On Bounded Posets Arising from Quantum Mechanical Measurements

Dorninger, Dietmar; Länger, Helmut

doi:10.1007/s10773-016-3068-x

Cited by 4 publications

(12 citation statements)

References 6 publications

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“…As shown in Dorninger and Länger (2016) up to isomorphism, the weakly structured sets of S-probabilities are exactly the bounded posets with an antitone involution in which the join of two disjoint elements exists and which have a full set of pseudostates, which in the light of Lemma 3.2 then reads Theorem 4.3 Up to isomorphism, the ∨-specific sets of Sprobabilities are exactly the bounded posets with an antitone involution in which the sum of two disjoint elements equals their join and which have a full set of pseudostates.…”

Section: Algebraic Representations Of Specific Sets Of S-probabilitiesmentioning

confidence: 91%

“…If (1), ( 2) and ( 7) hold, P is called a structured set of S-probabilities (cf. Dorninger and Länger (2016)), and if (1), ( 2) and ( 8) are satisfied P is known as a weakly structured set of S-probabilities (cf. Dorninger and Länger (2016)).…”

Section: Further Classes Of Specific Sets Of S-probabilitiesmentioning

confidence: 99%

“…Dorninger and Länger (2016)), and if (1), ( 2) and ( 8) are satisfied P is known as a weakly structured set of S-probabilities (cf. Dorninger and Länger (2016)). Now, we define the following classes of sets of Sprobabilities:…”

Section: Further Classes Of Specific Sets Of S-probabilitiesmentioning

confidence: 99%

“…Tkadlec (1991)). That P is lattice-ordered can be characterized by a simple criterion: According to Theorem 2.4, P is a concrete logic, and as shown in Dorninger and Länger (2016), a structured set of S-probabilities P which is a concrete logic is a lattice if and only if for all p, q ∈ P max( p, q) ∈ P (the maximum of the functions considered pointwise).…”

Section: Theorem 33 the Class Of Structured Sets Of S-probabilities Is A Proper Subclass Of The Class Of ∨-Specific Sets Of Sprobabilitiementioning

confidence: 99%

“…If for p, q ∈ P with p ∧ q = 0 the element p ∨ q exists in P, then a specific state on P satisfying (S5) if p, q ∈ P and p ∧q = 0 then s( p ∨q) = s( p)+s(q), is called a pseudostate on P (cf. Dorninger and Länger (2016)).…”

Section: Algebraic Representations Of Specific Sets Of S-probabilitiesmentioning

confidence: 99%

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On Boolean posets of numerical events

Dorninger

Länger

2021

Adv. in Comp. Int.

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With many physical processes in which quantum mechanical phenomena can occur, it is essential to take into account a decision mechanism based on measurement data. This can be achieved by means of so-called numerical events, which are specified as follows: Let S be a set of states of a physical system and p(s) the probability of the occurrence of an event when the system is in state $$s\in S$$ s ∈ S . A function $$p:S\rightarrow [0,1]$$ p : S → [ 0 , 1 ] is called a numerical event or alternatively, an S-probability. If a set P of S-probabilities is ordered by the order of real functions, it becomes a poset which can be considered as a quantum logic. In case the logic P is a Boolean algebra, this will indicate that the underlying physical system is a classical one. The goal of this paper is to study sets of S-probabilities which are not far from being Boolean algebras by means of the addition and comparison of functions that occur in these sets. In particular, certain classes of so-called Boolean posets of S-probabilities are characterized and related to each other and descriptions based on sets of states are derived.

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Section: Algebraic Representations Of Specific Sets Of S-probabilitiesmentioning

confidence: 91%

Section: Further Classes Of Specific Sets Of S-probabilitiesmentioning

confidence: 99%

Section: Further Classes Of Specific Sets Of S-probabilitiesmentioning

confidence: 99%

Section: Theorem 33 the Class Of Structured Sets Of S-probabilities Is A Proper Subclass Of The Class Of ∨-Specific Sets Of Sprobabilitiementioning

confidence: 99%

Section: Algebraic Representations Of Specific Sets Of S-probabilitiesmentioning

confidence: 99%

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On Boolean posets of numerical events

Dorninger

Länger

2021

Adv. in Comp. Int.

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Quantum Logics Defined by Sets of Numerical Events

Dorninger

Länger

2019

Reports on Mathematical Physics

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Identifying quantum logics by numerical events

Dorninger

2020

Mathematica Slovaca

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Let S be a set of states of a physical system and let p(s) be the probability of an occurrence of an event when the system is in the state s ∈ S. The function p from S to [0, 1] is called a numerical event, multidimensional probability or, alternatively, S-probability. Given a set of numerical events which has been obtained by measurements and not supposing any knowledge of the logical structure of the events that appear in the physical system, the question arises which kind of logic is inherent to the system under consideration. In particular, does one deal with a classical situation or a quantum one?In this survey several answers are presented. Starting by associating sets of numerical events to quantum logics we study structures that arise when S-probabilities are partially ordered by the order of functions and characterize those structures which indicate that one deals with a classical system. In particular, sequences of numerical events are considered that give rise to Bell-like inequalities. At the center of all studies there are so called algebras of S-probabilities, subsets of these and their generalizations. A crucial feature of these structures is that order theoretic properties can be expressed by the addition and subtraction of real functions entailing simplified algorithmic procedures.The study of numerical events and algebras of S-probabilities goes back to a cooperation of E. G. Beltrametti and M. J. Mączyński in 1991 and has since then resulted in a series of subsequent papers of physical interest the main results of which will be commented on and put in an appropriate context.

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On Bounded Posets Arising from Quantum Mechanical Measurements

Cited by 4 publications

References 6 publications

On Boolean posets of numerical events

On Boolean posets of numerical events

Quantum Logics Defined by Sets of Numerical Events

Identifying quantum logics by numerical events

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