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The Sasaki hook is not a [static] implicative connective but induces a backward [in time] dynamic one that assigns causes
 Int. Journ. of Theor. Physics
"... 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 socalled ‘causal duality ’ (Coecke, Moore and Stubbe 2001) for t ..."
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Cited by 13 (8 self)
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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 socalled ‘causal duality ’ (Coecke, Moore and Stubbe 2001) for the particular case of a quantum measurement with a projector as corresponding selfadjoint 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
Quantum informationflow, concretely, abstractly
 PROC. QPL 2004
, 2004
"... These ‘lecture notes ’ are based on joint work with Samson Abramsky. I will survey and informally discuss the results of [3, 4, 5, 12, 13] in a pedestrian not too technical way. These include: • ‘The logic of entanglement’, that is, the identification and abstract axiomatization of the ‘quantum info ..."
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Cited by 11 (5 self)
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These ‘lecture notes ’ are based on joint work with Samson Abramsky. I will survey and informally discuss the results of [3, 4, 5, 12, 13] in a pedestrian not too technical way. These include: • ‘The logic of entanglement’, that is, the identification and abstract axiomatization of the ‘quantum informationflow ’ which enables protocols such as quantum teleportation. 1 To this means we defined strongly compact closed categories which abstractly capture the behavioral properties of quantum entanglement. • ‘Postulates for an abstract quantum formalism ’ in which classical informationflow (e.g. token exchange) is part of the formalism. As an example, we provided a purely formal description of quantum teleportation and proved correctness in abstract generality. 2 In this formalism types reflect kinds, contra the essentially typeless von Neumann formalism [25]. Hence even concretely this formalism manifestly improves on the usual one. • ‘A highlevel approach to quantum informatics’. 3 Indeed, the above discussed work can be conceived as aiming to solve: von Neumann quantum formalism � highlevel language lowlevel language. I also provide a brief discussion on how classical and quantum uncertainty can be mixed in the above formalism (cf. density matrices). 4
Intuitionistic quantum logic of an nlevel system
, 2009
"... A decade ago, Isham and Butterfield proposed a topostheoretic approach to quantum mechanics, which meanwhile has been extended by Döring and Isham so as to provide a new mathematical foundation for all of physics. Last year, three of the present authors redeveloped and refined these ideas by combin ..."
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Cited by 8 (0 self)
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A decade ago, Isham and Butterfield proposed a topostheoretic approach to quantum mechanics, which meanwhile has been extended by Döring and Isham so as to provide a new mathematical foundation for all of physics. Last year, three of the present authors redeveloped and refined these ideas by combining the C*algebraic approach to quantum theory with the socalled internal language of topos theory (see arXiv:0709.4364). The goal of the present paper is to illustrate our abstract setup through the concrete example of the C*algebra Mn(C) of complex n × n matrices. This leads to an explicit expression for the pointfree quantum phase space Σn and the associated logical structure and Gelfand transform of an nlevel system. We also determine the pertinent nonprobabilisitic stateproposition pairing (or valuation) and give a very natural topostheoretic reformulation of the Kochen–Specker Theorem. In our approach, the nondistributive lattice P(Mn(C)) of projections in Mn(C) (which forms the basis of the traditional quantum logic of Birkhoff and von Neumann) is replaced by a specific distributive lattice O(Σn) of functions from the poset C(Mn(C))
Logic of Dynamics & Dynamics of Logic; Some Paradigm Examples
"... The development ofoperationalquantum logicpointsoutthatclassicalbooleanstructuresaretoo rigidtodescribe theactualand potentialpropertiesofquantum systems.Operationalquantum logic bearsupon basicaxiomswhich aremotivatedby empiricalfactsand assuch supportsthedynamicshiftfromclassicaltononclassicallog ..."
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Cited by 6 (5 self)
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The development ofoperationalquantum logicpointsoutthatclassicalbooleanstructuresaretoo rigidtodescribe theactualand potentialpropertiesofquantum systems.Operationalquantum logic bearsupon basicaxiomswhich aremotivatedby empiricalfactsand assuch supportsthedynamicshiftfromclassicaltononclassicallogic resultinginto a dynamicsoflogic.On theotherhand,a dynamic extensionofoperationalquantum logicallows ustoexpressdynamic 1 reasoninginthesensethatwecancapturehow actualpropertiespropagate, includingtheirtemporalcausalstructure.Itisinthissense thatpassingfromstaticoperationalquantum logictodynamicoperationalquantum logicresultsina truelogicofdynamicsthatprovides a unifledlogicaldescriptionofsystemswhich evolve orwhich aresubmittedtomeasurements. Whilefocusingon thequantalesemantics fordynamicoperationalquantum logic,we cananalyzethepointsof difierencewiththeexistingquantalesemanticsfor(non)commutative linearlogic.Linearlogicisheretobeconceivedasa resourcesensitive logiccapableofdealingwi...
References
"... The statical part of quantum logic can be described by projections on a Hilbert space, or more abstractly, by an orthomodular lattice (OML). There are good reasons to consider the Sasaki projection (generalizing the composition of projections) and its adjoint the Sasaki hook as the main structural ..."
