Tristan Benoist

Tristan Benoist

I am a researcher (CR CNRS) in mathematical physics in the probability team of Toulouse Mathematics Institute (IMT). I am also an associate researcher at neighboring Theoretical Physics Laboratory (LPT).

I am mainly interested in the mathematics of quantum system thermodynamics. I focus on definitions of thermodynamic quantities and their fluctuations, and study related questions in probability.

In 2021 I gave a masterclass to master students in mathematics in Toulouse. It consisted of a short presentation of a non exhaustive selection of topics relating probability and quantum mechanics. The support I used is available: Introduction to quantum mechanics for probabilists. It may be helpful to understand my research interests.

With Ion Nechita and Clément Pellegrini, we organize the IMT-LPT mathematical physics seminar. The schedule is available on indico.math.cnrs.

I am the coordinator of ANR project “Quantum Trajectories (QTraj)”. I am also a member of ANR project ESQuisses.

I am one of the organizers of the semester Mathematical Aspects of Quantum Mechanics from January to June 2024.

Several events are organized and you are welcome to participate.

Contact information


You may find or publish corrections or comments on PubPeer.


All my articles preprints are available on this arXiv page. You can also find my articles listed on my ORCID and this HAL page.

  1. T. Benoist, L. Bruneau and C. Pellegrini, Quantum trajectory of the one atom maser, (2024) arXiv:2403.20094 (preprint)

    Abstract: The evolution of a quantum system undergoing repeated indirect measurements naturally leads to a Markov chain on the set of states which is called a quantum trajectory. In this paper we consider a specific model of such a quantum trajectory associated to the one-atom maser model. It describes the evolution of one mode of the quantized electromagnetic field in a cavity interacting with two-level atoms. When the system is non-resonant we prove that this Markov chain admits a unique invariant probability measure. We moreover prove convergence in the Wasserstein metric towards this invariant measure. These results rely on a purification theorem: almost surely the state of the system approaches the set of pure states. Compared to similar results in the literature, the system considered here is infinite dimensional. While existence of an invariant measure is a consequence of the compactness of the set of states in finite dimension, in infinite dimension existence of an invariant measure is not free. Furthermore usual purification criterions in finite dimension have no straightforward equivalent in infinite dimension.

  2. T. Benoist, L. Bruneau, V. Jaksic, A. Panati, C.-A. Pillet, On the thermodynamic limit of two-times measurement entropy production, (2024) arXiv:2402.09380 (preprint)

    Abstract: We provide a justification, via the thermodynamic limit, of the modular formula for entropy production in two-times measurement proposed in [Benoist, Bruneau, Jakšić, Panati and Pillet - arXiv:2310.10582]. We consider the cases of open quantum systems in which all thermal reservoirs are either (discrete) quantum spin systems or free Fermi gases.

  3. T. Benoist, A. Hautecœur and C. Pellegrini, Quantum Trajectories. Spectral Gap, Quasi-compactness & Limit Theorems, (2024) arXiv:2402.03879 (preprint)

    Abstract: Quantum trajectories are Markov processes modeling the evolution of a quantum system subjected to repeated independent measurements. Inspired by the theory of random products of matrices, it has been shown that these Markov processes admit a unique invariant measure under a purification and an irreducibility assumptions. This paper is devoted to the spectral study of the underlying Markov operator. Using Quasi-compactness, it is shown that this operator admits a spectral gap and the peripheral spectrum is described in a precise manner. Next two perturbations of this operator are studied. This allows to derive limit theorems (Central Limit Theorem, Berry-Esseen bounds and Large Deviation Principle) for the empirical mean of functions of the Markov chain as well as the Lyapounov exponent of the underlying random dynamical system.

  4. T. Benoist, L. Bruneau, V. Jaksic, A. Panati, C.-A. Pillet, A note on two-times measurement entropy production and modular theory, Lett. Math. Phys. 114 32 (2024) (link, preprint)

    Abstract: Recent theoretical investigations of the two-times measurement entropy production (2TMEP) in quantum statistical mechanics have shed a new light on the mathematics and physics of the quantum-mechanical probabilistic rules. Among notable developments are the extensions of entropic fluctuation relations to quantum domain and discovery of a deep link between 2TMEP and modular theory of operator algebras. All these developments concerned the setting where the state of the system at the instant of the first measurement is the same as the state whose entropy production is measured. In this work we consider the case where these two states are different and link this more general 2TEMP to modular theory. The established connection allows us to show that under general ergodicity assumptions the 2TEMP is essentially independent of the choice of the system state at the instant of the first measurement due to a decoherence effect induced by the first measurement. This stability sheds a new light on the concept of quantum entropy production, and, in particular, on possible quantum formulations of the celebrated classical Gallavotti--Cohen Fluctuation Theorem which will be studied in the continuation of this work.

