Pareto-optimal cycles for power, efficiency and fluctuations of quantum heat engines using reinforcement learning

The full optimization of a quantum heat engine requires operating at high power, high efficiency, and high stability (i.e., low power fluctuations). However, these three objectives cannot be simultaneously optimized—as indicated by the so-called thermodynamic uncertainty relations—and a systematic approach to finding optimal balances between them including power fluctuations has, as yet, been elusive. Here we propose such a general framework to identify Pareto-optimal cycles for driven quantum heat engines that trade off power, efficiency, and fluctuations. We then employ reinforcement learning to identify the Pareto front of a quantum dot-based engine and find abrupt changes in the form of optimal cycles when switching between optimizing two and three objectives. We further derive analytical results in the fast- and slow-driving regimes that accurately describe different regions of the Pareto front

Characterizing (non-)Markovianity through Fisher Information

A non-isolated physical system typically loses information to its environment, and when such loss is irreversible the evolution is said to be Markovian. Non-Markovian effects are studied by monitoring how information quantifiers, such as the distance between physical states, evolve in time. Here we show that the Fisher information metric emerges as the natural object to study in this context; we fully characterize the relation between its contractivity properties and Markovianity, both from the mathematical and operational point of view. We prove, both for classical and quantum dynamics, that Markovianity is equivalent to the monotonous contraction of the Fisher metric at all points of the set of states. At the same time, operational witnesses of non-Markovianity based on the dilation of the Fisher distance cannot, in general, detect all non-Markovian evolutions, unless specific physical postprocessing is applied to the dynamics. Finally, we show for the first time that non-Markovian dilations of Fisher distance between states at any time correspond to backflow of information about the initial state of the dynamics at time 0, via Bayesian retrodiction.Comment: comments are welcome

Fundamental limits of metrology at thermal equilibrium

We consider the estimation of an unknown parameter $\theta$ through a quantum probe at thermal equilibrium. The probe is assumed to be in a Gibbs state according to its Hamiltonian $H_\theta$, which is divided in a parameter-encoding term $H^P_\theta$ and an additional, parameter-independent, control $H^C$. Given a fixed encoding, we find the maximal Quantum Fisher Information attainable via arbitrary $H^C$, which provides a fundamental bound on the measurement precision. Our bounds show that: (i) assuming full control of $H^C$, quantum non-commutativity does not offer any fundamental advantage in the estimation of $\theta$; (ii) an exponential quantum advantage arises at low temperatures if $H^C$ is constrained to have a spectral gap; (iii) in the case of locally-encoded parameters, the optimal sensitivity presents a Heisenberg-like $N^2$-scaling in terms of the number of particles of the probe, which can be reached with local measurements. We apply our results to paradigmatic spin chain models, showing that these fundamental limits can be approached using local two-body interactions. Our results set the fundamental limits and optimal control for metrology with thermal and ground state probes, including probes at the verge of criticality.Comment: Comments welcome

Physicality of evolution and statistical contractivity are equivalent notions of maps

Statistical quantifiers are generically required to contract under physical evolutions, following the intuition that information should be lost under noisy transformations. This principle is very relevant in statistics, and it even allows to derive uniqueness results based on it: by imposing their contractivity under any physical maps, the Chentsov-Petz theorem singles out a unique family of metrics on the space of probability distributions (or density matrices) called the Fisher information metrics. This result might suggest that statistical quantifiers are a derived concept, as their very definition is based on physical maps. The aim of this work is to disprove this belief. Indeed, we present a result dual to the Chentsov-Petz theorem, proving that among all possible linear maps, the only ones that contract the Fisher information are exactly the physical ones. This result shows that, contrary to the common opinion, there is no fundamental hierarchy between physical maps and canonical statistical quantifiers, as either of them can be defined in terms of the other

Quantum Fisher Information and its dynamical nature

The importance of the quantum Fisher information metric is testified by the number of applications that this has in very different fields, ranging from hypothesis testing to metrology, passing through thermodynamics. Still, from the rich range of possible quantum Fisher information, only a handful are typically used and studied. This review aims at collecting a number of results scattered in the literature that can be useful to people who begin the study of Fisher information and to those who are already working on it to have a more organic understanding of the topic. Moreover, we complement the review with new results about the relation between Fisher information and physical evolutions. Extending the study done in [1], we prove that all the physically realisable dynamics can be defined solely in terms of their relation with respect to the Fisher information metric. Moreover, other properties as Markovianity, retrodiction or detailed balance can be expressed in the same formalism. These results show a fact that was partially overseen in the literature, namely the inherently dynamical nature of Fisher information.Comment: 36 pages of main text, 15 of additional information, 12 of appendix, and one of inde

