Topologies on quantum effects

Ma, Zhihao; Zhu, Sen

doi:10.1016/s0034-4877(09)90043-x

Cited by 5 publications

(3 citation statements)

References 23 publications

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“…This notion can be generalized to arbitrary statistical theories. The generalization of the notion of PVM (which is based on projections) to an observable based on effects is called a positive operator valued measure (POVM)) [99][100][101][102][103][104][105] and, in QM, will be represented by a mapping (B(H) stands for the set of bounded operators in H).…”

Section: Quantal Effectsmentioning

confidence: 99%

Generalized Probabilities in Statistical Theories

Holik

Massri²,

Plastino

et al. 2021

Quantum Reports

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We discuss different formal frameworks for the description of generalized probabilities in statistical theories. We analyze the particular cases of probabilities appearing in classical and quantum mechanics and the approach to generalized probabilities based on convex sets. We argue for considering quantum probabilities as the natural probabilistic assignments for rational agents dealing with contextual probabilistic models. In this way, the formal structure of quantum probabilities as a non-Boolean probabilistic calculus is endowed with a natural interpretation.

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Section: Quantal Effectsmentioning

confidence: 99%

Generalized Probabilities in Statistical Theories

Holik

Massri²,

Plastino

et al. 2021

Quantum Reports

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show abstract

“…We say that a net {a α } α∈Λ of P is order convergent to a ∈ P if there exist two nets {u α } α∈Λ and {v α } α∈Λ of P such that a ↑ u α ≤ a α ≤ v α ↓ a. We denote order convergence as 24]). Lemma 3.5.…”

Section: The Jointly Continuity Of Sequential Productmentioning

confidence: 99%

Continuity of the sequential product of sequential quantum effect algebras

2016

Journal of Mathematical Physics

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In order to study quantum measurement theory, sequential product defined by A • B = A 1/2 BA 1/2 for any two quantum effects A, B is introduced. Physically motivated conditions ask the sequential product to be continuous with respect to the strong operator topology. In this paper, we study the continuity problems of the sequential product A • B = A 1/2 BA 1/2 with respect to the other important topologies, as norm topology, weak operator topology, order topology, interval topology, etc.

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“…A generalization of the above mechanism for computing probabilities is given by the notion of quantal effects and positive operator valued measures (POVM)) [75][76][77][78][79][80][81]. In quantum mechanics, a POVM is represented by a mapping…”

Section: A Elementary Tests In Quantum Mechanicsmentioning

confidence: 99%

Quantum Information as a Non-Kolmogorovian Generalization of Shannon’s Theory

Holik

Bosyk

Bellomo

2015

Entropy

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In this article, we discuss the formal structure of a generalized information theory based on the extension of the probability calculus of Kolmogorov to a (possibly) non-commutative setting. By studying this framework, we argue that quantum information can be considered as a particular case of a huge family of non-commutative extensions of its classical counterpart. In any conceivable information theory, the possibility of dealing with different kinds of information measures plays a key role. Here, we generalize a notion of state spectrum, allowing us to introduce a majorization relation and a new family of generalized entropic measures.

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Topologies on quantum effects

Cited by 5 publications

References 23 publications

Generalized Probabilities in Statistical Theories

Generalized Probabilities in Statistical Theories

Continuity of the sequential product of sequential quantum effect algebras

Quantum Information as a Non-Kolmogorovian Generalization of Shannon’s Theory

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