Sanov and central limit theorems for output statistics of quantum Markov chains

Horssen, Merlijn van; Guţǎ, Mădălin

doi:10.1063/1.4907995

Cited by 22 publications

(40 citation statements)

References 49 publications

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“…In the same spirit, we speculate that the above quantum equivalence results can contribute towards a better understanding of dynamical phase transitions on the level of quantum states. A possible application is the extension to continuous time of the Sanov Theorem for the empirical measure of multiple successive jumps, developed in [39]. In a different direction, the two ensembles set-up could be used to unify the existing system identification and asymptotic normality theory for discrete [40] and continuous [41] quantum Markov processes.…”

Section: Discussionmentioning

confidence: 99%

Equivalence of matrix product ensembles of trajectories in open quantum systems

et al. 2015

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The equivalence of thermodynamic ensembles is at the heart of statistical mechanics and central to our understanding of equilibrium states of matter. Recently, it has been shown that there is a formal connection between the dynamics of open quantum systems and the statistical mechanics in an extra dimension. This is established through the fact that an open system dynamics generates a Matrix Product state (MPS) which encodes the set of all possible quantum jump trajectories and permits the construction of generating functions in the spirit of thermodynamic partition functions. In the case of continuous-time Markovian evolution, such as that generated by a Lindblad master equation, the corresponding MPS is a so-called continuous MPS which encodes the set of continuous measurement records terminated at some fixed total observation time. Here we show that if one instead terminates trajectories after a fixed total number of quantum jumps, e.g. emission events into the environment, the associated MPS is discrete. This establishes an interesting analogy: The continuous and discrete MPS correspond to different ensembles of quantum trajectories, one characterised by total time the other by total number of quantum jumps. Hence they give rise to quantum versions of different thermodynamic ensembles, akin to "grand-canonical" and "isobaric", but for trajectories. Here we prove that these trajectory ensembles are equivalent in a suitable limit of long time or large number of jumps. This is in direct analogy to equilibrium statistical mechanics where equivalence between ensembles is only strictly established in the thermodynamic limit. An intrinsic quantum feature is that the equivalence holds only for all observables that commute with the number of quantum jumps.

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Section: Discussionmentioning

confidence: 99%

Equivalence of matrix product ensembles of trajectories in open quantum systems

et al. 2015

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“…Various schemes for quantum parameter estimation based on repetitive or continuous measurements have been studied: see e.g. [17][18][19][20][21][22][23][24][25]. Among them, analogous setups were analyzed in [19,24], where the problem was formalized in terms of quantum Markov chains.…”

Section: Introductionmentioning

confidence: 99%

“…[17][18][19][20][21][22][23][24][25]. Among them, analogous setups were analyzed in [19,24], where the problem was formalized in terms of quantum Markov chains. Specifically in [19] it has been shown that, under rather general assumptions, the statistics of the associated estimation problem converges asymptotically to a normal one, generalizing the similar results which were known to apply to purely classical settings [26][27][28].…”

Section: Introductionmentioning

confidence: 99%

Quantum estimation via sequential measurements

Burgarth

Giovannetti

et al. 2015

New J. Phys.

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The problem of estimating a parameter of a quantum system through a series of measurements performed sequentially on a quantum probe is analyzed in the general setting where the underlying statistics is explicitly non-i.i.d. We present a generalization of the central limit theorem in the present context, which under fairly general assumptions shows that as the number N of measurement data increases the probability distribution of functionals of the data (e.g., the average of the data) through which the target parameter is estimated becomes asymptotically normal and independent of the initial state of the probe. At variance with the previous studies (Guţă M 2011 Phys. Rev. A 83 062324; van Horssen M and Guţă M 2015 J. Math. Phys. 56 022109) we take a diagrammatic approach, which allows one to compute not only the leading orders in N of the moments of the average of the data but also those of the correlations among subsequent measurement outcomes. In particular our analysis points out that the latter, which are not available in usual i.i.d. data, can be exploited in order to improve the accuracy of the parameter estimation. An explicit application of our scheme is discussed by studying how the temperature of a thermal reservoir can be estimated via sequential measurements on a quantum probe in contact with the reservoir.See figure 1(a). In this context, the ultimate limits on the attainable precision in the estimation of g, optimized with respect to the general detection strategy, can be computed, resulting in the so-called quantum Cramér-Rao bound, which exhibits the functional dependence upon g r via the quantum Fisher information. See e.g. [2,3,[7][8][9][10][11][12].In many situations of physical interest, however, the possibility of reinitializing the setup to the same state is not necessarily guaranteed. In the present study we are going to consider a different scheme, in which a single probing system undergoes multiple applications of g L while being monitored during the process without being

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“…An important observation is that the bounds on fluctuations of a counting observable and its FPTs are controlled by the average dynamical activity, in analogy to the role played by the entropy production in the case of currents [1,13]. We hope these results will add to the growing body of work applying large deviation ideas and methods to the study of dynamics in driven systems [29][30][31][32][33][34][35][36][37][38][39][40][41], glasses [25,[42][43][44][45][46][47], protein folding and signaling networks [48][49][50][51], open quantum systems [52][53][54][55][56][57][58][59][60][61][62][63][64], and many other problems in nonequilibrium [65][66][67][68][69][70][71][72].…”

Section: Introductionmentioning

confidence: 99%

Simple bounds on fluctuations and uncertainty relations for first-passage times of counting observables

2017

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Recent large deviation results have provided general lower bounds for the fluctuations of time-integrated currents in the steady state of stochastic systems. A corollary are so-called thermodynamic uncertainty relations connecting precision of estimation to average dissipation. Here we consider this problem but for counting observables, i.e., trajectory observables which, in contrast to currents, are non-negative and nondecreasing in time (and possibly symmetric under time reversal). In the steady state, their fluctuations to all orders are bound from below by a Conway-Maxwell-Poisson distribution dependent only on the averages of the observable and of the dynamical activity. We show how to obtain the corresponding bounds for first-passage times (times when a certain value of the counting variable is first reached) and their uncertainty relations. Just like entropy production does for currents, dynamical activity controls the bounds on fluctuations of counting observables.

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Sanov and central limit theorems for output statistics of quantum Markov chains

Cited by 22 publications

References 49 publications

Equivalence of matrix product ensembles of trajectories in open quantum systems

Equivalence of matrix product ensembles of trajectories in open quantum systems

Quantum estimation via sequential measurements

Simple bounds on fluctuations and uncertainty relations for first-passage times of counting observables

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