Equivalence Classes and Local Asymptotic Normality in System Identification for Quantum Markov Chains

Guţǎ, Mădălin; Kiukas, Jukka

doi:10.1007/s00220-014-2253-0

Cited by 30 publications

(48 citation statements)

References 41 publications

(96 reference statements)

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“…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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“…These results thus provide a direct means of using MPS methods to study the resource requirements of quantum stochastic simulation, extending their relevance to the field of predictive modeling. Our approach complements other uses of tensor network methods for the description of classical systems with stochastic elements [35][36][37][38][39][40][41] and machine learning [42][43][44][45][46][47][48][49][50]. We extend these results in introducing causal structure, adapting MPS methods for predictive modeling.Predictive models.-Consider a system that generates an output x t sampled from a random variable X t at each time t.…”

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confidence: 90%

Matrix Product States for Quantum Stochastic Modeling

et al. 2018

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In stochastic modeling, there has been a significant effort towards finding predictive models that predict a stochastic process' future using minimal information from its past. Meanwhile, in condensed matter physics, matrix product states (MPS) are known as a particularly efficient representation of 1D spin chains. In this Letter, we associate each stochastic process with a suitable quantum state of a spin chain. We then show that the optimal predictive model for the process leads directly to an MPS representation of the associated quantum state. Conversely, MPS methods offer a systematic construction of the best known quantum predictive models. This connection allows an improved method for computing the quantum memory needed for generating optimal predictions. We prove that this memory coincides with the entanglement of the associated spin chain across the past-future bipartition.The quest for simple representations and models of the physical world, often phrased as Occam's famous razor, underlies most scientific pursuits. In this spirit, computational mechanics seeks the most memory-efficient predictive models for stochastic processes-models which track relevant past information about a process, in order to generate statistically faithful future predictions [1-5]. The classically minimal models, ε-machines, have been used in diverse contexts from neuroscience to nonequilibrium contextuality [6][7][8][9][10][11][12][13][14][15]. Recently, it was shown that quantum extensions of ε-machines can further reduce their memory [16], leading recent studies to find memoryefficient quantum means of predictive modeling [17][18][19][20][21][22][23][24][25][26][27].In condensed matter, on the other hand, simplicity is sought after for the description of quantum many-body systems. Tensor networks, such as matrix product states (MPS), for instance, provide an efficient and useful description of one-dimensional quantum systems-i.e., spin chains [28][29][30]. This has led to reliable and powerful numerical methods for probing and simulating properties of multi-partite systems, whose study would be otherwise intractable [31][32][33][34].In this Letter, we develop a connection between εmachines and the MPS representation, shown in Fig. 1. We associate each stochastic process with a suitable quantum state of a spin chain called q-sample, measurement of which generates the corresponding stochastic process. We then show that the classical ε-machine of the process leads to a systematic MPS representation of the q-sample. Conversely, applying MPS methods to this state allows us to construct a q-simulator for the associated stochastic process -the best known quantum model [17,21,22]. Lastly, we show that the entanglement across the past-future bipartition of the q-sample coincides exactly with the quantum memory requirements * Yangchengran92@gmail.com † quantum@felix-binder.net ‡ mgu@quantumcomplexity.org Fig. 1. This Letter connects the complexity of stochastic processes and the representational complexity of spin chains. These (i) lin...

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“…The estimation of the permutationally invariant part of the density matrix as an approximation to the true state is also relevant in certain physical models [26][27][28]. Similarly, the estimation of matrix product states [29] is particularly relevant for many-body systems, but also for estimating dynamical parameters of open systems [30,31].…”

Section: Introductionmentioning

confidence: 99%

Statistically efficient tomography of low rank states with incomplete measurements

2016

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The construction of physically relevant low dimensional state models, and the design of appropriate measurements are key issues in tackling quantum state tomography for large dimensional systems. We consider the statistical problem of estimating low rank states in the set-up of multiple ions tomography, and investigate how the estimation error behaves with a reduction in the number of measurement settings, compared with the standard ion tomography setup. We present extensive simulation results showing that the error is robust with respect to the choice of states of a given rank, the random selection of settings, and that the number of settings can be significantly reduced with only a negligible increase in error. We present an argument to explain these findings based on a concentration inequality for the Fisher information matrix. In the more general setup of random basis measurements we use this argument to show that for certain rank r states it suffices to measure in O r d log () bases to achieve the average Fisher information over all bases. We present numerical evidence for random states of up to eight atoms, which suggests that a similar behaviour holds in the case of Pauli bases measurements, for randomly chosen states. The relation to similar problems in compressed sensing is also discussed.

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Equivalence Classes and Local Asymptotic Normality in System Identification for Quantum Markov Chains

Cited by 30 publications

References 41 publications

Equivalence of matrix product ensembles of trajectories in open quantum systems

Equivalence of matrix product ensembles of trajectories in open quantum systems

Matrix Product States for Quantum Stochastic Modeling

Statistically efficient tomography of low rank states with incomplete measurements

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