arXiv subjects

Yan Pautrat

Publications and source records attributed to Yan Pautrat.

17 recordsLinked to original sources

arXiv2020

Ke Li's lemma for quantum hypothesis testing in general von Neumann algebras

A lemma stated by Ke Li in [arXiv:1208.1400] has been used in e.g. [arXiv:1510.04682,arXiv:1706.04590,arXiv:1612.01464,arXiv:1308.6503,arXiv:1602.08898] for various tasks in quantum hypothesis testing, data compression with quantum side information or quantum key distribution. This lemma was originally proven in finite dimension, with a direct extension to type I von Neumann algebras. Here we show that the use of modular theory allows to give more transparent meaning to the objects constructed by the lemma, and to prove it for general von Neumann algebras. This yields a new proof of quantum Stein's lemma with slightly weaker assumption, as well as immediate generalizations of its second order asymptotics, for example the main results in [arXiv:1510.04682] and [arXiv:1208.1400].

math-ph↗

arXiv2019

Invariant measure for stochastic Schr\"odinger equations

Quantum trajectories are Markov processes that describe the time-evolution of a quantum system undergoing continuous indirect measurement. Mathematically, they are defined as solutions of the so-called "Stochastic Schr\"odinger Equations", which are nonlinear stochastic differential equations driven by Poisson and Wiener processes. This paper is devoted to the study of the invariant measures of quantum trajectories. Particularly, we prove that the invariant measure is unique under an ergodicity condition on the mean time evolution, and a "purification" condition on the generator of the evolution. We further show that quantum trajectories converge in law exponentially fast towards this invariant measure. We illustrate our results with examples where we can derive explicit expressions for the invariant measure.

math-ph↗

arXiv2018

Heat conservation and fluctuations between quantum reservoirs in the Two-Time Measurement picture

This work concerns the statistics of the Two-Time Measurement definition of heat variation in each reservoir of a thermodynamic quantum system. We study the cumulant generating function of the heat flows in the thermodynamic and large-time limits. It is well-known that, if the system is time-reversal invariant, this cumulant generating function satisfies the celebrated Evans--Searles symmetry. We show in addition that, under appropriate ultraviolet regularity assumptions on the local interaction between the reservoirs, it satisfies a translation-invariance property, as proposed in [Andrieux et al. New J. Phys. 2009]. We particularly fix some proofs of the latter article where the ultraviolet condition was not mentioned. We detail how these two symmetries lead respectively to fluctuation relations and a statistical refinement of heat conservation for isolated thermodynamic quantum systems. As in [Andrieux \emph{et al.} New J. Phys. 2009], we recover the Fluctuation-Dissipation Theorem in the linear response theory, short of Green--Kubo relations. We illustrate the general theory on a number of canonical models.

math-ph↗

arXiv2018

Recurrence and transience of continuous-time open quantum walks

This paper is devoted to the study of continuous-time processes known as continuous-time open quantum walks (CTOQWs). A CTOQW represents the evolution of a quantum particle constrained to move on a discrete graph, but also has internal degrees of freedom modeled by a state (in the quantum mechanical sense), and contain as a special case continuous-time Markov chains on graphs. Recurrence and transience of a vertex are an important notion in the study of Markov chains, and it is known that all vertices must be of the same nature if the Markov chain is irreducible. In the present paper we address the corresponding results in the context of irreducible CTOQWs. Because of the "quantum" internal degrees of freedom, CTOQWs exhibit non standard behavior, and the classification of recurrence and transience properties obeys a "trichotomy" rather than the classical dichotomy. Essential tools in this paper are the so-called "quantum trajectories" which are jump stochastic differential equations which can be associated with CTOQWs.

