## Contexts in quantum, classical and partition logic (2006)

Venue: | In Handbook of Quantum Logic |

Citations: | 8 - 7 self |

### BibTeX

@INPROCEEDINGS{Svozil06contextsin,

author = {Karl Svozil},

title = {Contexts in quantum, classical and partition logic},

booktitle = {In Handbook of Quantum Logic},

year = {2006}

}

### OpenURL

### Abstract

Contexts are maximal collections of co-measurable observables “bundled together ” to form a “quasi-classical mini-universe. ” Different notions of contexts are discussed for classical, quantum and generalized urn–automaton systems. PACS numbers: 02.10.-v,02.50.Cw,02.10.Ud

### Citations

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Citation Context .... One of the most compact and comprehensive versions of the Kochen-Specker proof by contradiction in three-dimensional Hilbert space R3 has been given by Peres [50]. (For other discussions, see Refs. =-=[4, 14, 50, 51, 52, 53, 54, 55, 56, 57]-=-.) Peres’ version uses a 33-element set of lines without a two-valued state. The direction vectors of these lines arise by all permutations of coordinates from (0,0,1), (0,±1,1), (0,±1, √ 2), and (±1,... |

590 |
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Citation Context ... to speak of the “real physical existence” of different contexts, as knowledge of a single context makes impossible the measurement of all the other ones. Einstein-Podolski-Rosen (EPR) type arguments =-=[41]-=- utilizing a configuration sketched in Fig. 1 claim to be able to infer two different contexts counterfactually. One context is measured on one side of the setup, the other context on the other side o... |

552 | Speakable and Unspeakable in Quantum Mechanics - Bell - 1987 |

436 |
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Citation Context ...m context can equivalently be formalized by a single (nondegenerate) “maximal” selfadjoint operator C, such that all commuting, compatible co-measurable observables are functions thereof. (e.g., Ref. =-=[7]-=-, Sec. II.10, p. 90, English translation p. 173; Ref. [9], § 2; Ref. [10], pp. 227,228; Ref. [11], § 84). Note that mutually commuting opators have identical pairwise orthogonal sets of eigenvectors (... |

323 | Algorithmic Information Theory
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Citation Context ...on and measurement is the basis of quantum random number generators [76] which serve as a kind of “quantum random oracle” [77, 78]. It should be kept in mind that randomness, at least algorithmically =-=[79, 80, 81]-=-, does not come “for free,” thus exhibiting an amazing capacity of single quanta to support random outcomes. Alternatively, the unpredictable, erratic outcomes might, in the context translation [82] s... |

268 | The logic of quantum mechanics - Birkhoff, Neumann - 1936 |

223 |
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(Show Context)
Citation Context ...rvables; i.e., within Boolean subalgebras, which will be identified with blocks and contexts of Hilbert lattices. Nevertheless, some of their theorems formally take into account ensembles of contexts =-=[9]-=- for which a multitude of incompatible observables contribute. If theoretical physics is assumed to be a faithful representation of our experience, such an “empirical,” “operational” [37–39] logic der... |

156 |
Measures on the closed subspaces of a Hilbert space
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Citation Context ...nse, Gleason’s theorem can be understood as the functional analytic generalization of the generation of all classical probability distributions by a convex sum of the extreme cases. Gleason’s theorem =-=[55, 62, 63, 64, 65, 66]-=- is a derivation of the Born rule from fundamental assumptions about quantum probabilities, guided by the quasi–classical; i.e., Boolean, sub-parts of quantum theory. Essentially, the main assumption ... |

146 |
On the problem of hidden variables in quantum mechanics
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(Show Context)
Citation Context ...ed, among them contextuality, as well as the abandonment of classical omniscience and realism discussed below. 1. Contextuality Contextuality abandons the context independence of measurement outcomes =-=[1, 3, 4]-=- by supposing that it is wrong to assume (cf. Ref. [1], Sec. 5) that the result of an observation is independent of what observables are measured alongside of it. Bell [1, Sec. 5] states that the “...... |

