Typicality Approach Research Articles

Accuracy of the typicality approach using Chebyshev polynomials

Abstract Trace estimators allow us to approximate thermodynamic equilibrium observables with astonishing accuracy. A prominent representative is the finite-temperature Lanczos method (FTLM) which relies on a Krylov space expansion of the exponential describing the Boltzmann weights. Here we report investigations of an alternative approach which employs Chebyshev polynomials. This method turns out to be also very accurate in general, but shows systematic inaccuracies at low temperatures that can be traced back to an improper behavior of the approximated density of states with and without smoothing kernel. Applications to archetypical quantum spin systems are discussed as examples.

Quantum dynamics in the interacting Fibonacci chain

Quantum dynamics on quasiperiodic geometries has recently gathered significant attention in ultra-cold atom experiments where non trivial localised phases have been observed. One such quasiperiodic model is the so called Fibonacci model. In this tight-binding model, non-interacting particles are subject to on-site energies generated by a Fibonacci sequence. This is known to induce critical states, with a continuously varying dynamical exponent, leading to anomalous transport. In this work, we investigate whether anomalous diffusion present in the non-interacting system survives in the presence of interactions and establish connections to a possible transition towards a localized phase. We investigate the dynamics of the interacting Fibonacci model by studying real-time spread of density-density correlations at infinite temperature using the dynamical typicality approach. We also corroborate our findings by calculating the participation entropy in configuration space and investigating the expectation value of local observables in the diagonal ensemble.

Hilbert space average of transition probabilities.

The typicality approach and the Hilbert space averaging method as its technical manifestation are important concepts of quantum statistical mechanics. Extensively used for expectation values we extend them in this paper to transition probabilities. In this context we also find that the transition probability of two random uniformly distributed states is connected to the spectral statistics of the considered operator. Furthermore, within our approach we are capable to consider distributions of matrix elements between states that are not orthogonal. We will demonstrate our quite general result numerically for a kicked spin chain in the integrable resp. chaotic regime.

Relaxation Theory for Perturbed Many-Body Quantum Systems versus Numerics and Experiment.

An analytical prediction is established of how an isolated many-body quantum system relaxes towards its thermal longtime limit under the action of a time-independent perturbation, but still remaining sufficiently close to a reference case whose temporal relaxation is known. This is achieved within the conceptual framework of a typicality approach by showing and exploiting that the time-dependent expectation values behave very similarly for most members of a suitably chosen ensemble of perturbations. The predictions are validated by comparison with various numerical and experimental results from the literature.

An Efficient Top-$K$ Spatial Keyword Typicality and Semantic Query

Existing spatial keyword query processing models mainly consider the spatial proximity and text relevancy between spatial objects and spatial keyword query, which usually makes the top- k answer objects are similar to each other. However, the user hopes to obtain the top- k results that are typical and semantically related to his/her query intention. This paper proposes a top- k spatial keyword typicality and sematic querying approach which can expeditiously provide top- k typical and semantically related objects to the given query. The approach consists of two processing steps. During the offline step, we first analyze the location-semantic relationships between spatial objects by considering both the location similarity and document semantic relevancy between them. For measuring the semantic similarity between documents associated to the spatial objects, we propose two methods, the keyword coupling relationship-based document similarity measure and the Word2Vec-CNN-based document similarity measure. Then, the Gaussian probabilistic density-based estimation method is leveraged to find a few representative objects from the dataset and then the order/permutation of remaining objects in the dataset can be generated corresponding to each representative object. The objects in the permutation are ranked in descending order according to their location-semantic relationships to the representative object. When a spatial keyword query coming, the online processing step first computes the spatial proximity and semantic relevancy between the query and each representative object, and then a small number of orders generated in the offline step can be selected and used at querying time to facilitate top- k typical and semantically related object selection by using the threshold algorithm (TA). Results of a preliminary user study demonstrate our location-semantic relationship measuring method can capture the location similarity and semantic relevancy between spatial objects accurately. The efficiency of typicality analysis and TA-based top- k selection algorithm is also demonstrated.

A Joint Typicality Approach to Compute–Forward

This paper presents a joint typicality framework for encoding and decoding nested linear codes in multi-user networks. This framework provides a new perspective on compute–forward within the context of discrete memoryless networks. In particular, it establishes an achievable rate region for computing a linear combination over a discrete memoryless multiple-access channel (MAC). When specialized to the Gaussian MAC, this rate region recovers and improves upon the lattice-based compute–forward rate region of Nazer and Gastpar, thus providing a unified approach for discrete memoryless and Gaussian networks. Furthermore, our framework provides some valuable insights on establishing the optimal decoding rate region for compute–forward by considering joint decoders, progressing beyond most previous works that consider successive cancellation decoding. Specifically, this paper establishes an achievable rate region for simultaneously decoding two linear combinations of nested linear codewords from $K$ senders.

Dynamical Typicality Approach to Eigenstate Thermalization.

