0587662 20250526122257.020240716235959.9 1 s. E cav_un_epca*0587661VaxjoLinnaeus University2024 measurements topology numerical value cav_un_auth*0101124 Kárný Miroslav UTIA-B Adaptivní systémy Department of Adaptive Systems AS AS 40 A Ústav teorie informace a automatizace AV ČR, v. v. i. cav_un_auth*0469824 Gaj Aleksej UTIA-B Adaptivní systémy Department of Adaptive Systems AS AS CZ 30 Ústav teorie informace a automatizace AV ČR, v. v. i. cav_un_auth*0101092 Guy Tatiana Valentine UTIA-B Adaptivní systémy Department of Adaptive Systems AS AS Department of Adaptive Systems 30 Ústav teorie informace a automatizace AV ČR, v. v. i. https://library.utia.cas.cz/separaty/2024/AS/karny-0587662.pdf CA21169 EU-COST XE cav_un_auth*0452289 A quantitative observation assigns numerical values to a phenomen on 𝑝∈𝒑 e.g. a system s property To ensure a proper observation process, any hidden feedback must be avoided. It means that the u ncertainty 𝑢∈𝒖 affect ing the assignment must not depend on the phenomen on itself. Since quantification implicitly involves compar isons e.g. 𝑎 is smaller than 𝑏””, 𝑐 is more desired tha n 𝑑 etc.etc.)), it assume s the existence of a transitive and complete ordering ≼ on 𝒑 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 (≺,𝒑) 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 𝒪:(𝒑,𝒖)↦𝒐 has a matrix structure 𝒪=[𝑂(𝑝,𝑢)], 𝑝∈𝒑,𝑢∈𝒖 To mitigate the influence of indices corresponding to phenomenon and uncertainty , the s ingular value decomposition (SVD) is applied 𝑂=𝑆𝑉𝑁∗ w it h 𝑁∗ denoting transposition and conjugat ion of 𝑁, [ Structurally, this implies that the uncertainty modelling unitary matrix 𝑁 spans complex Hilbert s space. Subspaces of this space are projected onto quantitative observations in 𝒐. 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. Dvurečenskij . Gleasons Theorem and Its Applications Mathematics and Its Applications , vol 60 Kluwer Academic Publishers, Dordrecht/Boston/London, 1993. [4] A. Gaj and M. Kárný. Quantum like modelling of uncertainty in dynamic decision making. In Quantum Information and Probability: from Foundations to Engineering (QIP24), 2024 cav_un_auth*0469826 Quantum Information and Probability: from Foundations to Engineering (QIP24) 20240611 20240614 Vaxjo https://lnu.se/contentassets/d3ac9dd22aba45afa716071e96924335/qip-posters_2024--7juni.pdf SE BB 10000 10100 10103 2025 3 2 O 4 4o 4 20250526122245.4 4 20250526122257.0 PO RVO:67985556 https://hdl.handle.net/11104/0355028 1 S 2024 Soubory v repozitáři: 0587662.pdf SCOPUS PUBMED WOS cav_un_epca*0587661 Quantum Information and Probability: from Foundations to Engineering (QIP24) - Posters Vaxjo Linnaeus University 2024