Operational Galois Adjunctions

Coecke, Bob; Moore, David

doi:10.1007/978-94-017-1201-9_8

Cited by 8 publications

(7 citation statements)

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“…Since these are arbitrary meets, it also follows that the lattice has arbitrary joins (see e.g. [54]).…”

Section: Some (Idiosyncratic) Lessons From the Pastmentioning

confidence: 99%

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A Universe of Processes and Some of Its Guises

Coecke

2011

Deep Beauty

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Our starting point is a particular 'canvas' aimed to 'draw' theories of physics, which has symmetric monoidal categories as its mathematical backbone. In this paper we consider the conceptual foundations for this canvas, and how these can then be converted into mathematical structure.With very little structural effort (i.e. in very abstract terms) and in a very short time span the categorical quantum mechanics (CQM) research program, initiated by Abramsky and the author in [6], has reproduced a surprisingly large fragment of quantum theory [45,171,180,61,57,62,49,3,63]. It also provides new insights both in quantum foundations and in quantum information, for example in [59,60,50,51,80,65,52,81], and has even resulted in automated reasoning software called quantomatic [76,77,75] which exploits the deductive power of CQM, which is indeed a categorical quantum logic [78].In this paper we complement the available material by not requiring prior knowledge of category theory, and by pointing at connections to previous and current developments in the foundations of physics.This research program is also in close synergy with developments elsewhere, for example in representation theory [73], quantum algebra [177], knot theory [188], topological quantum field theory [133] and several other areas. 1 arXiv:1009.3786v1 [quant-ph]

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“…Since these are arbitrary meets, it also follows that the lattice has arbitrary joins (see e.g. [54]).…”

Section: Some (Idiosyncratic) Lessons From the Pastmentioning

confidence: 99%

“…is that map which assigns to each property after the measurement one before the measurement. 5 A survey in the light of the Geneva School approach is in [54] 6 More details on this are in [153].…”

Section: Measurement Among Other Processesmentioning

confidence: 99%

A Universe of Processes and Some of Its Guises

Coecke

2011

Deep Beauty

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“…The conjunction defined above provides L with a complete lattice structure, where, as already mentioned, the corresponding joins have not necessarily a disjunctive significance. 16 In particular can there be properties b ∈ L that do not imply that some a ∈ A is actual but that do imply the join A ∈ L to be actual, the so-called superposition principle of quantum 13 Reality is obviously in no way to be understood as synonym for 'locality and non-contextuality' as it is sometimes the case in some (from a philosophical perspective) slightly naive discourses on philosophy of physics.…”

Section: What Quantum Logicality Can Be Aboutmentioning

confidence: 99%

“…The conjunction defined above provides L with a complete lattice structure, where, as already mentioned, the corresponding joins have not necessarily a disjunctive significance. 16 In particular can there be properties b ∈ L that do not imply that some a ∈ A is actual but that do imply the join A ∈ L to be actual, the so-called superposition principle of quantum theory -see Aerts (1981) and Coecke (2001a) for a rigorous discussion on this matter. Let us stress that at this point, as argued in Coecke, Moore and Smets (2001a), the full physically derivable logical content that emerges from our operational setting consists of a consequence relation 17 ⊢ ⊆ P(L) × L , that extends the lattice ordering ≤ ⊆ L × L exploiting conjunctivity, i.e.…”

Section: What Quantum Logicality Can Be Aboutmentioning

confidence: 99%

The Sasaki Hook Is Not a [Static] Implicative Connective but Induces a Backward [in Time] Dynamic One That Assigns Causes

