The complete Heyting algebra of subsystems and contextuality

Vourdas, A.

doi:10.1063/1.4817855

Cited by 10 publications

(10 citation statements)

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“…We easily verify equations ( 21), (23) and (29). The diagonal ones σ µµ are density matrices describing the two orbits, and are important for section 6.…”

Section: First Example: the Set C(3 6 Z)mentioning

confidence: 90%

“…Using a quantum (Hilbert space) analogue of the derivation of the Frechet inequality, we prove logical Bell-like inequalities. They have been introduced in [22] and used in [23,24], in the context of multipartite entangled systems. In this paper they are used with our coherent states in a single quantum system.…”

Section: Logical Bell-like Inequalities For a Single Quantum System: ...mentioning

confidence: 99%

“…• C6: our coherent states violate logical Bell-like inequalities for a single quantum system: logical Bell-like inequalities have been introduced in [22] in the context of multipartite entangled systems, and used in [23,24]. In the context of multipartite systems they describe correlations between the various parties.…”

Section: Introductionmentioning

confidence: 99%

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Ultra-quantum coherent states in a single finite quantum system

Vourdas

2023

J. Phys. A: Math. Theor.

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A set of n coherent states is introduced in a quantum system with d-dimensional Hilbert space H(d). It is shown that they resolve the identity, and also have a discrete isotropy property. A ﬁnite cyclic group acts on the set of these coherent states, and partitions it into orbits. A n-tuple representation of arbitrary states in H(d), analogous to the Bargmann representation, is deﬁned. There are two other important properties of these coherent states which make them ‘ultra-quantum’. The ﬁrst property is related to the Grothendieck formalism which studies the ‘edge’ of the Hilbert space and quantum formalisms. Roughly speaking the Grothendieck theorem considers a ‘classical’ quadratic form C that uses complex numbers in the unit disc, and a ‘quantum’ quadratic form Q that uses vectors in the unit ball of the Hilbert space. It shows that if C < 1, the corresponding Q might take values greater than 1, up to the complex Grothendieck constant kG. Q related to these coherent states is shown to take values in the ‘Grothendieck region’ (1, kG), which is classically forbidden in the sense that C does not take values in it. The second property complements this, showing that these coherent states violate logical Bell-like inequalities (which for a single quantum system are quantum versions of the Frechet probabilistic inequalities). In this sense also, our coherent states are deep into the quantum region.

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“…We easily verify equations ( 21), (23) and (29). The diagonal ones σ µµ are density matrices describing the two orbits, and are important for section 6.…”

Section: First Example: the Set C(3 6 Z)mentioning

confidence: 90%

Section: Logical Bell-like Inequalities For a Single Quantum System: ...mentioning

confidence: 99%

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Ultra-quantum coherent states in a single finite quantum system

Vourdas

2023

J. Phys. A: Math. Theor.

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“…After the fundamental work by Einstein, Podolsky and Rosen [1] and also Schrödinger [2] it has been studied extensively in the literature [3]. It leads to strong correlations between various parties, which have been studied within the general area of Bell inequalities and contextuality [4][5][6][7][8][9][10][11][12][13][14][15]. Kolmogorov (classical) probabilities obey many inequalities, and in this paper we are interested in Boole inequalities, Chung-Erdös inequalities [16] and Frechet inequalities [17,18].…”

Section: Introductionmentioning

confidence: 99%

“…• Frechet and CHSH inequalities: we study the logical derivation [11][12][13] of CHSH (Clauser, Horne, Shimony and Holt [6]) type of inequalities. We prove that all rank one (factorizable) states obey CHSH inequalities, while some rank two (entangled) states violate them.…”

Section: Introductionmentioning

confidence: 99%

Probabilistic inequalities and measurements in bipartite systems

Vourdas¹

2019

J. Phys. A: Math. Theor.

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Various inequalities (Boole inequality, Chung-Erdös inequality, Frechet inequality) for Kolmogorov (classical) probabilities are considered. Quantum counterparts of these inequalities are introduced, which have an extra 'quantum correction' term, and which hold for all quantum states. When certain sufficient conditions are satisfied, the quantum correction term is zero, and the classical version of these inequalities holds for all states. But in general, the classical version of these inequalities is violated by some of the quantum states. For example in bipartite systems, classical Boole inequalities hold for all rank one (factorizable) states, and are violated by some rank two (entangled) states. A logical approach to CHSH inequalities (which are related to the Frechet inequalities), is studied in this context. It is shown that CHSH inequalities hold for all rank one (factorizable) states, and are violated by some rank two (entangled) states. The reduction of the rank of a pure state by a quantum measurement with both orthogonal and coherent projectors, is studied. Bounds for the average rank reduction are given. I. INTRODUCTIONEntanglement is an important feature of quantum mechanics. After the fundamental work by Einstein, Podolsky and Rosen[1] and also Schrödinger[2] it has been studied extensively in the literature [3]. It leads to strong correlations between various parties, which have been studied within the general area of Bell inequalities and contextuality [4][5][6][7][8][9][10][11][12][13][14][15].Kolmogorov (classical) probabilities obey many inequalities, and in this paper we are interested in Boole inequalities, Chung-Erdös inequalities [16] and Frechet inequalities [17,18]. Quantum probabilities are different from Kolmogorov probabilities, and we show in this paper that they obey quantum versions of these inequalities, that contain extra 'quantum correction' terms. This is related to the fact that Kolmogorov probabilities are intimately connected to Boolean (classical) logic formalized with set theory, while quantum probabilities are related to the Birkhoff-von Neumann (quantum) logic [21,22] formalized with subspaces of a Hilbert space.We will use the terms quantum (classical) probabilistic inequalities, for those that contain (do not contain) quantum corrections. Then:• By definition all quantum states obey the quantum probabilistic inequalities.• We give sufficient conditions for the quantum corrections to be zero, in which case the quantum probabilistic inequalities reduce to the usual classical probabilistic inequalities.• In general, classical probabilistic inequalities are violated by some quantum states. It is interesting to study such cases, because this highlights the difference between quantum and classical (Kolmogorov) probabilities.

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Subsystems via quantum motions

Shojaei-Fard

2024

Anal.Math.Phys.

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The complete Heyting algebra of subsystems and contextuality

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Cited by 10 publications

References 40 publications

Ultra-quantum coherent states in a single finite quantum system

Ultra-quantum coherent states in a single finite quantum system

Probabilistic inequalities and measurements in bipartite systems

Subsystems via quantum motions

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