106 | Speakable and unspeakable in quantum mechanics (Cambridge - Bell - 1993 |

98 |
Finite-Dimensional Vector spaces
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(Show Context)
Citation Context ...tor C, such that all commuting, compatible co-measurable observables are functions thereof. (e.g., Ref. [7], Sec. II.10, p. 90, English translation p. 173; Ref. [9], § 2; Ref. [10], pp. 227,228; Ref. =-=[11]-=-, § 84). Note that mutually commuting opators have identical pairwise orthogonal sets of eigenvectors (forming an orthonormal basis) which correspond to pairwise orthogonal projectors adding up to uni... |

91 |
Experimental Realization of EinsteinPodolsky-Rosen-Bohm Gedankenexperiment: A New Violation of Bell's Inequalities
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Citation Context ... values, depending on its context.” So far, despite some claims to have measured contextuality, there is no direct experimental evidence. Some experimental findings inspired by Bell-type inequalities =-=[70, 71, 72]-=-, the Kochen-Specker theorem [73, 74] as well as the Greenberger-Horne-Zeilinger theorem [75] measure incompatible contexts one after another; i.e., temporally sequentially, and not simultaneously. He... |

82 |
Discussion with Einstein on epistemological problems in atomic physics
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(Show Context)
Citation Context ...1, Sec. 5] states that the “... result of an observation may reasonably depend not only on the state of the system ... but also on the complete disposition of the apparatus.” Note also Bohr’s remarks =-=[2]-=- about “the impossibility of any sharp separation between the behavior of atomic objects and the interaction with the measuring instruments which serve to define the conditions under which the phenome... |

78 | Quantum Logic
- Svozil
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(Show Context)
Citation Context ...omething is true because it is useful might also be applicable (for instance in acrimonious divorces), though less formalized. Some of the material presented here has already been published elsewhere =-=[12]-=-, in particular the partition logic part [13], or the section on quantum probabilities [14]. Here we emphasize the importance of the notion of context, which may serve as a unifying principle for all ... |

77 |
Bellos theorem: experimental tests and implications
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(Show Context)
Citation Context ...e experience which represent the bounds on classical probabilities enumerated in Table III. For historical reasons, the bounds 17-18, 19-20, 21-22, and 23-24 are called the Clauser-Horne inequalities =-=[27, 28]-=-. They are equivalent (up to permutations of pi,qi), and are the only additional inequalities structurally different from the two-event “1–1” case. 9full facet inequality inequality for p1 = p2 = q1 ... |

69 |
Orthomodular structures as quantum logics
- Pták, Pulmannová
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Citation Context ...ubsets of Hilbert logics; and (iii) the alleged randomness of certain single outcomes. A. Hilbert lattices as quantum logics Quantum logic has been introduced by Garrett Birkhoff and John von Neumann =-=[7, 31, 32, 33, 34]-=- in the thirties. They organized it top-down, starting from the Hilbert space formalism of quantum mechanics. Certain entities of Hilbert spaces are identified with propositions, partial order relatio... |

58 |
The logic of modern physics
- Bridgman
- 1927
(Show Context)
Citation Context ...s of contexts [9] for which a multitude of incompatible observables contribute. If theoretical physics is assumed to be a faithful representation of our experience, such an “empirical,” “operational” =-=[37, 38, 39]-=- logic derives its justification by the phenomena themselves. In this sense, one of the main justifications for quantum logic is the construction of the logical and algebraic order of events based on ... |

57 |
Experimental Tests of Realistic Local Theories via Bell's Theorem
- Aspect, Grangier, et al.
- 1981
(Show Context)
Citation Context ... values, depending on its context.” So far, despite some claims to have measured contextuality, there is no direct experimental evidence. Some experimental findings inspired by Bell-type inequalities =-=[70, 71, 72]-=-, the Kochen-Specker theorem [73, 74] as well as the Greenberger-Horne-Zeilinger theorem [75] measure incompatible contexts one after another; i.e., temporally sequentially, and not simultaneously. He... |