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The statical part of quantum logic can be described by projections on a Hilbert space, or more abstractly, by an orthomodular lattice (OML). There are good reasons to consider the Sasaki projection (generalizing the composition of projections) and its adjoint the Sasaki hook as the main structural operations on the OML. In [1] Kröger used the binary variants of the Sasaki operations (called skew meet and skew join) to axiomatize OMLs as skew Boolean lattices. It is natural to seek common generalizations of frames and complete OML, called skew frames (and interpreted either as “linearized frames ” or “intuitionistic OMLs”). The skew frames should also appear as structures on right ideals of C*algebras. Similar structures have already been studied (e.g. [2, 3, 4]). Despite that the axiomatization of skew frames is not finished, we know that every skew frame is at least a left distributive idempotent semiquantale and contains an open subframe of central elements. The openness of the inclusion provides an existence of its left adjoint, interpreted as a central cover. Then, using a machinery of triads [5], we calculate also lattices representing the dynamical part of the system. In the case of OML the resulting structure is a Girard couple of quantales, so we obtain a link to the cyclic linear logic.
QUANTALOIDS DESCRIBING CAUSATION AND PRO PAGATION OF PHYSICAL PROPERTIES
, 2000
"... A general principle of ‘causal duality ’ for physical systems, lying at the base of representation theorems for both compound and evolving systems, is proved; formally it is encoded in a quantaloidal setting. Other particular examples of quantaloids and quantaloidal morphisms appear naturally within ..."
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A general principle of ‘causal duality ’ for physical systems, lying at the base of representation theorems for both compound and evolving systems, is proved; formally it is encoded in a quantaloidal setting. Other particular examples of quantaloids and quantaloidal morphisms appear naturally within this setting; as in the case of causal duality, they originate from primitive physical reasonings on the lattices of properties of physical systems. Furthermore, an essentially dynamical operational foundation for studying physical systems is outlined; complementary as it is to the existing static operational foundation, it leads to the natural axiomatization of ‘causal duality ’ in operational quantum logic. Key words: causal duality, property lattice, galois adjoint, quantaloid. 1.
Bohrification
, 2010
"... The aim of this chapter is to construct new foundations for quantum logic and quantum spaces. This is accomplished by merging algebraic quantum theory and topos theory (encompassing the theory of locales or frames, of which toposes in a sense form the ultimate generalization). In a nutshell, the rel ..."
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The aim of this chapter is to construct new foundations for quantum logic and quantum spaces. This is accomplished by merging algebraic quantum theory and topos theory (encompassing the theory of locales or frames, of which toposes in a sense form the ultimate generalization). In a nutshell, the relation between these fields is as follows. First, our mathematical interpretation of Bohr’s ‘doctrine of classical concepts ’ is that the empirical content of a quantum theory described by a noncommutative (unital) C*algebra A is contained in the family of its commutative (unital) C*algebras, partially ordered by inclusion. Seen as a category, the ensuing poset C(A) canonically defines the topos [C(A), Set] of covariant functors from C(A) to the category Set of sets and functions. This topos contains the ‘Bohrification ’ A of A, defined as the tautological functor C ↦ → C, as an internal commutative C*algebra. Second, according to the toposvalid Gelfand duality theorem of Banaschewski and Mulvey, A has a Gelfand spectrum Σ(A), which is a locale internal to the topos
Bohrification of operator algebras and quantum logic
, 2009
"... Following Birkhoff and von Neumann, quantum logic has traditionally been based on the lattice of closed linear subspaces of some Hilbert space, or, more generally, on the lattice of projections in a von Neumann algebra A. Unfortunately, the logical interpretation of these lattices is impaired by the ..."
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Following Birkhoff and von Neumann, quantum logic has traditionally been based on the lattice of closed linear subspaces of some Hilbert space, or, more generally, on the lattice of projections in a von Neumann algebra A. Unfortunately, the logical interpretation of these lattices is impaired by their nondistributivity and by various other problems. We show that a possible resolution of these difficulties, suggested by the ideas of Bohr, emerges if instead of single projections one considers elementary propositions to be families of projections indexed by a partially ordered set C(A) of appropriate commutative subalgebras of A. In fact, to achieve both maximal generality and ease of use within topos theory, we assume that A is a socalled Rickart C*algebra and that C(A) consists of all unital commutative Rickart C*subalgebras of A. Such families of projections form a Heyting algebra in a natural way, so that the associated propositional logic is intuitionistic: distributivity is recovered at the expense of the law of the excluded middle. Subsequently, generalizing an earlier computation for n×n matrices, we prove that the Heyting algebra thus associated to A arises as a basis for the internal Gelfand spectrum (in the sense of Banaschewski–Mulvey) of the “Bohrification ” A of A, which is a commutative Rickart C*algebra in the topos of functors from C(A) to the category of sets. We explain the relationship of this construction to partial Boolean algebras and Bruns–Lakser completions. Finally, we establish a connection between probability measure on the lattice of projections on a Hilbert space H and probability valuations on the internal Gelfand spectrum of A for A = B(H).