Measurement-induced randomness and structure in controlled qubit processes

Venegas-Li, Ariadna; Jurgens, Alexandra M.; Crutchfield, James P.

doi:10.1103/physreve.102.040102

Cited by 9 publications

(9 citation statements)

References 42 publications

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“…To close, we note that the structural tools and the entropyrate method introduced by this trilogy were put to practical application in two other previous works. One diagnosed the origin of randomness and structural complexity in quantum measurement [38]. The other exactly determined the thermodynamic functioning of Maxwellian information engines [39], when there had been no previous method for this kind of detailed and accurate analysis.…”

Section: Discussionmentioning

confidence: 99%

Ambiguity rate of hidden Markov processes

Jurgens

Crutchfield

2021

Phys. Rev. E

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The -machine is a stochastic process's optimal model-maximally predictive and minimal in size. It often happens that to optimally predict even simply defined processes, probabilistic models-including the -machine-must employ an uncountably infinite set of features. To constructively work with these infinite sets we map the -machine to a place-dependent iterated function system (IFS)-a stochastic dynamical system. We then introduce the ambiguity rate that, in conjunction with a process's Shannon entropy rate, determines the rate at which this set of predictive features must grow to maintain maximal predictive power over increasing horizons. We demonstrate, as an ancillary technical result that stands on its own, that the ambiguity rate is the (until now missing) correction to the Lyapunov dimension of an IFS's attracting invariant set. For a broad class of complex processes, this then allows calculating their statistical complexity dimension-the information dimension of the minimal set of predictive features.

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Section: Discussionmentioning

confidence: 99%

Ambiguity rate of hidden Markov processes

Jurgens

Crutchfield

2021

Phys. Rev. E

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“…To close, we note that the structural tools and the entropy-rate method introduced by this trilogy were put to practical application in two other previous works. One diagnosed the origin of randomness and structural complexity in quantum measurement [34]. The other exactly determined the thermodynamic functioning of Maxwellian information engines [35], when there had been no previous method for this kind of detailed and accurate classification.…”

Section: Discussionmentioning

confidence: 99%

Shannon Entropy Rate of Hidden Markov Processes

Jurgens,

Crutchfield

2020

Preprint

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The -machine is a stochastic process' optimal model-maximally predictive and minimal in size. It often happens that to optimally predict even simply-defined processes, probabilistic modelsincluding the -machine-must employ an uncountably-infinite set of features. To constructively work with these infinite sets we map the -machine to a place-dependent iterated function system (IFS)-a stochastic dynamical system. We then introduce the ambiguity rate that, in conjunction with a process' Shannon entropy rate, determines the rate at which this set of predictive features must grow to maintain maximal predictive power. We demonstrate, as a ancillary technical result which stands on its own, that the ambiguity rate is the (until now missing) correction to the Lyapunov dimension of an IFS's attractor. For a broad class of complex processes and for the first time, this then allows calculating their statistical complexity dimension-the information dimension of the minimal set of predictive features.

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“…In point of fact, these structural tools and the entropy-rate method introduced here have already been put to practical use in two prior works. One diagnosed the origin of randomness and structural complexity in quantum measurement [35]. The other exactly determined the thermodynamic functioning of Maxwellian information engines [36], when there had been no previous method for this.…”

Section: Computational Advantages and Disadvantagesmentioning

confidence: 99%

“…In point of fact, the following and its sequel [34] were proceeded by two companions that applied the theoretical results here to two, rather different, physical domains. The first analyzed the origin of randomness and structural complexity engendered by quantum measurement [35]. The second solved a longstanding problem on exactly determining the thermodynamic functioning of Maxwellian demons, aka information engines [36].…”

Section: Introductionmentioning

confidence: 99%

Shannon Entropy Rate of Hidden Markov Processes

Jurgens

Crutchfield

2021

J Stat Phys

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Hidden Markov chains are widely applied statistical models of stochastic processes, from fundamental physics and chemistry to finance, health, and artificial intelligence. The hidden Markov processes they generate are notoriously complicated, however, even if the chain is finite state: no finite expression for their Shannon entropy rate exists, as the set of their predictive features is generically infinite. As such, to date one cannot make general statements about how random they are nor how structured. Here, we address the first part of this challenge by showing how to efficiently and accurately calculate their entropy rates. We also show how this method gives the minimal set of infinite predictive features. A sequel addresses the challenge’s second part on structure.

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Measurement-induced randomness and structure in controlled qubit processes

Cited by 9 publications

References 42 publications

Ambiguity rate of hidden Markov processes

Ambiguity rate of hidden Markov processes

Shannon Entropy Rate of Hidden Markov Processes

Shannon Entropy Rate of Hidden Markov Processes

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