52 |
Violation of Bell’s inequality under strict Einstein locality conditions,” Phys
- Weihs, Jennewein, et al.
- 1998
(Show Context)
Citation Context ... values, depending on its context.” So far, despite some claims to have measured contextuality, there is no direct experimental evidence. Some experimental findings inspired by Bell-type inequalities =-=[70, 71, 72]-=-, the Kochen-Specker theorem [73, 74] as well as the Greenberger-Horne-Zeilinger theorem [75] measure incompatible contexts one after another; i.e., temporally sequentially, and not simultaneously. He... |

48 |
A Treatise Concerning the Principles of Human Knowledge
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(Show Context)
Citation Context ... have been measured, but actually have not been measured: in this view, it is wrong to assume that [86] “entities sometimes exist without being experienced by any finite mind.” Indeed, Bekeley states =-=[87]-=-, “For as to what is said of the absolute existence of unthinking things without any relation to their being perceived, that seems perfectly unintelligible. Their esse [[to be]] is percepi [[to be per... |

40 | The logico-algebraic approach to quantum mechanics - Hooker |

39 |
A foundational principle for quantum mechanics
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- 1999
(Show Context)
Citation Context ...nformation codable into a quantized system is given by a single block or context. If the block contains n atoms corresponding to n possible measurement outcomes, then the information content is a nit =-=[83, 84, 85]-=-. The information needs not be “located” at a particular particle, as it can be “distributed” over a multi–partite state. In this sense, the quantum system could be viewed as a kind of (possibly nonlo... |

36 |
Experimental Consequences of Objective Local Theories
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(Show Context)
Citation Context ...e experience which represent the bounds on classical probabilities enumerated in Table III. For historical reasons, the bounds 17-18, 19-20, 21-22, and 23-24 are called the Clauser-Horne inequalities =-=[27, 28]-=-. They are equivalent (up to permutations of pi,qi), and are the only additional inequalities structurally different from the two-event “1–1” case. 9full facet inequality inequality for p1 = p2 = q1 ... |

36 |
Die Logik nicht gleichzeitig entscheidbarer Aussagen
- Specker
- 1960
(Show Context)
Citation Context ... 19C. Probability theory 1. Kochen-Specker theorem Quantum logics of Hilbert space dimension greater than two have not a single two-valued state interpretable as consistent, overall truth assignment =-=[44]-=-. This is the gist of the beautiful construction of Kochen and Specker [9]. For similar theorems, see Refs. [45–49]. As a result of the nonexistence of two-valued states, the classical strategy to con... |

34 |
Two simple proofs of the Kochen-Specker theorem
- Peres
- 1991
(Show Context)
Citation Context ... of all two-valued states fails entirely. One of the most compact and comprehensive versions of the Kochen-Specker proof by contradiction in three-dimensional Hilbert space R3 has been given by Peres =-=[50]-=-. (For other discussions, see Refs. [4, 14, 50–57].) Peres’ version uses a 33-element set of lines without a two-valued state. The direction vectors of these lines arise by all permutations of coordin... |

31 |
Foundations of Quantum Mechanics (Addison-Wesley
- Jauch
- 1968
(Show Context)
Citation Context ...ubsets of Hilbert logics; and (iii) the alleged randomness of certain single outcomes. A. Hilbert lattices as quantum logics Quantum logic has been introduced by Garrett Birkhoff and John von Neumann =-=[7, 31, 32, 33, 34]-=- in the thirties. They organized it top-down, starting from the Hilbert space formalism of quantum mechanics. Certain entities of Hilbert spaces are identified with propositions, partial order relatio... |

