We explore the notion of generated entropy in open quantum systems. We focus on the study of the discrete-time quantum walk on the line, from the entropy production perspective. We argue that the evolution of the coin can be modeled as an open two-level system that exchanges energy with the lattice at some effective temperature that depends on the initial state. The entropy balance shows that there is a positive-entropy production during the evolution, in accordance with the second law of thermodynamics.
We explore the notion of generated entropy in open quantum systems. We focus on the study of the discrete-time quantum walk on the line, from the entropy production perspective. We argue that the evolution of the coin can be modeled as an open two-level system that exchanges energy with the lattice at some effective temperature that depends on the initial state. The entropy balance shows that there is a positive-entropy production during the evolution, in accordance with the second law of thermodynamics.
We explore the notion of generated entropy in open quantum systems. We focus on the study of the discrete-time quantum walk on the line, from the entropy production perspective. We argue that the evolution of the coin can be modeled as an open two-level system that exchanges energy with the lattice at some effective temperature that depends on the initial state. The entropy balance shows that there is a positive entropy production during the evolution, in accordance with the second law of thermodynamics.
Abstract. The coin-position entanglement generated by the evolution operator of a discrete–time quantum walk is quantified, using the von Neumann entropy of the reduced density operator (entropy of entanglement). In the case of a single walker, the entropy of entanglement converges, in the long time limit, to a well defined value which depends on the initial state. Exact expressions are obtained for local and non-local initial conditions. We also discuss the asymptotic bi-partite entanglement generated by non-separable coin operations for two coherent quantum walkers. In this case, the entropy of entanglement is observed to increase logarithmically with time. 1.
A consistent theory of non-equilibrium thermodynamics for Markovian open quantum systems has been developed in the late seventies in analogy with Classical Irreversible Thermodynamics. The time-evolution of these open systems is usually described by means of effective master equations in Lindblad form, that turn out to be reliable when there is a separation of time-scales between system and environment, such that memory effects are negligible and the so-called Markovian approximation is justified. In this framework, the variations of energy and entropy in the system are consistently described, distinguishing between heat and work contributions and providing a statement of the second law of thermodynamics as positivity of the entropy production.\nHowever, there is empirical evidence that many physical systems like photosynthetic complexes,\nopto-mechanical resonators and superconducting qubits, just to mention a few, experience more general non-Markovian dynamics. The formulation of the laws of thermodynamics in a non-Markovian setting is matter of current research and represents the main topic of the present work. As a first step, we show that the entropy production defined as in the Markovian case can be negative for a class of non-Markovian dynamics. We argue that this outcome should not be interpreted as a violation of the second law of thermodynamics because the environment must be explicitly taken into account in the balance of entropy in the non-Markovian setting. In order to justify this claim we adopt a more general point of view, studying a closed bipartite quantum system, such that each of the two subsystems plays the role of a finite environment out of equilibrium for the other one. We concentrate on the balance of energy first and construct an effective Hamiltonian for each subsystem using physically reasonable requirements; then we define heat and work as in the standard Markovian treatment, with the effective Hamiltonian replacing the free Hamiltonian. It turns out that, in our framework, the work power is perfectly balanced between subsystems, while the correlations can store a part of energy locally inaccessible and exchange it with both subsystems in the form of heat.\nConcerning the balance of entropy, a quite general formulation of the second law of thermodynamics can be given as follows: under the assumption of a factorized initial state for the compound system, the sum of the total variations of the entropies in the two subsystems is always nonnegative. We show with an explicit example that this general formulation does not correspond to the statement presented in the framework of Markovian master equations, which should not be considered a priori the second law of thermodynamics.\nIn the last part of the thesis we concentrate of the so-called fluctuation relations, that are results extending the thermodynamic formalism beyond the behavior of average quantities. After reviewing the main theoretical outcomes, such as the Jarzynski equality and the Crooks fluctuation theorem, we describe a proposal to access experimentally the work performed on an ensemble of diatomic molecules by a time-dependent electric field coupled with their vibrational degree of freedom. This procedure could then be used to test the quantum Jarzynski equality. With respect to the results so far appeared in the literature, in which the left-hand side of the equality is inferred from an experiment and the right-hand side is computed according to a model, in our proposed setting we should be able to estimate from the experiment both the left-hand side and the right-hand side of the equality, independently.
