Ben-Aroya et al. (Ben-Aroya et al. 2016) and later work on explicit two-source extractors, and we believe it is interesting in its own right.

Coarse Graining, Nonmaximal Entropy, and Power Laws.

We show that coarse graining produces significant and predictable effects on the entropy of states of equilibrium when the scale of coarse graining becomes comparable to that of density fluctuations. We demonstrate that a coarse-grained entropy typically evolves toward a state of effective equilibrium with a lower value than that of the state of maximum entropy theoretically possible. The finer the coarse graining, the greater the drop in effective entropy, and the more relevant the fluctuations around that. Fundamental considerations allow us to derive a remarkable power law that relates coarse graining to the effective entropy gap. Another power law is found that precisely relates the noise range of effective entropy fluctuations to coarse graining. We test both power laws with numerical simulations based on a well-studied two-dimensional lattice gas model. As expected, the effects of these power laws diminish as our description approaches a macroscopic level, eventually disappearing in the thermodynamic limit, where the maximum entropy principle is reasserted.

Entanglement and Majorana edge states in the Kitaev model

We investigate the von Neumann entanglement entropy and Schmidt gap in the ground state of the Kitaev model on the honeycomb lattice for square and cylindrical subsystems. We find that, for both the subsystems, the free-fermionic contribution to the entanglement entropy $S_E$ exhibits signatures of the phase transitions between the gapless and gapped phases. However, we find that $S_E$ does not show an expected monotonic behaviour as a function of the coupling $J_z$ between the suitably defined one-dimensional chains within the gapless phase for either geometry; moreover the system generically reaches a point of minimum entanglement within the gapless phase before the entanglement saturates or increases again until the gapped phase is reached. This may be attributed to the competition between the nearest neighbour correlation functions of the $x-x$ and $z-z$ bonds, and to the onset of gapless $\vec{k}$-modes in the bulk spectrum. On the other hand $S_E$ always monotonically varies with $J_z$ in the gapped phase independent of the subregion size or shape. Finally, further confirming the Li-Haldane conjecture, we find that the Schmidt gap $\Delta$ defined from the entanglement spectrum also signals the topological transitions but only if there are corresponding zero energy Majorana edge states that simultaneously appear or disappear across the transitions. We analytically corroborate some of our results on entanglement entropy, Schmidt gap and the bulk-edge correspondence using perturbation theory.

Fe(II) Spin Transition Materials Including an Amino-Ester 1,2,4-Triazole Derivative, Operating at, below, and above Room Temperature.

A new family of one-dimensional Fe(II) 1,2,4-triazole spin transition coordination polymers for which a modification of anion and crystallization solvent can tune the switching temperature over a wide range, including the room temperature region, is reported. This series of materials was prepared as powders after reaction of ethyl-4H-1,2,4-triazol-4-yl-acetate (αEtGlytrz) with an iron salt from a MeOH/H2O medium affording: [Fe(αEtGlytrz)3](ClO4)2 (1); [Fe(αEtGlytrz)3](ClO4)2·CH3OH (2); [Fe(αEtGlytrz)3](NO3)2·H2O (3); [Fe(αEtGlytrz)3](NO3)2 (4); [Fe(αEtGlytrz)3](BF4)2·0.5H2O (5); [Fe(αEtGlytrz)3](BF4)2 (6); and [Fe(αEtGlytrz)3](CF3SO3)2·2H2O (7). Their spin transition properties were investigated by (57)Fe Mossbauer spectroscopy, superconducting quantum interference device (SQUID) magnetometry, and differential scanning calorimetry (DSC). The temperature dependence of the high-spin molar fraction derived from (57)Fe Mössbauer spectroscopy in 1 reveals an abrupt single step transition between low-spin and high-spin states with a hysteresis loop of width 5 K (Tc(↑) = 296 K and Tc(↓) = 291 K). The properties drastically change with modification of anion and/or lattice solvent. The transition temperatures, deduced by SQUID magnetometry, shift to Tc(↑) = 273 K and Tc(↓) = 263 K for (2), Tc(↑) = 353 K and Tc(↓) = 333 K for (3), Tc(↑) = 338 K and Tc(↓) = 278 K for (4), T(↑) = 320 K and T(↓) = 305 K for (5), Tc(↑) = 106 K and Tc(↓) = 92 K for (6), and T(↑) = 325 K and T(↓) = 322 K for (7). Annealing experiments of 3 lead to a change of the morphology, texture, and magnetic properties of the sample. A dehydration/rehydration process associated with a spin state change was analyzed by a mean-field macroscopic master equation using a two-level Hamiltonian Ising-like model for 3. A new structural-property relationship was also identified for this series of materials [Fe(αEtGlytrz)3](anion)2·nSolvent based on Mössbauer and DSC measurements. The entropy gap associated with the spin transition and the volume of the inserted counteranion shows a linear trend, with decrease in entropy with increasing the size of the counteranion. The first materials of this substance class to display a complete spin transition in both spin states are also presented.