31 |
Information and Randomness:An Algorithmic perspective, 2 nd Ed.; Springer-Verlag
- Calude
- 2002
(Show Context)
Citation Context ...on and measurement is the basis of quantum random number generators [76] which serve as a kind of “quantum random oracle” [77, 78]. It should be kept in mind that randomness, at least algorithmically =-=[79, 80, 81]-=-, does not come “for free,” thus exhibiting an amazing capacity of single quanta to support random outcomes. Alternatively, the unpredictable, erratic outcomes might, in the context translation [82] s... |

29 |
Hidden variables and the two theorems of John Bell
- Mermin
- 1993
(Show Context)
Citation Context .... One of the most compact and comprehensive versions of the Kochen-Specker proof by contradiction in three-dimensional Hilbert space R3 has been given by Peres [50]. (For other discussions, see Refs. =-=[4, 14, 50, 51, 52, 53, 54, 55, 56, 57]-=-.) Peres’ version uses a 33-element set of lines without a two-valued state. The direction vectors of these lines arise by all permutations of coordinates from (0,0,1), (0,±1,1), (0,±1, √ 2), and (±1,... |

27 |
Correlation polytopes: their geometry and complexity
- Pitowsky
- 1991
(Show Context)
Citation Context ...ruth, respectively. If some events are independent, then their joint probability pq··· can be expressed as the product of their individual probabilities p, q, .... The associated correlation polytope =-=[16, 17, 18, 19, 20]-=- (see also Refs. [21, 22, 23]) is spanned by a convex combination of vertices, which are vectors of the form (p,q,..., pq,...), where the components are the individual probabilities of independent eve... |

27 | Bell-KochenSpecker theorem: A proof with 18 vectors
- Cabello, Estebaranz, et al.
- 1996
(Show Context)
Citation Context ... the orthogonality with (100). Now we can repeat the argument for case 1 in its mirrored form. The most compact way of deriving the Kochen-Specker theorem in four dimensions has been given by Cabello =-=[60, 61]-=-. It is depicted in Fig. 7. 2. Gleason’s derivation of the Born rule In view of the nonexistence of classical two-valued states on even finite superstructures of blocks or contexts associated with qua... |

26 |
On the theory of probabilities
- Boole
(Show Context)
Citation Context ...inctions. The associated inequalities must be obeyed by all classical probability distributions; they are bounds on classical (joint) probabilities termed “conditions of possible experience” by Boole =-=[15, 24]-=-. 1. Two-event “1–1” case Let us demonstrate the bounds on classical probabilities by the simplest nontrivial example of two propositions; e.g., E ≡“a particle detector aligned along direction a click... |

25 |
A physicist’s second reaction to Mengenlehre
- Bridgman
- 1934
(Show Context)
Citation Context ...s of contexts [9] for which a multitude of incompatible observables contribute. If theoretical physics is assumed to be a faithful representation of our experience, such an “empirical,” “operational” =-=[37, 38, 39]-=- logic derives its justification by the phenomena themselves. In this sense, one of the main justifications for quantum logic is the construction of the logical and algebraic order of events based on ... |

24 |
Greechie diagrams, nonexistence of measures in quantum logics and Kochen–Specker type constructions
- Svozil, Tkadlec
- 1996
(Show Context)
Citation Context ... divorces), though less formalized. Some of the material presented here has already been published elsewhere [12], in particular the partition logic part [13], or the section on quantum probabilities =-=[14]-=-. Here we emphasize the importance of the notion of context, which may serve as a unifying principle for all of the logics discussed. II. CLASSICAL CONTEXTS Logic is an ancient philosophical disciplin... |

24 |
Orthomodular lattices admitting no states
- Greechie
- 1971
(Show Context)
Citation Context ... least element 0; i.e., they lie “just above” 0 in a Hasse diagram of the partial order. A much more compact representation of the propositional calculus can be given in terms of its Greechie diagram =-=[40]-=-. In this representation, the emphasis is on Boolean subalgebras. Points “ ◦ ” represent the atoms. If they belong to the same Boolean subalgebra, they are connected by edges or smooth curves. The col... |

