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The Eigenstate Thermalization Hypothesis (ETH) has been highly influential in explaining thermodynamic behavior of closed quantum systems. As of yet, it is unclear whether and how the ETH applies to non-Hermitian systems. Here, we introduce a framework that extends the ETH to non-Hermitian systems. It hinges on a suitable choice of basis composed of right eigenvectors of the non-Hermitian model, a choice we motivate based on physical arguments. In this basis, and after correctly accounting for the nonorthogonality of non-Hermitian eigenvectors, expectation values of local operators reproduce the well-known ETH prediction for Hermitian systems. We illustrate the validity of the modified framework on non-Hermitian random-matrix and Sachdev--Ye--Kitaev models. Our results thus generalize the ETH to the non-Hermitian setting, and they illustrate the importance of the correct choice of basis to evaluate physical properties., Comment: 5+5 pages, 3+3 figures

Entanglement is assuming a central role in modern quantum many-body physics. Yet, for lattice gauge theories its certification remains extremely challenging. A key difficulty stems from the local gauge constraints underlying the gauge theory, which separate the full Hilbert space into a direct sum of subspaces characterized by different superselection rules. In this work, we develop the theoretical framework of entanglement witnessing for lattice gauge theories that takes this subtlety into account. We illustrate the concept at the example of a $\mathrm{U}(1)$ lattice gauge theory in 2+1 dimensions, without and with dynamical fermionic matter. As this framework circumvents costly state tomography, it opens the door to resource-efficient certification of entanglement in theoretical studies as well as in laboratory quantum simulations of gauge theories., Comment: 17+8 pages (single column), 4+2 figures; added references

While regexp matching is a powerful mechanism for finding patterns in data streams, regexp engines in general only find matches that do not overlap. Moreover, different forms of nondeterministic exploration, where symbols read are processed more than once, are often used, which can be costly in real-time matching. We present an algorithm that constructs from any regexp a Mealy machine that finds all matches and while reading each input symbol only once. The machine computed can also detect and distinguish different patterns or sub-patterns inside patterns. Additionally, we show how to compute a minimal Mealy machine via a variation of DFA minimization, by formalizing Mealy machines in terms of regular languages., Comment: Local technical report

Multipartite entanglement, such as witnessed through the quantum Fisher information (QFI), is a crucial resource for quantum technologies, but its experimental certification is highly challenging. Here, we propose an experimentally friendly protocol to measure the QFI. It relies on recording the short-time dynamics of simple observables after a quench from a thermal state, works for spins, bosons, and fermions, and can be implemented in standard cold-atom experiments and other platforms with temporal control over the system Hamiltonian. To showcase the protocol, we simulate it for the one-dimensional Fermi--Hubbard model. Further, we establish a family of bounds connecting the QFI to multipartite mode entanglement for fermionic systems, which enable the detection of multipartite entanglement at sizable temperatures. Our work paves a way to experimentally accessing entanglement for quantum enhanced metrology., Comment: 5+7 pages, 4+1 figures

Main results and techniques of the fractional calculus of variations are surveyed. We consider variational problems containing Caputo derivatives and study them using both indirect and direct methods. In particular, we provide necessary optimality conditions of Euler-Lagrange type for the fundamental, higher-order, and isoperimetric problems, and compute approximated solutions based on truncated Gr\"{u}nwald--Letnikov approximations of Caputo derivatives., Comment: This is a preprint of a paper whose final and definite form is in 'Handbook of Fractional Calculus with Applications. Vol 1: Basic Theory', De Gruyter. Submitted 29-March-2018; accepted, after a revision, 13-June-2018