math.PR↗

arXiv2017

Landauer's Principle for Trajectories of Repeated Interaction Systems

We analyze Landauer's principle for repeated interaction systems consisting of a reference quantum system $\mathcal{S}$ in contact with an environment $\mathcal{E}$ which is a chain of independent quantum probes. The system $\mathcal{S}$ interacts with each probe sequentially, for a given duration, and the Landauer principle relates the energy variation of $\mathcal{E}$ and the decrease of entropy of $\mathcal{S}$ by the entropy production of the dynamical process. We consider refinements of the Landauer bound at the level of the full statistics (FS) associated to a two-time measurement protocol of, essentially, the energy of $\mathcal{E}$. The emphasis is put on the adiabatic regime where the environment, consisting of $T \gg 1$ probes, displays variations of order $T^{-1}$ between the successive probes, and the measurements take place initially and after $T$ interactions. We prove a large deviation principle and a central limit theorem as $T \to \infty$ for the classical random variable describing the entropy production of the process, with respect to the FS measure. In a special case, related to a detailed balance condition, we obtain an explicit limiting distribution of this random variable without rescaling. At the technical level, we obtain a non-unitary adiabatic theorem generalizing that of [Commun. Math. Phys. (2017) 349: 285] and analyze the spectrum of complex deformations of families of irreducible completely positive trace-preserving maps.

math-ph↗

arXiv2017

Invariant Measure for Quantum Trajectories

We study a class of Markov chains that model the evolution of a quantum system subject to repeated measurements. Each Markov chain in this class is defined by a measure on the space of matrices. It is then given by a random product of correlated matrices taken from the support of the defining measure. We give natural conditions on this support that imply that the Markov chain admits a unique invariant probability measure. We moreover prove the geometric convergence towards this invariant measure in the Wasserstein metric. Standard techniques from the theory of products of random matrices cannot be applied under our assumptions, and new techniques are developed, such as maximum likelihood-type estimations.

math.PR↗

arXiv2016

Passage times, exit times and Dirichlet problems for open quantum walks

We consider open quantum walks on a graph, and consider the random variables defined as the passage time and number of visits to a given point of the graph. We study in particular the probability that the passage time is finite, the expectation of that passage time, and the expectation of the number of visits, and discuss the notion of recurrence for open quantum walks. We also study exit times and exit probabilities from a finite domain, and use them to solve Dirichlet problems and to determine harmonic measures. We consider in particular the case of irreducible open quantum walks. The results we obtain extend those for classical Markov chains.

math-ph↗

arXiv2016

Contractivity properties of a quantum diffusion semigroup

We consider a quantum generalization of the classical heat equation, and study contractivity properties of its associated semigroup. We prove a Nash inequality and a logarithmic Sobolev inequality. The former leads to an ultracontractivity result. This in turn implies that the largest eigenvalue and the purity of a state with positive Wigner function, evolving under the action of the semigroup, decrease at least inverse polynomially in time, while its entropy increases at least logarithmically in time.

quant-ph↗

arXiv2016

On entropy production of repeated quantum measurements I. General theory

We study entropy production (EP) in processes involving repeated quantum measurements of finite quantum systems. Adopting a dynamical system approach, we develop a thermodynamic formalism for the EP and study fine aspects of irreversibility related to the hypothesis testing of the arrow of time. Under a suitable chaoticity assumption, we establish a Large Deviation Principle and a Fluctuation Theorem for the EP.

math-ph↗

arXiv2015

Second-order asymptotics for quantum hypothesis testing in settings beyond i.i.d. - quantum lattice systems and more

Quantum Stein's Lemma is a cornerstone of quantum statistics and concerns the problem of correctly identifying a quantum state, given the knowledge that it is one of two specific states ($\rho$ or $\sigma$). It was originally derived in the asymptotic i.i.d. setting, in which arbitrarily many (say, $n$) identical copies of the state ($\rho^{\otimes n}$ or $\sigma^{\otimes n}$) are considered to be available. In this setting, the lemma states that, for any given upper bound on the probability $\alpha_n$ of erroneously inferring the state to be $\sigma$, the probability $\beta_n$ of erroneously inferring the state to be $\rho$ decays exponentially in $n$, with the rate of decay converging to the relative entropy of the two states. The second order asymptotics for quantum hypothesis testing, which establishes the speed of convergence of this rate of decay to its limiting value, was derived in the i.i.d. setting independently by Tomamichel and Hayashi, and Li. We extend this result to settings beyond i.i.d.. Examples of these include Gibbs states of quantum spin systems (with finite-range, translation-invariant interactions) at high temperatures.