24 |
Generalized urn models
- Wright
- 1990
(Show Context)
Citation Context ...alued states to allow embeddings into Boolean algebras. 27A. Partition logic The empirical logics (i.e., the propositional calculi) associated with the generalized urn models suggested by Ron Wright =-=[88, 89]-=-, and automaton logics (APL) [12, 90–93] are equivalent (cf. Refs. [12, p.145] and [13]) and can be subsumed by partition logics. The logical equivalence of automaton models with generalized urn model... |

22 |
Quantum Probability—Quantum Logic
- Pitowsky
- 1989
(Show Context)
Citation Context ...ruth, respectively. If some events are independent, then their joint probability pq··· can be expressed as the product of their individual probabilities p, q, .... The associated correlation polytope =-=[16, 17, 18, 19, 20]-=- (see also Refs. [21, 22, 23]) is spanned by a convex combination of vertices, which are vectors of the form (p,q,..., pq,...), where the components are the individual probabilities of independent eve... |

19 |
Optimal tests of quantum nonlocality
- Pitowsky, Svozil
- 2001
(Show Context)
Citation Context ...panned by the vertices corresponding to the rows enumerated in Table II. The bounds 17-18, 19-20, 21-22, and 23-24 are the Clauser-Horne inequalities. 103. Six event “3–3” case A similar calculation =-=[20]-=- for six events E1,E2,E3,F1,F2,F3 depicted in Fig. 1 yields an additional independent [29, 30] inequality for their probabilities p1, p2, p3,q1,q2,q3 and their joint probabilities of the type p1q1 + p... |

18 |
Realism: A Prolegomenon to the Philosophy of Quantum Mechanics
- Redhead, Incompleteness
- 1987
(Show Context)
Citation Context ...ed, among them contextuality, as well as the abandonment of classical omniscience and realism discussed below. 1. Contextuality Contextuality abandons the context independence of measurement outcomes =-=[1, 3, 4]-=- by supposing that it is wrong to assume (cf. Ref. [1], Sec. 5) that the result of an observation is independent of what observables are measured alongside of it. Bell [1, Sec. 5] states that the “...... |

18 |
Specker: ”Logical Structures Arising in Quantum Theory
- Kochen, P
- 1965
(Show Context)
Citation Context ...mplication relation and operations such as “and,” “or,” and the negation. Thereby, as we shall see, the resulting logical structures are “nonclassical,” in particular ”nonboolean.” Kochen and Specker =-=[35, 36]-=- suggested to consider only relations and operations among compatible, co-measurable observables; i.e., within Boolean subalgebras, which will be identified with blocks and contexts of Hilbert lattice... |

18 |
Experimental test of quantum nonlocality in three-photon Greenberger-Horne-Zeilinger entanglement
- Pan, Bouwmeester, et al.
(Show Context)
Citation Context ... there is no direct experimental evidence. Some experimental findings inspired by Bell-type inequalities [70–72], the KochenSpecker theorem [73, 74] as well as the Greenberger-Horne-Zeilinger theorem =-=[75]-=- measure incompatible contexts one after another; i.e., temporally sequentially, and not simultaneously. Hence, different contexts can only be measured on different particles. A more direct test of co... |

18 |
Haunted” measurements in quantum theory
- Greenberger, YaSin
- 1989
(Show Context)
Citation Context ... mechanics even more if reversible systems are considered; where an experiment can be “undone” only by investing all the information gained from previous experiments (without being able to copy these)=-=[97, 98]-=-. All incompatible blocks or contexts are pasted together to form the partition logic. These pasting still allow a sufficient number of two-valued states for the construction of probabilities based up... |