Coecke

Smets²

2004

International Journal of Theoretical Physics

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In this paper we argue that the Sasaki adjunction, which formally encodes the logicality that different authors tried to attach to the Sasaki hook as a 'quantum implicative connective', has a fundamental dynamic nature and encodes the so-called 'causal duality' (Coecke, Moore and Stubbe 2001) for the particular case of a quantum measurement with a projector as corresponding self-adjoint operator. In particular: The action of the Sasaki hook (a S → −) for fixed antecedent a assigns to some property "the weakest cause before the measurement of actuality of that property after the measurement", i.e. (a S → b) is the weakest property that guarantees actuality of b after performing the measurement represented by the projector that has the 'subspace a' as eigenstates for eigenvalue 1 , say, the measurement that 'tests' a . From this we conclude that the logicality attributable to quantum systems contains a fundamentally dynamic ingredient: Causal duality actually provides a new dynamic interpretation of orthomodularity. We also reconsider the status of the Sasaki hook within 'dynamic (operational) quantum logic' (DOQL), what leads us to the claim made in the title of this paper. More explicitly, although (as many argued in the past) the Sasaki hook should not be seen as an implicative hook, the formal motivation that persuaded others to do so, i.e. the Sasaki adjunction, does have a physical significance (in terms of causal duality). It is within the context of DOQL that we can then derive that the labeled dynamic hooks (forwardly and backwardly) that encode quantum measurements act on properties as (a 1, taking values in the 'disjunctive extension' DI(L) of the property lattice L (Coecke 2001a) , where a ∈ L is the tested property and (− → L −) is the Heyting implication that lives on DI(L) . Since these hooks (− ϕa → −) and (− ϕa ← −) extend to DI(L) × DI(L) they constitute internal operations . In an even more radical perspective one could say that the transition from either classical or constructive/intuitionistic logic to quantum logic entails besides the introduction of an additional unary connective 'operational resolution' (Coecke 2001a) the shift from a binary connective implication to a ternary connective where two of the arguments refer to qualities of the system and the third, the new one, to an obtained outcome (in a measurement). QUANTUM LOGICALITYWe claim that logical considerations on quantum behavior and as such, further development of the research field, have been 'corrupted' by two features. Once these two features are neutralized, the way towards an essentially dynamic quantum logic (i.e. a unified logic of 'changes' both for classical and quantum systems), or otherwise put, a true quantum process semantics, is opened. Moreover, the solution to the second 'corrupt feature' indicates that the logicality encoded in pure quantum theory is of a fundamental dynamic nature. Structures somewhat similar to those emerging in the context of categorical grammar (Lambek 1958), linear logic (Girard 1987...

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“…For sake of completeness we mention other work that has contributed to advances in the approach that is less directly of importance for what we do in this paper, but shows how the approach is developing into other directions as well [52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81].…”

Section: How We Will Proceedmentioning

confidence: 99%

Being and Change: Foundations of a Realistic Operational Formalism

Aerts¹

2002

Probing the Structure of Quantum Mechanics

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The aim of this article is to represent the general description of an entity by means of its states, contexts and properties. The entity that we want to describe does not necessarily have to be a physical entity, but can also be an entity of a more abstract nature, for example a concept, or a cultural artifact, or the mind of a person, etc..., which means that we aim at very general description. The effect that a context has on the state of the entity plays a fundamental role, which means that our approach is intrinsically contextual. The approach is inspired by the mathematical formalisms that have been developed in axiomatic quantum mechanics, where a specific type of quantum contextuality is modelled. However, because in general states also influence contextwhich is not the case in quantum mechanics -we need a more general setting than the one used there. Our focus on context as a fundamental concept makes it possible to unify 'dynamical change' and 'change under influence of measurement', which makes our approach also more general and more powerful than the traditional quantum axiomatic approaches. For this reason an experiment (or measurement) is introduced as a specific kind of context. Mathematically we introduce a state context property system as the structure to describe an entity by means of its states, contexts and properties. We also strive from the start to a categorical setting and derive the morphisms between * Published as: D. Aerts, "Being and change: foundations of a realistic operational formalism", in Probing the Structure of Quantum Mechanics: Nonlinearity, Nonlocality, Computation and Axiomatics, eds. D. Aerts, M. Czachor and T. Durt, World Scientific, Singapore (2002). 1 state context property systems from a merological covariance principle. We introduce the category SCOP with as elements the state context property systems and as morphisms the ones that we derived from this merological covariance principle. We introduce property completeness and state completeness and study the operational foundation of the formalism.

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Operational Galois Adjunctions

Cited by 8 publications

References 30 publications

A Universe of Processes and Some of Its Guises

A Universe of Processes and Some of Its Guises

The Sasaki Hook Is Not a [Static] Implicative Connective but Induces a Backward [in Time] Dynamic One That Assigns Causes

Being and Change: Foundations of a Realistic Operational Formalism

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