Entanglement structure of the two-channel Kondo model

Two electronic channels competing to screen a single impurity spin, as in the two-channel Kondo model, are expected to generate a ground state with nontrivial entanglement structure. We exploit a spin-chain representation of the two-channel Kondo model to probe the ground-state block entropy, negativity, tangle, and Schmidt gap, using a density matrix renormalization group approach. In the presence of symmetric coupling to the two channels we confirm field-theory predictions for the boundary entropy difference, $\ln (g_{UV}/g_{IR})=\ln(2)/2$, between the ultraviolet and infrared limits and the leading $\ln(x)/x$ impurity correction to the block entropy. The impurity entanglement, $S_{\text{imp}}$, is shown to scale with the characteristic length $\xi_{2CK}$. We show that both the Schmidt gap and the entanglement of the impurity with one of the channels $-$ as measured by the negativity$-$ faithfully serve as order parameters for the impurity quantum phase transition appearing as a function of channel asymmetry, allowing for explicit determination of critical exponents, $\nu\!\approx\! 2$ and $\beta \!\approx\! 0.2$. Remarkably, we find the emergence of tripartite entanglement only in the vicinity of the critical channel-symmetric point.

Dynamics of entanglement entropy and entanglement spectrum crossing a quantum phase transition

We study the time evolution of entanglement entropy and entanglement spectrum in a finite-size system which crosses a quantum phase transition at different speeds. We focus on the Ising model with a time-dependent magnetic field, which is linearly tuned on a time scale $\tau$ . The time evolution of the entanglement entropy displays different regimes depending on the value of $\tau$, showing also oscillations which depend on the instantaneous energy spectrum. The entanglement spectrum is characterized by a rich dynamics where multiple crossings take place with a gap-dependent frequency. Moreover, we investigate the Kibble-Zurek scaling of entanglement entropy and Schmidt gap.

A REALIZATION OF THE HOLOGRAPHIC ENTROPY

The entropy bound for the conventional quantum field theory is known as A3/4. Thus, there is an entropy gap from A3/4 to the holographic entropy A. In this paper, we show that the entropy gap can be overcome by introducing internal degrees of freedom to the system. Using a state counting method, we show explicitly how the familiar holographic thermodynamics with area entropy can be realized. It may help in the understanding of the microscopic origin of the holographic entropy.

Universal finitary codes with exponential tails

In 1977, Keane and Smorodinsky showed that there exists a finitary homomorphism from any finite-alphabet Bernoulli process to any other finite-alphabet Bernoulli process of strictly lower entropy. In 1996, Serafin proved the existence of a finitary homomorphism with finite expected coding length. In this paper, we construct such a homomorphism in which the coding length has exponential tails. Our construction is source-universal, in the sense that it does not use any information on the source distribution other than the alphabet size and a bound on the entropy gap between the source and target distributions. We also indicate how our methods can be extended to prove a source-specific version of the result for Markov chains.

Quantum spherical spin model on hypercubic lattices

We present an alternative treatment to the quantum spherical spin model on $(d\ensuremath{\ge}2)$-dimensional hypercubic lattices, focusing on the effects of quantum $(g)$ and thermal $(T)$ fluctuations, under a uniform magnetic field $h$, on the correlation function, correlation length, entropy, specific heat, and energy gap in the excitation spectrum. Explicit expressions for such quantities are provided close to the $d\ensuremath{\ge}2$ quantum ($g={g}_{c}$, $T=0$) and $d\ensuremath{\ge}3$ thermal $[T={T}_{c}(g)]$ phase transitions in $h=0$, including the low-$T$ quantum regimes near the quantum critical point. In particular, the calculation of the correlation function and correlation length generalizes the results on the $g=0$ classical spherical model. At $T=0$, the zero-field system is gapless at and below ${g}_{c}$; however, a gap opens in the quantum-disordered ground state, $g&gt;{g}_{c}$. Conversely, the null gap for $T\ensuremath{\le}{T}_{c}(g\ensuremath{\ne}0)$ becomes finite as $T\ensuremath{\rightarrow}{T}_{c}(g\ensuremath{\ne}0{)}^{+}$; thus, quantum fluctuations suppress the critical prefactors of observables near ${T}_{c}(g\ensuremath{\ne}0)$, though they are irrelevant to the universality class shared with the gapless classical spherical model. The results on the entropy and specific heat in $g\ensuremath{\ne}0$ circumvent the drawback in classical spherical models concerning the third law of thermodynamics, as $T\ensuremath{\rightarrow}0$.

Directional Ordering of Fluctuations in a Two-Dimensional Compass Model

In the Mott insulating phase of the transition metal oxides, the effective orbital-orbital interaction is directional both in orbital space and in real space. We discuss a classical realization of directional coupling in two dimensions. Despite extensive degeneracy of the ground state, the model exhibits partial orbital ordering in the form of directional ordering of fluctuations at low temperatures stabilized by an entropy gap. Transition to the disordered phase is shown to be in the Ising universality class through exact mapping and multicanonical Monte Carlo simulations.