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arXiv:2206.14223 · Annales Henri Poincaré

Concentration Inequalities for Output Statistics of Quantum Markov Processes

Federico Girotti, Juan P. Garrahan, Mădălin Guţǎ

Published Mar 27, 2023• 11 citations
arXivHTMLPDF

Abstract

Abstract We derive new concentration bounds for time averages of measurement outcomes in quantum Markov processes. This generalizes well-known bounds for classical Markov chains, which provide constraints on finite-time fluctuations of time-additive quantities around their averages. We employ spectral, perturbation and martingale techniques, together with non-commutative $$L_2$$ L 2 theory, to derive: (i) a Bernstein-type concentration bound for time averages of the measurement outcomes of a quantum Markov chain, (ii) a Hoeffding-type concentration bound for the same process, (iii) a generalization of the Bernstein-type concentration bound for counting processes of continuous-time quantum Markov processes, (iv) new concentration bounds for empirical fluxes of classical Markov chains which broaden the range of applicability of the corresponding classical bounds beyond empirical averages. We also suggest potential application of our results to parameter estimation and consider extensions to reducible quantum channels, multi-time statistics and time-dependent measurements, and comment on the connection to so-called thermodynamic uncertainty relations.

DOI
10.1007/s00023-023-01286-1

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Abstract We derive new concentration bounds for time averages of measurement outcomes in quantum Markov processes. This generalizes well-known bounds for classical Markov chains, which provide constraints on finite-time fluctuations of time-additive quantities around their averages. We employ spectral, perturbation and martingale techniques, together with non-commutative L2L_2L2​ L 2 theory, to derive: (i) a Bernstein-type concentration bound for time averages of the measurement outcomes of a quantum Markov chain, (ii) a Hoeffding-type concentration bound for the same process, (iii) a generalization of the Bernstein-type concentration bound for counting processes of continuous-time quantum Markov processes, (iv) new concentration bounds for empirical fluxes of classical Markov chains which broaden the range of applicability of the corresponding classical bounds beyond empirical averages. We also suggest potential application of our results to parameter estimation and consider extensions to reducible quantum channels, multi-time statistics and time-dependent measurements, and comment on the connection to so-called thermodynamic uncertainty relations.
Martingale concentration inequalities constitute a powerful mathematical tool in the analysis of problems in a wide variety of fields ranging from probability and statistics to information theory and machine learning. Here, we apply techniques borrowed from this field to quantum hypothesis testing, which is the problem of discriminating quantum states belonging to two different sequences, (ρn}n and (σn}n. We obtain achievability bounds on the finite blocklength type II Steinand Hoeffding errors which, for i.i.d. states, are in general tighter than the corresponding bounds obtained by Audenaert, Mosonyi, and Verstraete. We also derive finite blocklength bounds and moderate deviation results for pairs of sequences of correlated states satisfying a (non-homogeneous) factorization property. Examples of such sequences include Gibbs states of spin chains with translation-invariant finite range interaction, as well as finitely correlated quantum states. We apply our results to find bounds on the capacity of a certain class of classical-quantum channels with memory, which satisfy a so-called channel factorization property-both in the finite blocklength and moderate deviation regimes.
In this paper, we consider the statistics of repeated measurements on the output of a quantum Markov chain. We establish a large deviations result analogous to Sanov’s theorem for the multi-site empirical measure associated to finite sequences of consecutive outcomes of a classical stochastic process. Our result relies on the construction of an extended quantum transition operator (which keeps track of previous outcomes) in terms of which we compute moment generating functions, and whose spectral radius is related to the large deviations rate function. As a corollary to this, we obtain a central limit theorem for the empirical measure. Such higher level statistics may be used to uncover critical behaviour such as dynamical phase transitions, which are not captured by lower level statistics such as the sample mean. As a step in this direction, we give an example of a finite system whose level-1 (empirical mean) rate function is independent of a model parameter while the level-2 (empirical measure) rate is not.