quant-ph↗

arXiv2015

Landauer's Principle in Repeated Interaction Systems

We study Landauer's Principle for Repeated Interaction Systems (RIS) consisting of a reference quantum system $\mathcal{S}$ in contact with a structured environment $\mathcal{E}$ made of a chain of independent quantum probes; $\mathcal{S}$ interacts with each probe, for a fixed duration, in sequence. We first adapt Landauer's lower bound, which relates the energy variation of the environment $\mathcal{E}$ to a decrease of entropy of the system $\mathcal{S}$ during the evolution, to the peculiar discrete time dynamics of RIS. Then we consider RIS with a structured environment $\mathcal{E}$ displaying small variations of order $T^{-1}$ between the successive probes encountered by $\mathcal{S}$, after $n\simeq T$ interactions, in keeping with adiabatic scaling. We establish a discrete time non-unitary adiabatic theorem to approximate the reduced dynamics of $\mathcal{S}$ in this regime, in order to tackle the adiabatic limit of Landauer's bound. We find that saturation of Landauer's bound is equivalent to a detailed balance condition on the repeated interaction system, reflecting the non-equilibrium nature of the repeated interaction system dynamics. This is to be contrasted with the generic saturation of Landauer's bound known to hold for continuous time evolution of an open quantum system interacting with a single thermal reservoir in the adiabatic regime.

math-ph↗

arXiv2015

Irreducible decompositions and stationary states of quantum channels

For a quantum channel (completely positive, trace-preserving map), we prove a generalization to the infinite dimensional case of a result by Baumgartner and Narnhofer. This result is, in a probabilistic language, a decomposition of a general quantum channel into its irreducible positive recurrent components. This decomposition is related with a communication relation on the reference Hilbert space. This allows us to describe the full structure of invariant states of a quantum channel, and of their supports.

math-ph↗

arXiv2015

Full statistics of energy conservation in two times measurement protocols

The first law of thermodynamics states that the average total energy current between different reservoirs vanishes at large times. In this note we examine this fact at the level of the full statistics of two times measurement protocols also known as the Full Counting Statistics. Under very general conditions, we establish a tight form of the first law asserting that the fluctuations of the total energy current computed from the energy variation distribution are exponentially suppressed in the large time limit. We illustrate this general result using two examples: the Anderson impurity model and a 2D spin lattice model.

math-ph↗

arXiv2014

Homogeneous Open Quantum Random Walks on a lattice

We study Open Quantum Random Walks for which the underlying graph is a lattice, and the generators of the walk are translation-invariant. We consider the quantum trajectory associated with the OQRW, which is described by a position process and a state process. We obtain a central limit theorem and a large deviation principle for the position process, and an ergodic result for the state process. We study in detail the case of homogeneous OQRWs on a lattice, with internal space $h={\mathbb C}^2$.

math.PR↗

arXiv2014

Open Quantum Random Walks: reducibility, period, ergodic properties

We study the analogues of irreducibility, period, and communicating classes for open quantum random walks, as defined by Attal et al. (J. Stat. Phys., 2012). We recover results similar to the standard ones for Markov chains, in terms of ergodic behavior, decomposition into irreducible subsystems, and characterization of stationary states.

math.PR↗

arXiv2011

Entropic Fluctuations in Quantum Statistical Mechanics. An Introduction

These lecture notes provide an elementary introduction, within the framework of finite quantum systems, to recent developments in the theory of entropic fluctuations.

math-ph↗

arXiv2004

From n+1-level atom chains to n-dimensional noises

In quantum physics, the state space of a countable chain of (n+1)-level atoms becomes, in the continuous field limit, a Fock space with multiplicity n. In a more functional analytic language, the continuous tensor product space over R of copies of the space C^{n+1} is the symmetric Fock space Gamma_s(L^2(R;C^n)). In this article we focus on the probabilistic interpretations of these facts. We show that they correspond to the approximation of the n-dimensional normal martingales by means of obtuse random walks, that is, extremal random walks in R^n whose jumps take exactly n+1 different values. We show that these probabilistic approximations are carried by the convergence of the basic matrix basis a^i_j(p) of $\otimes_N \CC^{n+1}$ to the usual creation, annihilation and gauge processes on the Fock space.

math.PR↗