  5. T. Benoist, J.-L. Fatras and C. Pellegrini, Limit theorems for Quantum Trajectories, Stoch. Process. Their Appl. 164 (2023) 288 - 310 (link, preprint)

    Abstract: Quantum trajectories are Markov processes modeling the evolution of a quantum system subjected to repeated independent measurements. Under purification and irreducibility assumptions, these Markov processes admit a unique invariant measure — see Benoist et al. (2019). In this article we prove finer limit theorems such as Law of Large Numbers (LLN), Functional Central Limit Theorem, Law of Iterated Logarithm and Moderate Deviation Principle. The proof of the LLN is based on Birkhoff’s ergodic theorem and an analysis of harmonic functions. The other theorems are proved using martingale approximation of empirical sums.

  6. T. Benoist, M. Fraas and J. Fröhlich, The appearance of particle tracks in detectors -- II: the semi-classical realm, J. Math. Phys. 63 (2022) 062101 (link, preprint)

    Abstract: The appearance of tracks, close to classical orbits, left by charged quantum particles propagating inside a detector, such as a cavity periodically illuminated by light pulses, is studied for a family of idealized models. In the semi-classical regime, which is reached when one considers highly energetic particles, we present a detailed, mathematically rigorous analysis of this phenomenon. If the Hamiltonian of the particles is quadratic in position- and momentum operators, as in the examples of a freely moving particle or a particle in a homogeneous external magnetic field, we show how symmetries, such as spherical symmetry, of the initial state of a particle are broken by tracks consisting of infinitely many approximately measured particle positions and how, in the classical limit, the initial position and velocity of a classical particle trajectory can be reconstructed from the observed particle track.

  7. T. Benoist, L. Hänggli and C. Rouzé, Deviation bounds and concentration inequalities for quantum noises, Quantum 6 (2022) 772 (link, preprint)

    Abstract: We provide a stochastic interpretation of non-commutative Dirichlet forms in the context of quantum filtering. For stochastic processes motivated by quantum optics experiments, we derive an optimal finite time deviation bound expressed in terms of the non-commutative Dirichlet form. Introducing and developing new non-commutative functional inequalities, we deduce concentration inequalities for these processes. Examples satisfying our bounds include tensor products of quantum Markov semigroups as well as Gibbs samplers above a threshold temperature.

  8. T. Benoist, C. Bernardin, R. Chétrite, R. Chhaibi, J. Najnudel and C. Pellegrini, Emergence of jumps in quantum trajectories via homogeneization, Commun. Math. Phys. 387 (2021) 1821 - 1867 (link, preprint)

    Abstract: In the strong noise regime, we study the homogeneization of quantum trajectories i.e. stochastic processes appearing in the context of quantum measurement. When the generator of the average semi-group can be separated into three distinct time scales, we start by describing a homogenized limiting semi-group. This result is of independent interest and is formulated outside of the scope of quantum trajectories. Going back to the quantum context, we show that, in the Meyer-Zheng topology, the time-continuous quantum trajectories converge weakly to the discontinuous trajectories of a pure jump Markov process. Notably, this convergence cannot hold in the usual Skorokhod topology.

  9. T. Benoist, N. Cuneo, V. Jakšić and C.-A. Pillet, On entropy production of repeated quantum measurements II. Examples, J. Stat. Phys. 182 (2021) 44 (link, preprint)

    Abstract: We illustrate the mathematical theory of entropy production in repeated quantum measurement processes developed in a previous work by studying examples of quantum instruments displaying various interesting phenomena and singularities. We emphasize the role of the thermodynamic formalism, and give many examples of quantum instruments whose resulting probability measures on the space of infinite sequences of outcomes (shift space) do not have the (weak) Gibbs property. We also discuss physically relevant examples where the entropy production rate satisfies a large deviation principle but fails to obey the central limit theorem and the fluctuation-dissipation theorem. Throughout the analysis, we explore the connections with other, a priori unrelated topics like functions of Markov chains, hidden Markov models, matrix products and number theory.

