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Observables are proper models of measurements
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SYSNO ASEP 0587662 Document Type A - Abstract R&D Document Type O - OstatnΓ Title Observables are proper models of measurements Author(s) KΓ‘rnΓ½, Miroslav (UTIA-B) RID, ORCID
Gaj, Aleksej (UTIA-B) ORCID
Guy, Tatiana Valentine (UTIA-B) RID, ORCIDNumber of authors 3 Source Title Quantum Information and Probability: from Foundations to Engineering (QIP24) - Posters. - Vaxjo : Linnaeus University, 2024 Number of pages 1 s. Publication form Online - E Action Quantum Information and Probability: from Foundations to Engineering (QIP24) Event date 11.06.2024 - 14.06.2024 VEvent location Vaxjo Country SE - Sweden Event type WRD Language eng - English Country SE - Sweden Keywords measurements ; topology ; numerical value Subject RIV BB - Applied Statistics, Operational Research OECD category Statistics and probability Institutional support UTIA-B - RVO:67985556 Annotation A quantitative observation assigns numerical values to a phenomen on πβπ e.g. a system s property To ensure a proper observation process, any hidden feedback must be avoided. It means that the u ncertainty π’βπ affect ing the assignment must not depend on the phenomen on itself. Since quantification implicitly involves compar isons e.g. π is smaller than πββ, π is more desired tha n π etc.etc.)), it assume s the existence of a transitive and complete ordering βΌ on π It can be shown, that i ts completeness is always attainable under uncertainty. The result [1] implies existence of a continuous, ordering preserving, quantitative observation iff the topology of open intervals in (βΊ,π) does not require more complexity than the natural order ing of real numbers . Hence , it is possible to distinguish a countable number of realizations of the quantitatively described phenomenon and a countable number of uncertainties that can be associated. Therefore , the observation mapping πͺ:(π,π)β¦π has a matrix structure πͺ=[π(π,π’)], πβπ,π’βπ To mitigate the influence of indices corresponding to phenomenon and uncertainty , the s ingular value decomposition (SVD) is applied π=πππβ w it h πβ denoting transposition and conjugat ion of π, [ Structurally, this implies that the uncertainty modelling unitary matrix π spans complex Hilbert s space. Subspaces of this space are projected onto quantitative observations in π. These subspaces represent the relevant, distinguishable random events . Thus, the quantitative observation is to be handled as an observable [ 3]. Th e proposed work elaborates on and discusses this idea The twin work [4] addresses this viewpoint within the context of decision making. It demonstrates that a probabilistic model applied to subspaces model ling uncertainties is appropriate. The present study suggests that the findings of [4] are applicable to any quantitative observation (measurement).
[1]G. Debreu. Representation of a pr eference ordering by a numerical function. In R.M. Thrall,
C.H. Coombs, and R.L. Davis, editors, Decision Processes 159 65, Wiley, 1954.
[2 ] G.H. Golub and C.F. Van Loan. Matrix Computations . Johns Hopkins , Univ. Press, 2012.
[3] A. DvureΔenskij . Gleasons Theorem and Its Applications Mathematics and Its
Applications , vol 60 Kluwer Academic Publishers, Dordrecht/Boston/London, 1993.
[4] A. Gaj and M. KΓ‘rnΓ½. Quantum like modelling of uncertainty in dynamic decision making. In
Quantum Information and Probability: from Foundations to Engineering (QIP24), 2024
Workplace Institute of Information Theory and Automation Contact MarkΓ©ta VotavovΓ‘, votavova@utia.cas.cz, Tel.: 266 052 201. Year of Publishing 2025
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