• quantum information
• ergotropy
• thermodynamics

In contrast from classical thermodynamics, in which the properties of a macroscopic system in equilibrium are specified by a small number of degrees of freedom, a finite quantum system may retain memory of its initial conditions for an indefinite time (indeed, the only case in which a finite quantum system can thermalize under unitary evolution is if its initial state $\rho$ has already the spectrum of a state of thermal equilibrium).
As a consequence, the amount of work that we can extract coupling a quantum system to a thermal bath (the free energy of the system) is, in general, larger than the work that we can extract from the system alone, that is its ergotropy.
The discrepancy between the latter two quantities disappears in the macroscopic limit of infinitely many non-interacting copies of the system, in which we retrieve the classical formula for the maximum extractable work. At formal level, the ergotropy of a system characeterized by the Hamiltonian is a functional of the quantum state of the system defined by maximising the variation of its mean energy under an arbitrary evolution induced by the an arbitrary unitary operator.

In the present thesis we shall study the distribution of the extracted work with respect to the Haar measure of the unitrary group. In particular I shall prove that when the dimension of the Hilbert space is big, under a very general hypothesis the distribution is a gaussian with respect to the Haar measure, and I will estimate the error of this approximation for every moment of the distribution.

As a second topic we shall consider the extraction of work from more copies of the system. The main result we obtain in this regime is to show that when we increase the number of copies the variance of the extracted work decreases, and the distribution becomes better approximable to a normal distribution. In particular this will allows us to find an analytical expression that quantifies, as a function of the number of copies N, the maximum discrepance with the classical macroscopic case. We will also look at the more general problem of correlated systems. Quantum correlations are a feature of quantum mechanics which gives rise to phenomena that are not accountable by classical physics.
In presence of correlations, the thermodynamical behaviour of finite quantum systems festures some remarkable differences from the behaviour of classical systems; for example, it has been shown that in this regime the Clasius inequality may be violated.
Whereas from non correlated quantum system we can extract less work than in the classical case, in presence of quantum correlations we can on the contrary hope to beat the classical limit, and extract more work.
When we allow for a correlated initial state, we will see that we can indeed extract more work than in the classical case. To express how much the state is correlated, I will use as a measure the minimum rank necessary to represent the state as a (translationally symmetric) matrix product operator. In the limit of large N, I will be able to find some bounds for the work that, in the best case, can be extracted from such a correlated state.



%This thesis is about how the energy of a quantum system can change with unitary evolutions.


%Chapter~\ref{chap: introduction} is meant to be an informal introduction to the concept of ergotropy and to the problem of work extraction from quantum systems, with a a brief survey of the existing literature.
%In particular, we shall introduce the fundamental quantity $W_U (\rho; H)$ which, for a quantum system intialized
%into the state $\rho$ and characterized by the Hamiltonian $H$, measures the variation of the mean energy under the evolution induced by the unitary operator $U$, providing an estimation of the work extracted from the system \sidecite{Lenard1978} . The maximum work that we can extract from a system initialised in the quantum state $\rho$ is given by the maximum possible value of $W_U (\rho; H)$, which we call the \emph{ergotropy} $\mathcal{E}(\rho; H)$ \sidecite{PASSIVE3}.

%In the other chapters, I will derive new results.

%In chapter \ref{chap:moments}, I will study the distribution of $W_{U} (\rho; H)$ with respect to the Haar measure of the unitrary group.
%Random unitaries are a resource required for a lot of quantum algorithm \cite{Hayashi2005, Scott2008, BrandaoHorodecki2008}.
%For example, the \emph{randomized benchmark} protocol is a method to test the error rate of a quantum circuit, requiring it to perform a sequence of random operations \sidecite{Emerson2005}. Versions of the randomized benchmark are used by the companies IBM \cite{McKay2019} and Microsoft \cite{Helsen2019} to test the functionality of their experimental quantum computing hardware.
%Other applications of random unitaries include quantum cryptography \sidecite{DiVincenzo2002} and the simulation of many body physics \cite{Nahum2017, Jonay2018}.
%However, implementing a random unitary is not an easy task. Actually, only a small subset of quantum operations (called the \emph{Clifford gates} \cite{Gottesman1998}) are easy to implement in an actual circuit - the complexity of a quantum circuit is often measured with the number of non-Clifford gates it requires \cite{Veitch2014}.
%To overcome these difficulties, it has been theorized the possibility of circuits that simulate a random unitary up to a certain moment of the distribution \sidecite{Gross2007}. A circuit which is able to emulate a uniform distribution up to the $t$-th moment is called an \emph{unitary t-design}. If we want to realise a $t$-design with $t > 3$, it is still necessary to use non-Clifford gates \cite{Sawicki2017, Bannai2018}, but only a small amount of them \sidecite{Haferkamp2020}.
%The results of chapter~\ref{chap:moments} are useful for charachterising an ideal source of uniformly distributed unitary operations. Furthermore, they are valid also for its approximations which are used in actual quantum computing, the $t$-designs which match the uniform unitary distribution up to the $t$-th moment.
%Indeed, I will estimate, for all the moments of the distribution, the maximum error that we make in replacing the distribution of the energy with a gaussian distribution.