Bounds on the capacity and power of quantum batteries

Quantum batteries, composed of quantum cells, are expected to outperform their classical analogs. The origin of such advantages lies in the role of quantum correlations, which may arise during the charging and discharging processes performed on the battery. In this theoretical work, we introduce a systematic characterization of the relevant quantities of quantum batteries, i.e., the capacity and the power, in relation to such correlations. For these quantities, we derive upper bounds for batteries that are a collection of non-interacting quantum cells with fixed Hamiltonians. The capacity, that is, a bound on the stored or extractable energy, is derived with the help of the energy-entropy diagram, and this bound is respected as long as the charging and discharging processes are entropy preserving. While studying power, we consider a geometric approach for the evolution of the battery state in the energy eigenspace of the battery Hamiltonian. Then, an upper bound for power is derived for arbitrary charging process, in terms of the Fisher information and the energy variance of the battery. The former quantifies the speed of evolution, and the latter encodes the non-local character of the battery state. Indeed, due to the fact that the energy variance is bounded by the multipartite entanglement properties of batteries composed of qubits, we establish a fundamental bound on power imposed by quantum entanglement. We also discuss paradigmatic models for batteries that saturate the bounds both for the stored energy and the power. Several experimentally realizable quantum batteries, based on integrable spin chains, the Lipkin-Meshkov-Glick and the Dicke models, are also studied in the light of these newly introduced bounds.We acknowledge the Spanish Ministry MINECO (National Plan 15 Grant: FISICATEAMO No. FIS2016-79508- P, SEVERO OCHOA No. SEV-2015-0522, FPI), European Social Fund, Fundació Cellex, Generalitat de Catalunya (AGAUR Grant No. 2017 SGR 1341 and CERCA/Program), EU FEDER, ERC AdG OSYRIS, EU FETPRO QUIC, and the National Science Centre, Poland-Symfonia Grant No. 2016/20/W/ST4/00314. M.N.B. gratefully acknowledges financial supports from Max-Planck Institute fellowship and from SERB-DST, Government of India, and A.R. thanks supports from CELLEX-ICFO-MPQ fellowship. We also thank M. Polini for fruitful discussions and for drawing our attention to Ref. [34], and an anonymous referee for motivating the finding of Corollary.Postprint (author's final draft

Entanglement in Many Body Quantum Systems

THESIS SUMMARYTEXT:This thesis is made of two parts. In the first one, the issue of entanglement in many body systems is addressed. The concept of entanglement and some of the recent progress on the study of entropy of entanglement in many body quantum systems are reviewed. Emphasis is placed on the scaling properties of entropy for one-dimensional models at quantum phase transitions. Then, we focus on the area-law scaling of the entanglement entropy. An explicit computation in arbitrary dimensions of the entanglement entropy of the ground state of a discretized scalar free field theory that shows the expected area law result is also presented. For this system, it is shown that area law scaling is a manifestation of a deeper reordering of the vacuum produced by majorization relations.To finish this first part, the issue of how simple can a quantum system be such as to give a highly entangled ground state is addressed. In particular, we propose a Hamiltonian of a XX model with a ground state whose entropy scales linearly with the size of the block. It provides a simple example of a one dimensional system of spin-1/2 particles with nearest neighbour interactions that violates area-law for the entanglement entropy.The second part of this thesis deals with the problem of simulating quantum mechanics for highly entangled systems. Two different approaches to this issue are considered. One consists of using ultra-cold atoms systems as quantum simulators. With this aim, some experimental techniques related to cold atoms that allow to simulate strongly correlated many body quantum systems are reviewed an explicit example of simulation is presented. In particular, we analyze how to achieve a Mott state of Laughlin wave functions in an optical lattice and study the consequences of considering anharmonic corrections to each single site potential expansion that were not taken into account until now.Finally, a different approach to simulate strongly correlated systems is considered: to use small quantum computers to simulate them. An explicit quantum algorithm that creates the Laughlin state for an arbitrary number of particles n in the case of falling fraction equal to one is presented. We further prove the optimality of the circuit using permutation theory arguments and we compute exactly how entanglement develops along the action of each gate. We also discuss its experimental feasibility decomposing the qudits and the gates in terms of qubits and two qubit-gates as well as the generalization to arbitrary falling fraction.KEYWORDS: Entanglement, Many body quantum systems, Quantum Information Condensed Matter, Cold atoms, Spin chains, Quantum simulator, Quantum computation."Entrellaçament quàntic en sistemes de molts cossos"TEXT:Aquesta tesi està composada per dues parts. En la primera, adrecem la qüestió de l'entrellaçament quàntic en els sistemes de molts cossos. Així, introduïm primer el concepte d'entrellaçament i revisem els progressos recents sobre aquest camp. A continuació, ens centrem la llei d'àrea per l'entropia d'entrellaçament i presentem un càlcul explícit d'aquesta entropia per a l'estat fonamental d'un camp escalar no interactuant obtenint la llei d'àrea esperada. Finalment, acabem aquesta part presentant un sistema molt senzill 1-dimensional que tot i tenir interaccions locals mostra una llei de volum per l'entropia.En la segona part de la tesi tractem el problema de la simulació de sistemes quàntics altament entrellaçats. Considerem dos possibles vies per tractar aquest problema. Una d'elles consisteix en la utilització d'àtoms ultra-freds com a simuladors quàntics. En particular, analitzem un mètode per obtenir un estat producte de funcions d'ona de Laughlin en un xarxa òptica i estudiem les conseqüències de considerar la correcció anharmònica de l'expansió del potencial a cada pou de la xarxa. Finalment, considerem una altra aproximació a la simulació de sistemes fortament correlacionats: utilitzar petits ordinadors quàntics per a simular-los. Per il.lustrar aquest tipus de simulació, presentem un algoritme quàntic que crea un estat de Laughlin per un nombre arbitrari de partícules i en el cas de fracció d'ocupació 1. </i