Thermodynamic tradeoff relations quantify the fundamental concept of "no free lunch" in the physical world, suggesting that faster and more precise physical processes come at a higher thermodynamic cost. The key elements in these trade-off relations are the thermodynamic uncertainty relation and speed limit, which are closely tied to information inequalities from which other trade-off relations are derived. Concentration inequalities are relations that complement information inequalities in statistical analyses and have been widely used in various fields. However, their role in thermodynamic trade-off relations remains unclear. This Letter develops thermodynamic concentration inequalities that provide bounds for the distribution of observables in quantum and classical Markov processes. We derive a set of trade-off relations that generalize speed limits and thermodynamic uncertainty relations from the developed thermodynamic concentration inequalities. The derived trade-off relations hold under minimal assumptions of the underlying physical processes. This Letter clarifies the role of concentration inequalities in thermodynamics, paving the way for deriving new trade-off relations.
Quantum functional inequalities (e.g., the logarithmic Sobolev and Poincaré inequalities) have found widespread application in the study of the behavior of primitive quantum Markov semigroups. The classical counterparts of these inequalities are related to each other via a so-called transportation cost inequality of order 2 (TC2). The latter inequality relies on the notion of a metric on the set of probability distributions called the Wasserstein distance of order 2. (TC2) in turn implies a transportation cost inequality of order 1 (TC1). In this paper, we introduce quantum generalizations of the inequalities (TC1) and (TC2), making use of appropriate quantum versions of the Wasserstein distances, one recently defined by Carlen and Maas and the other defined by us. We establish that these inequalities are related to each other, and to the quantum modified logarithmic Sobolev- and Poincaré inequalities, as in the classical case. We also show that these inequalities imply certain concentration-type results for the invariant state of the underlying semigroup. We consider the example of the depolarizing semigroup to derive concentration inequalities for any finite dimensional full-rank quantum state. These inequalities are then applied to derive upper bounds on the error probabilities occurring in the setting of finite blocklength quantum parameter estimation.
This thesis is the fruit of investigations on the extension of ideas of Markov chain mixing to the quantum setting, and its application to problems of dissipative engineering. A Markov chain describes a statistical process where the probability of future events depends only on the state of the system at the present point in time, but not on the history of events. Very many important processes in nature are of this type, therefore a good understanding of their behavior has turned out to be very fruitful for science. Markov chains always have a non-empty set of limiting distributions (stationary states). The aim of Markov chain mixing is to obtain (upper and/or lower) bounds on the number of steps it takes for the Markov chain to reach a stationary state. The natural quantum extensions of these notions are density matrices and quantum channels. We set out to develop a general mathematical framework for studying quantum Markov chain mixing. We introduce two new distance measures into the quantum setting; the quantum χ2divergence and Hilbert’s projective metric. The quantum χ2 divergence allows us to extend the class of functional techniques called Poincare inequalities to the quantum setting. In the process, we identify the appropriate framework for discussing notions such as detailed balance and χ2-mixing: monotone Riemannian metrics on matrix spaces. This insight allows us to characterize the connection between spectral properties of a (primitive) quantum channel and its mixing time. Within the same framework we also derive a restricted quantum version of the celebrated conductance bound (Cheeger’s inequality). We then consider Hilbert’s projective metric a well known tool in Perron-Frobenius theory in the context of quantum information theory. As a classic tool for analyzing convergence of positive maps on cones, Hilbert’s projective metric is especially good at providing existence proofs of maps between two spaces (or cones). We relate this measure to distinguishability measures on restricted cones of operators, and analyze the observable loss of information after the application of a quantum channel under a restrict set of measurements. Various notions of contractivity of quantum channels are revealed through Hilbert’s metric. Still on the topic of quantum Markov chains, we introduce the notion of cutoff phenomenon to the quantum setting. The cutoff phenomenon describes the situation when a Markov chain does not converge for a potentially long time, and then at a specific point in time abruptly converges to equilibrium. In the thermodynamic limit, the convergence profile will then look like a step function. We show that this type of convergence behavior occurs, as measured in trace-norm, when a system is subject to a product channel, and the noise on each product is modeled by a one-parameter semigroup of quantum channels. Finally, we consider three independent tasks of dissipative engineering. The first task, which is aimed at experimental realization in the near future, consists in dissipatively
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.