Quantum magic, which accounts for the nonstabilizer content of a state, is essential for universal quantum computation beyond classically simulable resources. However, the way magic builds up during structured unitary dynamics remains largely open. Here, we investigate the generation and evolution of quantum magic in discrete-time quantum walks (DTQWs)—a simple, tunable unitary model realizable on a wide range of architectures. We use stabilizer Rényi entropy as a measure of quantum magic and investigate single- and two-walker quantum walks on a one-dimensional lattice, considering a wide range of initial coin states. Our results reveal that DTQWs can dynamically generate significant magic, with the amount and structure dependent on the initial state of the coin. In the case of a single walker, we found a nontrivial and complementary relationship between magic and entanglement at long times. In the two-walker setting, even states close to stabilizer form can evolve into highly magical states when subjected to quantum walk protocols. The generation of magic can occur independently of entanglement growth, emphasizing its distinct role as a quantum resource. We further show that the dynamical magic generation is robust under realistic noise—specifically, decoherence in the coin degree of freedom—demonstrating that this process persists across noisy settings. Finally, we discuss an experimentally feasible scheme to measure magic using current technology. Our findings position DTQWs as accessible and controllable platforms for producing quantum magic, offering a fresh perspective on their role in quantum information processing and reliable quantum computation.
We consider the temporal evolution of the entropy S of an isolated system from an initial non-equilibrium state 2· of entropy S = S i to the equilibrium state of maximum entropy S = S max > S i. The application of usual density matrix theory of quantum mechanics leads us to the surprising result P = dS/dt = 0. The entropy does not change in time. This result is valid in case of the equilibrium state. However it is wrong for a non-equilibrium state, because the entropy of the isolated system could not spontaneously increase from S i to S max by entropy production P > 0. This contradicts the second law of thermodynamics. We avoid this discrepancy by changing from quantum mechanics to quantum thermodynamics. Now the entropy production P = dS/dt and the time t are the relevant observables. In order to treat the temporal development of S and P, we at first prepare an initial state i of entropy production P i at time t < 0. Subsequently at ϑ t ≥ 0 the initial state i interacts with a continuum of final states f, which transfers the entropy production from i to f. The temporal evolution of the entropy production P(t) of our system is treated by means of the time-dependent perturbation theory. For that reason we split the operator of entropy production P = P 0 + W into two parts. The operator P 0 describes the unperturbed basic vectors of entropy production | i > and | f >, while the perturbation operator W determines the interaction between | i > and | f >. We expand the time-dependent state function < t | ψ > of the isolated system into the complete set of stationary eigenvectors | i > and | f > and calculate the time-dependent expansion coefficients c i(t) and c(f, t) by means of the dynamic equation of quantum thermodynamics − i k (d/dt) < t | ψ > = < t | P | ψ >. Here k means the Boltzmann constant, which is characteristic of quantum thermodynamics and represents the atomic entropy unit. The expansion coefficients are given by a system of coupled differential equations, from which we deduce an integral-differential equation for the coefficient c i(t) of the initial state i. Unfortunately it has been impossible to obtain an analytic solution of the complicated equation. However approximations are available. Accordingly, in the vicinity of the equilibrium state, the probability of still finding the initial state i of entropy production P i at time t is given by ω i(t) = | c i(t) |2 = exp (−Lt). The rate constant L ≥ σ depends on the density ρ(P f) of final states f of entropy production P f near by P i, and on the squared modulus of the perturbation matrix elements W if = < i | W | f >, which determines the interaction between | i > and | f >. The entropy production P(t) = P i ω i(t) = P i exp (−ϑLt) of the isolated system decreases exponentially and irreversibly with proceeding time t. Its rate-determining step is the transfer of entropy production from i to f. The integral over the entropy production of the final states f vanishes. Therefore the continuum states f do not contribute to the total entropy production P(t). The temporal evolution of the entropy S(t) can easily be calculated by integration of P(t). In conclusion, the entropy production of a non-equilibrium state is finite P(t) > 0 and only vanishes in the case of the equilibrium state P(∞) = 0. Thus, quantum thermodynamics agrees with the second law of thermodynamics.