Quantum Thermodynamics: control and non-Markovianity

This Master’s Degree thesis is located in the field of Quantum Thermodynamics and its links with Information Theory. The starting purpose of the work is to study how memory effects (non-Markovian dynamics) can affect the thermodynamic performances of a quantum system. Incidentally we work out an optimal control theory for a class of simple thermal machinesperforming Carnot cycles. The mathematical treatment is based on tools from Open Quantum System Dynamics (OQS): evolution is described, under the most general physical assumptions, by means of Completely Positive Trace-preserving linear maps, acting on the density matrix of the system under consideration. We stand by the standard approach which mathematically identifies the Markovian character of a quantum process with its CP-divisibility. Under this assumption a standard form can be derived for the Master equation generating the evolution, that is the Gorini-Kossakowski-Sudarshan-Lindblad form. Based especially on OQS, our study of Quantum Thermodynamics focuses on the analysis of the performance of thermal engines. In this framework thermalisation effects induced by the interaction with a thermal reservoir will be modelled with Master Equations having the Gibbs state as the only stationary state. To simulate sensible restraints on the system, it will be also assumed external control on dynamical parameters such as the coupling constant to the thermal baths, as well as the local Hamiltonian of the system. Once defined the basic thermodynamic quantities and potentials, we review the main quantum thermodynamic cycles, i.e. the Quantum Carnot Cycle and the Quantum Otto Cycle. In the above setting we develop an optimal control theory for simple quantum engines such as qubits in a wide class of dynamical models. For this kind of study it is necessary to depart from the idealizations of quasistatic processes and analyse Finite Time Thermodynamics (FTT), which we study using a perturbative technique for the study of slowly controlled open quantum systems, called Slow-Driving approximation (S-D). The perturbative solution is very useful for it neglects initial conditions and it is relatively simple to compute, hence we employ it to detect the optimal working points of quantum thermal machines. For the paradigmatic case of a quantum Carnot cycle we find that it is possible to pinpoint three main features which determine, in the S-D scheme, the thermodynamic performance: • the shape of the protocol on the control parameters, • the speed of the protocol (for fixed shape), • a model dependent S-D correction amplitude, namely how large the perturbative correction to the steady state solution is. The S-D technique allows us to perform optimisation on the shape and duration of the control. The last contribution encodes, for a given protocol "how distant" the Slow-Driving corrections are from the quasistatic case. It is a model dependent amplitude which we then exploit as the main figure of merit in the study e.g. of non-Markovian models. We define a precise setup for the study of non-Markovian influence on a quantum thermodynamic system: we imagine to have an engine S in contact with the thermal baths, allowing some degrees of freedom of the reservoirs (namely, a system C for the cold bath and H for the hot one) to be taken into account explicitly and couple with S by an interaction Hamiltonian. In the simplest model we study H, S and C are qubits which interact via an exchange Hamiltonian between S and C (and S and H ). When coupled to the cold reservoirs, on both S and C the action of the bath is given by GKSL thermalising dissipators, and the controller has the ability to change the energy gap of the local Hamiltonian of S; the situation is specular when coupled to the hot bath. We study this model in the Slow-Driving approximation; in particular we look for the expression of the S-D amplitude as a function of the setup parameters, to study how the rates defining the GKSL operators influence it. We find that in a vast region of parameters space the introduction of SH and SC coupling can have improving effects on the FTT performance. We also examine the exact solution of an Otto cycle, confirming that the presence of the n-M enhances the power output

Geometric Optimisation of Quantum Thermodynamic Processes

Differential geometry offers a powerful framework for optimising and characterising finite-time thermodynamic processes, both classical and quantum. Here, we start by a pedagogical introduction to the notion of thermodynamic length. We review and connect different frameworks where it emerges in the quantum regime: adiabatically driven closed systems, time-dependent Lindblad master equations, and discrete processes. A geometric lower bound on entropy production in finite-time is then presented, which represents a quantum generalisation of the original classical bound. Following this, we review and develop some general principles for the optimisation of thermodynamic processes in the linear-response regime. These include constant speed of control variation according to the thermodynamic metric, absence of quantum coherence, and optimality of small cycles around the point of maximal ratio between heat capacity and relaxation time for Carnot engines