  10. M. Ballesteros, T. Benoist, M. Fraas, and J. Fröhlich, The appearance of particle tracks in detectors, Commun. Math. Phys. 385 (2021) 429 – 463 (link, preprint)

    Abstract: The phenomenon that a quantum particle propagating in a detector, such as a Wilson cloud chamber, leaves a track close to a classical trajectory is analyzed. We introduce an idealized quantum-mechanical model of a charged particle that is periodically illuminated by pulses of laser light resulting in repeated indirect measurements of the approximate position of the particle. For this model we present a mathematically rigorous analysis of the appearance of particle tracks, assuming that the Hamiltonian of the particle is quadratic in the position- and momentum operators, as for a freely moving particle or a harmonic oscillator.

  11. T. Benoist, M. Fraas, Y. Pautrat and C. Pellegrini, Invariant measure for stochastic Schrödinger equations, Ann. Henri Poincaré 22 (2021) 347 – 374 (link, preprint)

    Abstract: 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 Schrödinger 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.

  12. T. Benoist, A. Panati and Y. Pautrat, Heat conservation and fluctuations for open quantum systems in the Two-Time, J. Stat. Phys. 178 (2020) 893 – 925 (link, preprint)

    Abstract: his 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 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.

  13. T. Benoist, A. Panati and R. Raquépas, Control of fluctuations and heavy tails for heat variation in the two-time measurement framework, Ann. Henri Poincaré 20 (2019) 631 (link, preprint)

    Abstract: We study heat fluctuations in the two-time measurement framework. For bounded perturbations, we give sufficient ultraviolet regularity conditions on the perturbation for the moments of the heat variation to be uniformly bounded in time, and for the Fourier transform of the heat variation distribution to be analytic and uniformly bounded in time in a complex neighborhood of 0. On a set of canonical examples, with bounded and unbounded perturbations, we show that our ultraviolet conditions are essentially necessary. If the form factor of the perturbation does not meet our assumptions, the heat variation distribution exhibits heavy tails. The tails can be as heavy as preventing the existence of a fourth moment of the heat variation.

  14. T. Benoist, F. Gamboa and C. Pellegrini, Quantum non demolition measurements: parameter estimation for mixtures of multinomials, Electron. J. Stat. 12(1) (2018) 555 – 571 (link, preprint)

    Abstract: In Quantum Non Demolition measurements, the sequence of observations is distributed as a mixture of multinomial random variables. Parameters of the dynamics are naturally encoded into this family of distributions. We show the local asymptotic mixed normality of the underlying statistical model and the consistency of the maximum likelihood estimator. Furthermore, we prove the asymptotic optimality of this estimator as it saturates the usual Cramér Rao bound.

  15. T. Benoist, V. Jaksic and C.-A. Pillet, Energy statistics in open harmonic networks, J. Stat. Phys. 168 (2017) 1016 – 1030 (link, preprint)

    Abstract: We relate the large time asymptotics of the energy statistics in open harmonic networks to the variance-gamma distribution and prove a full Large Deviation Principle. We consider both Hamiltonian and stochastic dynamics, the later case including electronic RC networks. We compare our theoretical predictions with the experimental data obtained by Ciliberto et al. [Phys. Rev. Lett. 110, 180601 (2013)].

  16. T. Benoist, M. Fraas, Y. Pautrat and C. Pellegrini, Invariant measure for quantum trajectories, Probab. Theory Relat. Fields 174 (2019) 307 – 334 (link, preprint)

    Abstract: 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.

  17. T. Benoist, I. Nechita, On bipartite unitary matrices generating subalgebra-preserving quantum operations, Linear Algebra Appl. 521 (2018) 70 – 103 (link, preprint)

    Abstract: We study the structure of bipartite unitary operators which generate via the Stinespring dilation theorem, quantum operations preserving some given matrix algebra, independently of the ancilla state. We characterize completely the unitary operators preserving diagonal, block-diagonal, and tensor product algebras. Some unexpected connections with the theory of quantum Latin squares are explored, and we introduce and study a Sinkhorn-like algorithm used to randomly generate quantum Latin squares.

  18. T. Benoist, V. Jaksic, Y. Pautrat, C.-A. Pillet, On entropy production of repeated quantum measurements I. General theory, Commun. Math. Phys. 357(1) (2018) 77 – 123 (link, preprint)

    Abstract: 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.

