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94
Automaton Logic
 International Journal of Theoretical Physics
, 1996
"... The experimental logic of Moore and Mealy type automata is investigated. key words: automaton logic; partition logic; comparison to quantum logic; intrinsic measurements 1 ..."
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Cited by 79 (47 self)
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The experimental logic of Moore and Mealy type automata is investigated. key words: automaton logic; partition logic; comparison to quantum logic; intrinsic measurements 1
A Partial Order on Classical and Quantum States
, 2002
"... We introduce a partial order on classical and quantum states which reveals that these sets are actually domains: Directed complete partially ordered sets with an intrinsic notion of approximation. The operational significance of the orders involved conclusively establishes that physical information ..."
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Cited by 19 (6 self)
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We introduce a partial order on classical and quantum states which reveals that these sets are actually domains: Directed complete partially ordered sets with an intrinsic notion of approximation. The operational significance of the orders involved conclusively establishes that physical information has a natural domain theoretic structure. In the same
NonOrthomodular Models for Both Standard Quantum Logic and Standard Classical Logic: Repercussions for Quantum Computers
 Helv. Phys. Acta
, 1999
"... Abstract. It is shown that propositional calculuses of both quantum and classical logics are noncategorical. We find that quantum logic is in addition to an orthomodular lattice also modeled by a weakly orthomodular lattice and that classical logic is in addition to a Boolean algebra also modeled by ..."
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Cited by 17 (12 self)
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Abstract. It is shown that propositional calculuses of both quantum and classical logics are noncategorical. We find that quantum logic is in addition to an orthomodular lattice also modeled by a weakly orthomodular lattice and that classical logic is in addition to a Boolean algebra also modeled by a weakly distributive lattice. Both new models turn out to be nonorthomodular. We prove the soundness and completeness of the calculuses for the models. We also prove that all the operations in an orthomodular lattice are fivefold defined. In the end we discuss possible repercussions of our results to quantum computations and quantum computers.
Why John von Neumann did not Like the Hilbert Space Formalism of Quantum Mechanics (and What he Liked Instead)
"... this paper is to give an answer to the above question. More precisely, we only aim at clarifying one aspect of von Neumann's dissatisfaction with the Hilbert space formalism: the one which is related to the contrast between the standard Hilbert space formalism and the `case II 1 ' structure { and th ..."
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Cited by 13 (1 self)
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this paper is to give an answer to the above question. More precisely, we only aim at clarifying one aspect of von Neumann's dissatisfaction with the Hilbert space formalism: the one which is related to the contrast between the standard Hilbert space formalism and the `case II 1 ' structure { and the analysis is not complete even in this direction (see the end of the nal Section for a few more advantages von Neumann saw in the II 1 structure). Other issues related to von Neumann's criticism of the standard formalism of quantum mechanics { for instance the question of why von Neumann considered it necessary to replace the axioms of the Hilbert space formalism by the axioms of Jordan algebras [24] { are entirely beyond the scope of the present investigation.
Density conditions for quantum propositions
 Journal of Mathematical Physics
, 1996
"... be formulated in terms of projective geometry. In threedimensional Hilbert space, elementary logical propositions are associated with onedimensional subspaces, corresponding to points of the projective plane. It is shown that, starting with three such propositions corresponding to some basis {⃗u, ..."
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Cited by 12 (11 self)
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be formulated in terms of projective geometry. In threedimensional Hilbert space, elementary logical propositions are associated with onedimensional subspaces, corresponding to points of the projective plane. It is shown that, starting with three such propositions corresponding to some basis {⃗u, ⃗v, ⃗w}, successive application of the binary logical operation (x, y) ↦ → (x ∨ y) ⊥ generates a set of elementary propositions which is countable infinite and dense in the projective plane if and only if no vector of the basis {⃗u, ⃗v, ⃗w} is orthogonal to the other ones. PACS numbers: 02.10.Ab,02.40.k
Quantum logic in dagger kernel categories
 Order
"... This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples of interest, such as categories of relations, partial inject ..."
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Cited by 9 (9 self)
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This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples of interest, such as categories of relations, partial injections, Hilbert spaces (also modulo phase), and Boolean algebras, and (2) have interesting categorical/logical/ordertheoretic properties, in terms of kernel fibrations, such as existence of pullbacks, factorisation, orthomodularity, atomicity and completeness. For instance, the Sasaki hook and andthen connectives are obtained, as adjoints, via the existentialpullback adjunction between fibres. 1
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 8 (3 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
Poisson spaces with a transition probability
 Reviews of Mathematical Physics
, 1997
"... The common structure of the space of pure states P of a classical or a quantum mechanical system is that of a Poisson space with a transition probability. This is a topological space equipped with a Poisson structure, as well as with a function p: P × P → [0, 1], with certain properties. The Poisson ..."
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Cited by 8 (1 self)
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The common structure of the space of pure states P of a classical or a quantum mechanical system is that of a Poisson space with a transition probability. This is a topological space equipped with a Poisson structure, as well as with a function p: P × P → [0, 1], with certain properties. The Poisson structure is connected with the transition probabilities through unitarity (in a specific formulation intrinsic to the given context). In classical mechanics, where p(ρ,σ) = δρσ, unitarity poses no restriction on the Poisson structure. Quantum mechanics is characterized by a specific (complex Hilbert space) form of p, and by the property that the irreducible components of P as a transition probability space coincide with the symplectic leaves of P as a Poisson space. In conjunction, these stipulations determine the Poisson structure of quantum mechanics up to a multiplicative constant (identified with Planck’s constant).