We find the structure of generators of norm continuous quantum Markov semigroups on B(h) that are symmetric with respect to the scalar product tr(ρ^{1/2}xρ^{1/2}y) induced by a faithful normal invariant state invariant state ρand satisfy two quantum generalisations of the classical detailed balance condition related with this non-commutative notion of symmetry: the so-called standard detailed balance condition and the standard detailed balance condition with an antiunitary time reversal.
The principle of detailed balance is at the basis of equilibrium physics and is equivalent to the Kubo-Martin-Schwinger (KMS) condition (under quite general assumptions). In the present paper we prove that a large class of open quantum systems satisfies a dynamical generalization of the detailed balance condition ({\it dynamical detailed balance}) expressing the fact that all the micro-currents, associated to the Bohr frequencies are constant. The usual (equilibrium) detailed balance condition is characterized by the property that this constant is identically zero. From this we deduce a simple…
We propose a definition of infinite dimensional Choi-Jamiolkowski state associated with a completely positive trace preserving map. We introduce the notion of Theta-KMS adjoint of a quantum Markov semigroup, which is identified with the time reversed semigroup. The break down of Theta-KMS symmetry (or Theta-standard quantum detailed balance in the sense of Fagnola-Umanita) is measured by means of the von Neumann relative entropy of the Choi-Jamiolkowski states associated with the semigroup and its Theta-KMS adjoint.
We give an explicit entropy production formula for a class of quantum Markov semigroups, arising in the weak coupling limit of a system coupled with reservoirs, whose generators $\mathcal{L}$ are sums of other generators $\mathcal{L}_ω$ associated with positive Bohr frequencies $ω$ of the system. As a consequence, we show that any such semigroup satisfies the quantum detailed balance condition with respect to an invariant state if and only if all semigroups generated by each $\mathcal{L}_ω$ so do with respect to the same invariant state.
The concept of balance between two state preserving quantum Markov semigroups on von Neumann algebras is introduced and studied as an extension of conditions appearing in the theory of quantum detailed balance. This is partly motivated by the theory of joinings. Balance is defined in terms of certain correlated states (couplings), with entangled states as a specific case. Basic properties of balance are derived and the connection to correspondences in the sense of Connes is discussed. Some applications and possible applications, including to non-equilibrium statistical mechanics, are briefly e…
We characterize all Gaussian dynamical semigroups in continuous variables quantum systems of n-bosonic modes which have a thermal Gibbs state as a stationary solution. This is performed through an explicit relation between the diffusion and dissipation matrices, which characterize the semigroup dynamic, and the covariance matrix of the thermal equilibrium state. We also show that Alicki's quantum detailed-balance condition, based on a Gelfand-Naimark-Segal inner product, allows the determination of the temperature dependence of the diffusion and dissipation matrices, and the identification of …
We show that uniformly accelerated detectors can display genuinely thermal features even if the Kubo-Martin-Schwinger (KMS) condition fails to hold. These features include satisfying thermal detailed balance and having a Planckian response identical to cases in which the KMS condition is satisfied. In this context, we discuss that satisfying the KMS condition for accelerated trajectories is just sufficient but not necessary for the Unruh effect to be present in a given quantum field theory. Furthermore, we extract the necessary and sufficient conditions for the response function of an accelera…
Markovian master equations (formally known as quantum dynamical semigroups) can be used to describe the evolution of a quantum state $ρ$ when in contact with a memoryless thermal bath. This approach has had much success in describing the dynamics of real-life open quantum systems in the lab. Such dynamics increase the entropy of the state $ρ$ and the bath until both systems reach thermal equilibrium, at which point entropy production stops. Our main result is to show that the entropy production at time $t$ is bounded by the relative entropy between the original state and the state at time $2t$…
We study a class of finite dimensional quantum dynamical semigroups exp(tL) whose generators L are sums of Lindbladians satisfying the detailed balance condition. Such semigroup arise in the weak coupling (van Hove) limit of Hamiltonian dynamical systems describing open quantum systems out of equilibrium. We prove a general entropic fluctuation theorem for this class of semigroups by relating the cumulant generating function of entropy transport to the spectrum of a family of deformations of the generator L. We show that, besides the celebrated Evans-Searles symmetry, this cumulant generating …
On entropy production of repeated quantum measurements III. Quantum detailed balance
Univ Toulouse, INUC, UT2J, INSA Toulouse, TSECNRS, IMT（图卢兹大学）；Université Paris Cité and Sorbonne Université, CNRSLaboratoire de Probabilités, Statistique et Modélisation（巴黎西岱大学和索邦大学）；Dipartimento di Matematica, Politecnico di Milano（米兰理工大学数学系）；Université de Toulon, CNRS, CPT, UMR 7332（土伦大学）；Aix-Marseille Univ, CNRS, CPT, UMR 7332（艾克斯-马赛大学）
We consider a class of quantum dissipative semigroup on a von-Neumann algebra which admits a normal invariant state. We investigate asymptotic behavior of the dissipative dynamics and their relation to that of the canonical Markov shift. In case the normal invariant state is also faithful, we also extend the notion of `quantum detailed balance' introduced by Frigerio-Gorini and prove that forward weak Markov process and backward weak Markov process are equivalent by an anti-unitary operator.
The master equation for a probability density function (pdf) driven by Lévy noise, if conditioned to conform with the principle of detailed balance, admits a transformation to a contractive strongly continuous semigroup dynamics. Given a priori a functional form of the semigroup potential, we address the ground-state reconstruction problem for generic Lévy-stable semigroups, for {\em all} values of the stability index $μ\in (0,2)$. That is known to resolve an invariant pdf for confined Lévy flights (e.g. the former jump-type process). Jeopardies of the procedure are discussed, with a focus on:…
We introduce a new framework that yields spectral bounds on norms of functions of transition maps for finite, homogeneous Markov chains. The techniques employed work for bounded semigroups, in particular for classical as well as for quantum Markov chains and they do not require additional assumptions like detailed balance, irreducibility or aperiodicity. We use the method in order to derive convergence bounds that improve significantly upon known spectral bounds. The core technical observation is that power-boundedness of transition maps of Markov chains enables a Wiener algebra functional cal…
Quantum Gibbs Sampling in Infinite Dimensions: Generation, Mixing Times and Circuit Implementation
Bocconi University（博科尼大学）；Inria, Télécom Paris – LTCI, Institut Polytechnique de Paris（法国国家信息与自动化研究所，巴黎电信学院-通信与图像处理实验室，巴黎理工学院）；Institute for Quantum Information, RWTH Aachen University（亚琛工业大学量子信息研究所）；Inria, ENS de Lyon, Université Claude Bernard Lyon 1, LIP（法国国家信息与自动化研究所，里昂高等师范学校，克劳德·贝尔纳里昂第一大学，计算与信息科学实验室）
We generalize the notions of the non-commutative Poincaré and modified log-Sobolev inequalities for primitive quantum Markov semigroups (QMS) to not necessarily primitive ones. These two inequalities provide estimates on the decoherence time of the evolution. More precisely, we focus on an algebraic definition of environment-induced decoherence in open quantum systems which happens to be generic on finite dimensional systems and describes the asymptotic behavior of any QMS. An essential tool in our analysis is the explicit structure of the decoherence-free algebra generated by the QMS, a centr…