  19. T. Benoist, M. Fraas, V. Jaksic, C.-A. Pillet, Full statistics of erasure processes: Isothermal adiabatic theory and a statistical Landauer principle, Rev. Roumaine Math. Pures Appl. 62 (2017) 259 – 286 (link, preprint)

    Abstract: We study driven finite quantum systems in contact with a thermal reservoir in the regime in which the system changes slowly in comparison to the equilibration time. The associated isothermal adiabatic theorem allows us to control the full statistics of energy transfers in quasi-static processes. Within this approach, we extend Landauer's Principle on the energetic cost of erasure processes to the level of the full statistics and elucidate the nature of the fluctuations breaking Landauer's bound.

  20. T. Benoist, C. Pellegrini, F. Ticozzi, Exponential Stability of Subspaces for Quantum Stochastic Master Equations, Ann. Henri Poincaré, 18 (2017) 2045-2074 (link, preprint)

    Abstract: We study the stability of quantum pure states and, more generally, subspaces for stochastic dynamics that describe continuously-monitored systems. We show that the target subspace is almost surely invariant if and only if it is invariant for the average evolution, and that the same equivalence holds for the global asymptotic stability. Moreover, we prove that a strict linear Lyapunov function for the average evolution always exists, and latter can be used to derive sharp bounds on the Lyapunov exponents of the associated semigroup. Nonetheless, we also show that taking into account the measurements can lead to an improved bound on stability rate for the stochastic, non-averaged dynamics. We discuss explicit examples where the almost sure stability rate can be made arbitrary large while the average one stays constant.

  21. T. Benoist, V. Jaksic, A. Panati, Y. Pautrat, C.-A. Pillet, Full statistics of energy conservation in two times measurement protocols, Phys. Rev. E 92 (2015) 032115. (link, preprint)

    Abstract: 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.

  22. T. Benoist, C. Pellegrini, Large Time Behavior and Convergence Rate for Quantum Filters Under Standard Non Demolition Conditions, Commun. Math. Phys. 331 (2014) 703 – 723. (link, preprint)

    Abstract: A quantum system S undergoing continuous time measurement is usually described by a jump-diffusion stochastic differential equation. Such an equation is called a stochastic master equation and its solution is called a quantum trajectory. This solution describes actually the evolution of the state of S. In the context of Quantum Non Demolition measurement, we investigate the large time behavior of this solution. It is rigorously shown that, for large time, this solution behaves as if a direct Von Neumann measurement has been performed at time 0. In particular the solution converges to a random pure state which is related to the wave packet reduction postulate. Using theory of Girsanov transformation, we determine precisely the exponential rate of convergence towards this random state. The important problem of state estimation (used in experiment) is also investigated.

  23. M. Bauer, D. Bernard, T. Benoist, Iterated stochastic measurements, J. Phys. A: Math. Theor. 45 (2012) 494020. (link, preprint)

    Abstract: We describe a measurement device principle based on discrete iterations of Bayesian updating of system state probability distributions. Although purely classical by nature, these measurements are accompanied with a progressive collapse of the system state probability distribution during each complete system measurement. This measurement scheme finds applications in analysing repeated non-demolition indirect quantum measurements. We also analyse the continuous time limit of these processes, either in the Brownian diffusive limit or in the Poissonian jumpy limit. In the quantum mechanical framework, this continuous time limit leads to Belavkin equations which describe quantum systems under continuous measurements.

  24. M. Bauer, T. Benoist, D. Bernard, Repeated Quantum Non-Demolition Measurements: Convergence and Continuous Time Limit, Ann. Henri Poincaré 14 (2013) 639 – 679. (link, preprint)

    Abstract: We analyze general enough models of repeated indirect measurements in which a quantum system interacts repeatedly with randomly chosen probes on which Von Neumann direct measurements are performed. We prove, under suitable hypotheses, that the system state probability distribution converges after a large number of repeated indirect measurements, in a way compatible with quantum wave function collapse. Similarly a modified version of the system density matrix converges. We show that the convergence is exponential with a rate given by some relevant mean relative entropies. We also prove that, under appropriate rescaling of the system and probe interactions, the state probability distribution and the system density matrix are solutions of stochastic differential equations modeling continuous-time quantum measurements. We analyze the large time convergence of these continuous-time processes and prove convergence.

Ph.D. thesis

Open quantum systems and quantum stochastic processes (defended in September 2014, supervised by Denis Bernard)

Note: The almost sure convergence rate of Equation (5.205) derived in Section 5.5.4 is too optimistic. It has been corrected in (Benoist, Pellegrini, Ticozzi 2017).

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Full CV

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