This book intends to deepen the study of the fractional calculus, giving special emphasis to variable-order operators. It is organized in two parts, as follows. In the first part, we review the basic concepts of fractional calculus (Chapter 1) and of the fractional calculus of variations (Chapter 2). In Chapter 1, we start with a brief overview about fractional calculus and an introduction to the theory of some special functions in fractional calculus. Then, we recall several fractional operators (integrals and derivatives) definitions and some properties of the considered fractional derivatives and integrals are introduced. In the end of this chapter, we review integration by parts formulas for different operators. Chapter 2 presents a short introduction to the classical calculus of variations and review different variational problems, like the isoperimetric problems or problems with variable endpoints. In the end of this chapter, we introduce the theory of the fractional calculus of variations and some fractional variational problems with variable-order. In the second part, we systematize some new recent results on variable-order fractional calculus of (Tavares, Almeida and Torres, 2015, 2016, 2017, 2018). In Chapter 3, considering three types of fractional Caputo derivatives of variable-order, we present new approximation formulas for those fractional derivatives and prove upper bound formulas for the errors. In Chapter 4, we introduce the combined Caputo fractional derivative of variable-order and corresponding higher-order operators. Some properties are also given. Then, we prove fractional Euler-Lagrange equations for several types of fractional problems of the calculus of variations, with or without constraints., Comment: The final authenticated version of this preprint is available online as a SpringerBrief in Applied Sciences and Technology at [https://doi.org/10.1007/978-3-319-94006-9]. In this version some typos, detected by the authors while reading the galley proofs, were corrected, SpringerBriefs in Applied Sciences and Technology, Springer, Cham, 2019

In this paper, we discuss the existence and uniqueness of solutions of a boundary value problem for a fractional differential equation of order $\alpha\in(2,3)$, involving a general form of fractional derivative. First, we prove an equivalence between the Cauchy problem and the Volterra equation. Then, two results on the existence of solutions are proven, and we end with some illustrative examples., Comment: This is a preprint of a paper whose final and definite form is with "Bull. Malays. Math. Sci. Soc."

In this paper, the fractional order Hegselmann-Krause type model with leadership is studied.We seek an optimal control strategy for the system to reach a consensus in such a way that the control mechanism is included in the leader dynamics. Necessary optimality conditions are obtained by the use of a fractional counterpart of Pontryagin Maximum Principle. The effectiveness of the proposed control strategy is illustrated by numerical examples., Comment: This is a preprint of a paper whose final and definite form is in Theory and Applications of Non-integer Order Systems, Lecture Notes in Electrical Engineering, Springer Verlag

In this paper we present a new type of fractional operator, which is a generalization of the Caputo and Caputo--Hadamard fractional derivative operators. We study some properties of the operator, namely we prove that it is the inverse operation of a generalized fractional integral. A relation between this operator and a Riemann--Liouville type is established. We end with a fractional Gronwall inequality type, which is useful to compare solutions of fractional differential equations., Comment: This is a preprint of a paper whose final and definite form is with "Mathematical Inequalities & Applications"

We study two generalizations of fractional variational problems by considering higher-order derivatives and a state time delay. We prove a higher-order integration by parts formula involving a Caputo fractional derivative of variable order and we establish several necessary optimality conditions for functionals containing a combined Caputo derivative of variable fractional order. Because the endpoint is considered to be free, we also deduce associated transversality conditions. In the end, we consider functionals with a time delay and deduce corresponding optimality conditions. Some examples are given to illustrate the new results. Computational aspects are discussed using the open source software package Chebfun., Comment: This is a preprint of a paper whose final and definite form is with 'Journal of Computational and Applied Mathematics', ISSN: 0377-0427. Submitted 15-Feb-2017; Revised 17-Apr-2017; Accepted 21-Apr-2017

The aim of this work is to show, based on concrete data observation, that the choice of the fractional derivative when modelling a problem is relevant for the accuracy of a method. Using the least squares fitting technique, we determine the order of the fractional differential equation that better describes the experimental data, for different types of fractional derivatives., Comment: This is a preprint of a paper whose final and definite form is with "Applicable Analysis and Discrete Mathematics"

We study fractional variational problems of Herglotz type of variable order. Necessary optimality conditions, described by fractional differential equations depending on a combined Caputo fractional derivative of variable order, are proved. Two different cases are considered: the fundamental problem, with one independent variable, and the general case, with several independent variables. We end with some illustrative examples of the results of the paper., Comment: This is a preprint of a paper whose final and definite form is with 'Discrete and Continuous Dynamical Systems -- Series S' (DCDS-S), ISSN 1937-1632 (print), ISSN 1937-1179 (online), available at [https://www.aimsciences.org/journals/home.jsp?journalID=15]. Paper Submitted 30-July-2016; Revised 03-Feb-2017; Accepted 27-March-2017