Quantum speed limits set an upper bound to the rate at which a quantum system can evolve and as such can be used to analyze the scrambling of information. To this end, we consider the survival probability of a thermofield double state under unitary time-evolution which is related to the analytic continuation of the partition function. We provide an exponential lower bound to the survival probability with a rate governed by the inverse of the energy fluctuations of the initial state. Further, we elucidate universal features of the non-exponential behavior at short and long times of evolution that follow from the analytic properties of the survival probability and its Fourier transform, both for systems with a continuous and for systems with a discrete energy spectrum. We find the spectral form factor in a number of illustrative models, notably we obtain the exact answer in the Gaussian unitary ensemble for any N with excellent agreement with recent numerical studies. We also discuss the relationship of our findings to models of black hole information loss, such as the

We compute the entanglement between separated blocks in certain spin models showing that at criticality this entanglement is a function of the ratio of the separation to the length of the blocks and can be written as a product of a power law and an exponential decay. It thereby interpolates between the entanglement of individual spins and blocks of spins. It captures features of correlation functions at criticality as well as the monogamous nature of entanglement. We exemplify invariant features of this entanglement to microscopic changes within the same universality class. We find this entanglement to be invariant with respect to simultaneous scale transformations of the separation and the length of the blocks. As a corollary, this study estimates the entanglement between separated regions of those quantum fields to which the considered spin models map at criticality.Correlations have long been a central object of study in condensed matter with attention recently drawn to entanglement -unique correlations possible only in quantum mechanics. The presence of entanglement inside condensed matter systems [1,2] is also experimentally supported [3]. These quantum correlations become particularly interesting at quantum phase transitions (QPT), which occur at zero temperature as the relative strengths of interactions in a many-body system are varied [4]. At a QPT, in general, the entanglement between individual spins is non-zero only for very small separations between the spins [5] ( Fig. 1, panel (a)). On the other hand, the entanglement between adjacent blocks of spins (which cannot by definition, have a separation) diverges with the length of the blocks [6, 7, 8] ( Fig. 1, panel (b)). However, an intermediate situation where one considers the entanglement between blocks of spins which are separated (i.e., non-complementary) is an open problem. It would be interesting to study how this entanglement scales, i.e., varies with the size of the blocks ∆ and their separation x, at a QPT. Here, we conduct such a study (see, Fig.1, panel (c)) and find that both the short ranged nature of spin-spin entanglement and divergent nature of adjacent block entanglement can be recovered qualitatively as limiting cases of the expression for the entanglement of non-complementary blocks. This can be viewed as an interpolation between known behaviors of entanglement in the aforementioned limits. Though, what we compute is a form of bipartite entanglement, it also captures aspects of the multipartite entanglement in the system, as we will discuss later.Note that two other information theoretic quantities involving non-complementary blocks of a quantum many body system have recently drawn much attention. One of them, the von Neumann entropy of disjoint blocks quantifies the entanglement of these blocks with respect to their complement, but not that between them [9,10]. The second, mutual information, does quantify the correlations between the blocks, but cannot be considered as a measure of entanglement (i.e., the "quantum" part of the corr...

Characterizing the work statistics of driven complex quantum systems is generally challenging because of the exponential growth with the system size of the number of transitions involved between different energy levels. We consider the quantum work distribution associated with the driving of chaotic quantum systems described by random matrix Hamiltonians and characterize exactly the work statistics associated with a sudden quench for arbitrary temperature and system size. Knowledge of the work statistics yields the Loschmidt echo dynamics of an entangled state between two copies of the system of interest, the thermofield double state. This echo dynamics is dictated by the spectral form factor. We discuss its relation to frame potentials and its use to assess information scrambling.Quantum thermodynamics has become a research field in bloom, building on the dialogue between technological progress and foundations of physics [1,2]. In the presence of thermal and quantum fluctuations, familiar concepts from traditional thermodynamics at the macroscale, such as heat and work, become stochastic variables. Their analysis has been a fertile ground of inquiry, leading to the discovery of fluctuation theorems [3][4][5][6][7] and time-work uncertainty relations [8][9][10].In the quantum domain, even for isolated systems governed by unitary dynamics, work is not considered to be an observable. The definition of work requires two projective measurements, one at the beginning and another at the end of the physical pro-Aurélia Chenu: achenu@dipc.org,

A general method to build the entanglement renormalization (cMERA) for interacting quantum field theories is presented. We improve upon the well-known Gaussian formalism used in free theories through a class of variational non-Gaussian wavefunctionals for which expectation values of local operators can be efficiently calculated analytically and in a closed form. The method consists of a series of scale-dependent nonlinear canonical transformations on the fields of the theory under consideration. Here, the λ φ 4 and the sine-Gordon scalar theories are used to illustrate how nonperturbative effects far beyond the Gaussian approximation are obtained by considering the energy functional and the correlation functions of the theory.In recent years, tensor networks, a new and powerful class of variational states, have proved to be very useful in addressing both static and dynamical aspects of a wide number of interacting many-body systems. They represent a class of systematic variational ansätze which, through the Rayleigh-Ritz variational principle, provide an elegant approximation to the ground state of an interacting theory by systematically identifying those degrees of freedom that are actually relevant for observable physics. These variational ansätze are nonperturbative and can be applied both in the lattice and in the continuum. As an example, the Multiscale Entanglement Renormalization Ansatz (MERA), a variational real-space renormalization scheme on the quantum state, represents the wavefunction of the quantum system at different length scales [1].A continuous version of MERA, known as cMERA, was proposed in [2] for free field theories. It consists of building a scale-dependent representation of the ground state wavefunctional through a scale-dependent linear canonical transformation of the fields of the theory. Namely, the renormalization in scale is generated by a quadratic operator, and thus, the resulting state is given by a Gaussian wavefunctional. Despite this fact obviously limits the interest of this trial state for interacting quantum field theories (QFT), the Gaussian ansatz has been used in cMERA and correctly reproduces correlation functions and entanglement entropy in free field theories [3, 4]. Furthermore, as the Gaussian cMERA is currently studied as a possible realization of holography [5][6][7][8][9][10], it is timely to develop interacting versions of cMERA in order to advance in this program. In [11], the Gaussian cMERA was applied to interacting bosonic and fermionic field theories. In [12], authors developed some techniques to build systematic perturbative calculations of cMERA circuits but restricted to the weakly interacting regime.Our aim here is to provide a non-perturbative method to build truly non-Gaussian cMERA wavefunctionals for interacting QFTs. A justifiable way of doing so would be to formulate a perturbative expansion for which the Gaussian wavefunction appears in its first order [13][14][15][16].Unfortunately, with these methods, expectation values of operators cannot be cal...

In this work, a non-Gaussian cMERA tensor network for interacting quantum field theories (icMERA) is presented. This consists of a continuous tensor network circuit in which the generator of the entanglement renormalization of the wavefunction is nonperturbatively extended with nonquadratic variational terms. The icMERA circuit nonperturbatively implements a set of scale dependent nonlinear transformations on the fields of the theory, which suppose a generalization of the scale dependent linear transformations induced by the Gaussian cMERA circuit. Here we present these transformations for the case of self-interacting scalar and fermionic field theories. Finally, the icMERA tensor network is fully optimized for the λφ 4 theory in (1 + 1) dimensions. This allows us to evaluate, nonperturbatively, the connected parts of the two-and four-point correlation functions. Our results show that icMERA wavefunctionals encode proper non-Gaussian correlations of the theory, thus providing a new variational tool to study phenomena related with strongly interacting field theories.