We consider the set of all initial states within a microcanonical energy shell of an isolated many-body quantum system, which exhibit an arbitrary but fixed nonequilibrium expectation value for some given observable A. On the condition that this set is not too small, it is shown by means of a dynamical typicality approach that most such initial states exhibit thermalization if and only if A satisfies the so-called weak eigenstate thermalization hypothesis (wETH). Here, thermalization means that the expectation value of A spends most of its time close to the microcanonical value after initial transients have died out. The wETH means that, within the energy shell, most eigenstates of the pertinent system Hamiltonian exhibit very similar expectation values of A.

Typicality approach to the optical conductivity in thermal and many-body localized phases

We study the frequency dependence of the optical conductivity $\text{Re} \, \sigma(\omega)$ of the Heisenberg spin-$1/2$ chain in the thermal and near the transition to the many-body localized phase induced by the strength of a random $z$-directed magnetic field. Using the method of dynamical quantum typicality, we calculate the real-time dynamics of the spin-current autocorrelation function and obtain the Fourier transform $\text{Re} \, \sigma(\omega)$ for system sizes much larger than accessible to standard exact-diagonalization approaches. We find that the low-frequency behavior of $\text{Re} \, \sigma(\omega)$ is well described by $\text{Re} \, \sigma(\omega) \approx \sigma_\text{dc} + a \, |\omega|^\alpha$, with $\alpha \approx 1$ in a wide range within the thermal phase and close to the transition. We particularly detail the decrease of $\sigma_\text{dc}$ in the thermal phase as a function of increasing disorder for strong exchange anisotropies. We further find that the temperature dependence of $\sigma_\text{dc}$ is consistent with the existence of a mobility edge.

Typicality in spin-network states of quantum geometry

In this work, we extend the so-called typicality approach, originally formulated in statistical mechanics contexts, to $SU(2)$-invariant spin-network states. Our results do not depend on the physical interpretation of the spin network; however, they are mainly motivated by the fact that spin-network states can describe states of quantum geometry, providing a gauge-invariant basis for the kinematical Hilbert space of several background-independent approaches to quantum gravity. The first result is, by itself, the existence of a regime in which we show the emergence of a typical state. We interpret this as the proof that in that regime there are certain (local) properties of quantum geometry which are "universal". Such a set of properties is heralded by the typical state, of which we give the explicit form. This is our second result. In the end, we study some interesting properties of the typical state, proving that the area law for the entropy of a surface must be satisfied at the local level, up to logarithmic corrections which we are able to bound.

Towards application of one-class classification methods to medical data.

In the problem of one-class classification (OCC) one of the classes, the target class, has to be distinguished from all other possible objects, considered as nontargets. In many biomedical problems this situation arises, for example, in diagnosis, image based tumor recognition or analysis of electrocardiogram data. In this paper an approach to OCC based on a typicality test is experimentally compared with reference state-of-the-art OCC techniques—Gaussian, mixture of Gaussians, naive Parzen, Parzen, and support vector data description—using biomedical data sets. We evaluate the ability of the procedures using twelve experimental data sets with not necessarily continuous data. As there are few benchmark data sets for one-class classification, all data sets considered in the evaluation have multiple classes. Each class in turn is considered as the target class and the units in the other classes are considered as new units to be classified. The results of the comparison show the good performance of the typicality approach, which is available for high dimensional data; it is worth mentioning that it can be used for any kind of data (continuous, discrete, or nominal), whereas state-of-the-art approaches application is not straightforward when nominal variables are present.

Beyond storm and stress: Typicality, transactions, timing, and temperament to account for adolescent change.

Adolescent "storm and stress" has been a focal yet controversial developmental framework for over a century. In the present article, we challenge the current perspective that storm and stress is neither ubiquitous nor inevitable but probable. Instead, we argue, storm and stress is a vestigial developmental framework, and we propose a more comprehensive approach to understanding adolescent-typical changes based on six premises: (1) The biological changes of adolescence are inevitable and ubiquitous; (2) adolescent biological changes drive various mechanisms of adolescent behavior; (3) adolescent biological changes are shaped by environmental influences; (4) individual differences in adolescent emotional-behavior changes are domain specific and vary in intensity; (5) there are individual differences in the age of onset and duration of periods of adolescent change; and (6) individual differences in the duration and intensity of transitions in emotional arousal are functionally modulated by burgeoning emotion regulation skills. We conclude with the more comprehensive 4T (typicality, transactions, temperament, and timing) approach and suggestions to guide adolescent research in the 21st century.

Loop-closing: A typicality approach

Loop-closing has long been identified as a critical issue when building maps from local observations. Topological mapping methods abstract the problem of how loops are closed from the problem of how to determine the metrical layout of places in the map and how to deal with noisy sensors. The typicality problem refers to the identification of new classes in a general classification context. This typicality concept is used in this paper to help a robot acquire a topological representation of the environment during its exploration phase. The problem is addressed using the INCA statistic which follows a distance-based approach. In this paper we describe a place recognition approach based on match testing by means of the INCA test. We describe the theoretical basis of the approach and present extensive experimental results performed in both a simulated and a real robot-environment system; Behaviour Based philosophy is used to construct the whole control architecture. Obtained results show the validity of the approach.