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Multi-point Gaussian states, quadratic-exponential cost functionals, and large deviations estimates for linear quantum stochastic systems

Vladimirov, Igor; Petersen, Ian; James, Matthew

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This paper is concerned with risk-sensitive performance analysis for linear quantum stochastic systems interacting with external bosonic fields. We consider a cost functional in the form of the exponential moment of the integral of a quadratic polynomial of the system variables over a bounded time interval. Such functionals are related to more conservative behaviour and robustness of systems with respect to statistical uncertainty, which makes the challenging problems of their computation and...[Show more]

dc.contributor.authorVladimirov, Igor
dc.contributor.authorPetersen, Ian
dc.contributor.authorJames, Matthew
dc.date.accessioned2019-10-17T00:51:10Z
dc.identifier.issn0095-4616
dc.identifier.urihttp://hdl.handle.net/1885/177007
dc.description.abstractThis paper is concerned with risk-sensitive performance analysis for linear quantum stochastic systems interacting with external bosonic fields. We consider a cost functional in the form of the exponential moment of the integral of a quadratic polynomial of the system variables over a bounded time interval. Such functionals are related to more conservative behaviour and robustness of systems with respect to statistical uncertainty, which makes the challenging problems of their computation and minimization practically important. To this end, we obtain an integro-differential equation for the time evolution of the quadratic–exponential functional, which is different from the original quantum risk-sensitive performance criterion employed previously for measurement-based quantum control and filtering problems. Using multi-point Gaussian quantum states for the past history of the system variables and their first four moments, we discuss a quartic approximation of the cost functional and its infinite-horizon asymptotic behaviour. The computation of the asymptotic growth rate of this approximation is reduced to solving two algebraic Lyapunov equations. Further approximations of the cost functional, based on higher-order cumulants and their growth rates, are applied to large deviations estimates in the form of upper bounds for tail distributions. We discuss an auxiliary classical Gaussian–Markov diffusion process in a complex Euclidean space which reproduces the quantum system variables at the level of covariances but has different fourth-order cumulants, thus showing that the risk-sensitive criteria are not reducible to quadratic–exponential moments of classical Gaussian processes. The results of the paper are illustrated by a numerical example and may find applications to coherent quantum risk-sensitive control problems, where the plant and controller form a fully quantum closed-loop system, and other settings with nonquadratic cost functionals.
dc.description.sponsorshipThis work is supported by the Air Force Office of Scientific Research (AFOSR) and Office of Naval Research Global (ONRG) under Agreement Number FA2386-16-1-4065 and the Australian Research Council (ARC) under Grant DP180101805.
dc.format.extent55 pages
dc.format.mimetypeapplication/pdf
dc.language.isoen_AU
dc.publisherSpringer
dc.rights© Springer Science+Business Media, LLC, part of Springer Nature 2018
dc.sourceApplied Mathematics and Optimization
dc.subjectrisk-sensitive performance
dc.subjectlinear quantum
dc.subjectstochastic systems
dc.subjectbosonic fields
dc.subjectcost functional
dc.subjectexponential moment
dc.subjectquadratic polynomial
dc.subjectintegro-differential equation
dc.titleMulti-point Gaussian states, quadratic-exponential cost functionals, and large deviations estimates for linear quantum stochastic systems
dc.typeJournal article
local.description.notesImported from ARIES
local.identifier.citationvolumeOnline
dc.date.issued2018-07-24
local.identifier.absfor010503 - Mathematical Aspects of Classical Mechanics, Quantum Mechanics and Quantum Information Theory
local.identifier.absfor020603 - Quantum Information, Computation and Communication
local.identifier.absfor010203 - Calculus of Variations, Systems Theory and Control Theory
local.identifier.ariespublicationu1038773xPUB3
local.publisher.urlhttps://www.springernature.com/gp/products/journals
local.type.statusAccepted Version
local.contributor.affiliationVladimirov, Igor, College of Engineering and Computer Science, The Australian National University
local.contributor.affiliationPetersen, Ian, College of Engineering and Computer Science, The Australian National University
local.contributor.affiliationJames, Matthew, College of Engineering and Computer Science, The Australian National University
dc.relationhttp://purl.org/au-research/grants/arc/DP180101805
local.bibliographicCitation.issue1
local.bibliographicCitation.startpage83
local.bibliographicCitation.lastpage137
local.identifier.doi10.1007/s00245-018-9512-y
local.identifier.absseo970108 - Expanding Knowledge in the Information and Computing Sciences
local.identifier.absseo970109 - Expanding Knowledge in Engineering
local.identifier.absseo970101 - Expanding Knowledge in the Mathematical Sciences
dc.date.updated2021-11-28T07:27:56Z
dcterms.accessRightsOpen Access
dc.provenancehttp://sherpa.ac.uk/romeo/issn/0095-4616/ Author can archive post-print (ie final draft post-refereeing). Author's post-print on institutional repository or funder designated repository after 12 months embargo from first online publication (Sherpa/Romeo as of 22/10/2019)
CollectionsANU Research Publications

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