17 |
Boole’s ‘conditions od possible experience’ and the quantum puzzle
- George
- 1994
(Show Context)
Citation Context ...ruth, respectively. If some events are independent, then their joint probability pq··· can be expressed as the product of their individual probabilities p, q, .... The associated correlation polytope =-=[16, 17, 18, 19, 20]-=- (see also Refs. [21, 22, 23]) is spanned by a convex combination of vertices, which are vectors of the form (p,q,..., pq,...), where the components are the individual probabilities of independent eve... |

17 | Selecta (Birkhäuser - Specker - 1990 |

16 |
The physical Church-Turing thesis and physical computational complexity
- Pitowsky
- 1990
(Show Context)
Citation Context ...et of vertices by solving the so called hull problem. The inequalities coincide with Boole’s “conditions of possible experience.” The hull problem is algorithmically solvable but computationally hard =-=[26]-=-. In the above example, the “conditions of possible experience” are given by the inequalities enumerated in Table Ib). One of their consequences are bounds on joint occurrences of events. Suppose, for... |

16 |
The calculus of partial propositional functions
- Kochen, Specker
- 1965
(Show Context)
Citation Context ...mplication relation and operations such as “and,” “or,” and the negation. Thereby, as we shall see, the resulting logical structures are “nonclassical,” in particular ”nonboolean.” Kochen and Specker =-=[35, 36]-=- suggested to consider only relations and operations among compatible, co-measurable observables; i.e., within Boolean subalgebras, which will be identified with blocks and contexts of Hilbert lattice... |

16 |
Gleason’s Theorem and Its Applications
- Dvurečenskij
- 1992
(Show Context)
Citation Context ...nse, Gleason’s theorem can be understood as the functional analytic generalization of the generation of all classical probability distributions by a convex sum of the extreme cases. Gleason’s theorem =-=[55, 62, 63, 64, 65, 66]-=- is a derivation of the Born rule from fundamental assumptions about quantum probabilities, guided by the quasi–classical; i.e., Boolean, sub-parts of quantum theory. Essentially, the main assumption ... |

15 |
Boolean embeddings of orthomodular sets and quantum logic
- Zierler, Schlessinger
- 1965
(Show Context)
Citation Context ...ave not a single two-valued state interpretable as consistent, overall truth assignment [44]. This is the gist of the beautiful construction of Kochen and Specker [9]. For similar theorems, see Refs. =-=[45, 46, 47, 48, 49]-=-. As a result of the nonexistence of two-valued states, the classical strategy to construct probabilities by a convex combination of all two-valued states fails entirely. One of the most compact and c... |

15 |
Complementarity and the quantum eraser
- Herzog, Kwiat, et al.
- 1995
(Show Context)
Citation Context ... mechanics even more if reversible systems are considered; where an experiment can be “undone” only by investing all the information gained from previous experiments (without being able to copy these)=-=[97, 98]-=-. All incompatible blocks or contexts are pasted together to form the partition logic. These pasting still allow a sufficient number of two-valued states for the construction of probabilities based up... |

14 |
An Elementay Proof of Gleason’s Theorem
- Cooke, M, et al.
- 1984
(Show Context)
Citation Context ...nse, Gleason’s theorem can be understood as the functional analytic generalization of the generation of all classical probability distributions by a convex sum of the extreme cases. Gleason’s theorem =-=[55, 62, 63, 64, 65, 66]-=- is a derivation of the Born rule from fundamental assumptions about quantum probabilities, guided by the quasi–classical; i.e., Boolean, sub-parts of quantum theory. Essentially, the main assumption ... |

14 |
The state of the pentagon. A nonclassical example
- Wright
- 1978
(Show Context)
Citation Context ...alued states to allow embeddings into Boolean algebras. 27A. Partition logic The empirical logics (i.e., the propositional calculi) associated with the generalized urn models suggested by Ron Wright =-=[88, 89]-=-, and automaton logics (APL) [12, 90–93] are equivalent (cf. Refs. [12, p.145] and [13]) and can be subsumed by partition logics. The logical equivalence of automaton models with generalized urn model... |