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140
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 82 (46 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
ORTHOMODULARITY IN INFINITE DIMENSIONS; A THEOREM Of M. Solèr
, 1995
"... Maria Pia Solèr has recently proved that an orthomodular form that has an infinite orthonormal sequence is real, complex, or quaternionic Hilbert space. This paper provides an exposition of her result, and describes its consequences for Baer ∗rings, infinitedimensional projective geometries, ort ..."
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Cited by 24 (0 self)
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Maria Pia Solèr has recently proved that an orthomodular form that has an infinite orthonormal sequence is real, complex, or quaternionic Hilbert space. This paper provides an exposition of her result, and describes its consequences for Baer ∗rings, infinitedimensional projective geometries, orthomodular lattices, and Mackey’s quantum logic.
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 22 (7 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 18 (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 ' structur ..."
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Cited by 17 (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.
Decompositions in quantum logic
 Transactions of the AMS
, 1996
"... In 1996, Harding showed that the binary decompositions of any algebraic, relational, or topological structure X form an orthomodular poset Fact X. Here, we begin an investigation of the structural properties of such orthomodular posets of decompositions. We show that a finite set S of binary decompo ..."
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Cited by 12 (6 self)
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In 1996, Harding showed that the binary decompositions of any algebraic, relational, or topological structure X form an orthomodular poset Fact X. Here, we begin an investigation of the structural properties of such orthomodular posets of decompositions. We show that a finite set S of binary decompositions in Fact X is compatible if and only if all the binary decompositions in S can be built from a common nary decomposition of X. This characterization of compatibility is used to show that for any algebraic, relational, or topological structure X, the orthomodular poset Fact X is regular. Special cases of this result include the known facts that the orthomodular posets of splitting subspaces of an inner product space are regular, and that the orthomodular posets constructed from the idempotents of a ring are regular. This result also establishes the regularity of the orthomodular posets that Mushtari constructs from bounded modular lattices, the orthomodular posets one constructs from the subgroups of a group, and the orthomodular posets one constructs from a normed group with operators. Moreover, all these orthomodular posets are regular for the same reason. The characterization of compatibility is also used to show that for any structure X, the finite Boolean subalgebras of Fact X correspond to finitary direct product decompositions of the structure X. For algebraic and relational structures X, this result is extended to show that the Boolean subalgebras of Fact X correspond to representations of the structure X as the global sections of a sheaf of structures over a Boolean space. The above results can be given a physical interpretation as well. Assume that the true or false questions 4 of a quantum mechanical system correspond to binary direct product decompositions of the state space of the system, as is the case with the usual von Neumann interpretation of quantum mechanics. Suppose S is a subset of 4. Then a necessary and sufficient condition that all questions in S can be answered simultaneously is that any two questions in S can be answered simultaneously. Thus, regularity in quantum mechanics follows from the assumption that questions correspond to decompositions. 1.
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
OPERATIONAL QUANTUM LOGIC: AN OVERVIEW
, 2000
"... The term quantum logic has different connotations for different people, having been considered as everything from a metaphysical attack on classical reasoning to an exercise in abstract algebra. Our aim here is to give a uniform presentation of what we call operational quantum logic, highlighting bo ..."
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Cited by 12 (5 self)
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The term quantum logic has different connotations for different people, having been considered as everything from a metaphysical attack on classical reasoning to an exercise in abstract algebra. Our aim here is to give a uniform presentation of what we call operational quantum logic, highlighting both its concrete physical origins and its purely mathematical structure. To orient readers new to this subject, we shall recount some of the historical development of quantum logic, attempting to show how the physical and mathematical sides of the subject have influenced and enriched one another.
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 11 (6 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