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24
Concrete Domains
 Theoretical Computer Science
, 1993
"... This paper introduces the theory of a particular kind of computation domains called concrete domains. The purpose of this theory is to find a satisfactory framework for the notions of coroutine computation and sequentiality of evaluation. Diagrams are emphasized because I believe that an important ..."
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Cited by 35 (1 self)
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This paper introduces the theory of a particular kind of computation domains called concrete domains. The purpose of this theory is to find a satisfactory framework for the notions of coroutine computation and sequentiality of evaluation. Diagrams are emphasized because I believe that an important part of learning lattice theory is the acquisition of skill in drawing diagrams. George Gratzer 1 Domains of computation In general, we follow Scott's approach [Sco70]. To every syntactic object one associates a semantic object which is found in an appropriate semantic domain. For technical details, we follow [Mil73] and [Plo78] rather than Scott. Definition 1.1 A partial order is a pair ! D; ? where D is a nonempty set and is a binary relation satisfying: i) 8x 2 D x x (reflexivity) ii) 8x; y 2 D x y; y x ) x = y (antisymmetry) iii) 8x; y; z 2 D x y; y z ) x z (transitivity) One writes x ! y when x y and x 6= y. Two elements x and y are comparable when either x y or y x. W...
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 23 (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.
Contextual logic for quantum systems
 Journal of Mathematical Physics
, 2005
"... In this work we build a quantum logic that allows us to refer to physical magnitudes pertaining to different contexts from a fixed one without the contradictions with quantum mechanics expressed in nogo theorems. This logic arises from considering a sheaf over a topological space associated to the ..."
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Cited by 6 (4 self)
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In this work we build a quantum logic that allows us to refer to physical magnitudes pertaining to different contexts from a fixed one without the contradictions with quantum mechanics expressed in nogo theorems. This logic arises from considering a sheaf over a topological space associated to the Boolean sublattices of the ortholattice of closed subspaces of the Hilbert space of the physical system. Differently to standard quantum logics, the contextual logic maintains a distributive lattice structure and a good definition of implication as a residue of the conjunction.
Property lattices for independent quantum systems
 Reports on Mathematical Physics
"... We consider the description of two independent quantum systems by a complete atomistic ortholattice (caolattice) L. It is known that since the two systems are independent, no Hilbert space description is possible, i.e. L ̸ = P(H), the lattice of closed subspaces of a Hilbert space (theorem 1). We ..."
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Cited by 6 (5 self)
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We consider the description of two independent quantum systems by a complete atomistic ortholattice (caolattice) L. It is known that since the two systems are independent, no Hilbert space description is possible, i.e. L ̸ = P(H), the lattice of closed subspaces of a Hilbert space (theorem 1). We impose five conditions on L. Four of them are shown to be physically necessary. The last one relates the orthogonality between states in each system to the orthocomplementation of L. It can be justified if one assumes that the orthogonality between states in the total system induces the orthocomplementation of L. We prove that if L satisfies these five conditions, then L is the separated product proposed by Aerts in 1982 to describe independent quantum systems (theorem 2). Finally, we give strong arguments to exclude the separated product and therefore our last condition. As a consequence, we ask whether among the calattices that satisfy our first four basic necessary conditions, there exists an orthocomplemented one different from the separated product.
2002, Linearity and compound physical systems: the case of two separated spin 1/2 entities
 In D. Aerts, M. Czachor and T. Durt (Eds.), Probing the Structure of Quantum Mechanics: Nonlinearity, Nonlocality, Probability and Axiomatics
"... We illustrate some problems that are related to the existence of an underlying linear structure at the level of the property lattice associated with a physical system, for the particular case of two explicitly separated spin 1/2 objects that are considered, and mathematically described, as one compo ..."
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Cited by 2 (2 self)
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We illustrate some problems that are related to the existence of an underlying linear structure at the level of the property lattice associated with a physical system, for the particular case of two explicitly separated spin 1/2 objects that are considered, and mathematically described, as one compound system. It is shown that the separated product of the property lattices corresponding with the two spin 1/2 objects does not have an underlying linear structure, although the property lattices associated with the subobjects in isolation manifestly do. This is related at a fundamental level to the fact that separated products do not behave well with respect to the covering law (and orthomodularity) of elementary lattice theory. In addition, we discuss the orthogonality relation associated with the separated product in general and consider the related problem of the behavior of the corresponding Sasaki projections as partial state space mappings. 1
Equations, states, and lattices of infinitedimensional Hilbert space
 Int. J. Theor. Phys
, 2000
"... Abstract. We provide several new results on quantum state space, on lattice of subspaces of an infinite dimensional Hilbert space, and on infinite dimensional Hilbert space equations as well as on connections between them. In particular we obtain an nvariable generalized orthoarguesian equation whi ..."
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Cited by 2 (2 self)
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Abstract. We provide several new results on quantum state space, on lattice of subspaces of an infinite dimensional Hilbert space, and on infinite dimensional Hilbert space equations as well as on connections between them. In particular we obtain an nvariable generalized orthoarguesian equation which holds in any infinite dimensional Hilbert space. Then we strengthen Godowski’s result by showing that in an ortholattice on which strong states are defined Godowski’s equations as well as the orthomodularity hold. We also prove that all 6 and 4variable orthoarguesian equations presented in the literature can be reduced to new 4 and 3variable ones, respectively and that Mayet’s examples follow from Godowski’s equations. To make a breakthrough in testing these massive equations we designed several novel algorithms for generating Greechie diagrams with an arbitrary number of blocks and atoms (currently testing with up to 50) and for automated checking of equations on them. A way of obtaining complex infinite dimensional Hilbert space from the Hilbert lattice equipped with several additional conditions and without invoking the notion of state is presented. Possible repercussions of the results to quantum computing problems are discussed.
Scopes and limits of modality in quantum mechanics
, 2008
"... We develop an algebraic frame for the simultaneous treatment of actual and possible properties of quantum systems. We show that, in spite of the fact that the language is enriched with the addition of a modal operator to the orthomodular structure, contextuality remains a central feature of quantum ..."
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Cited by 1 (1 self)
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We develop an algebraic frame for the simultaneous treatment of actual and possible properties of quantum systems. We show that, in spite of the fact that the language is enriched with the addition of a modal operator to the orthomodular structure, contextuality remains a central feature of quantum systems. 1
Is Quantum Logic a Logic
 Handbook of Quantum Logic and Quantum Structures, volume Quantum Logic
, 2008
"... Is a Quantum Logic a Logic? [1] in which they strengthen a previous negative ..."
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Cited by 1 (1 self)
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Is a Quantum Logic a Logic? [1] in which they strengthen a previous negative
The role of the modular pairs in the category of complete orthomodular lattice
 Lett. Math. Phys., this issue. (Received July
, 1978
"... ABSTRACT. We study the modular pairs of a complete orthomodular lattice i.e. a CROC. We propose the concept of mmorphism as a mapping which preserves the lattice structure, the orthogonality and the property to be a modular pair. We give a characterization of the mmorphisms in the case of the comp ..."
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Cited by 1 (1 self)
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ABSTRACT. We study the modular pairs of a complete orthomodular lattice i.e. a CROC. We propose the concept of mmorphism as a mapping which preserves the lattice structure, the orthogonality and the property to be a modular pair. We give a characterization of the mmorphisms in the case of the complex ttilbert space to justify this concept.
Equations and state and lattice properties that hold in infinite dimensional hilbert space
 International J. Theoretical Physics
"... Abstract. We provide several new results on quantum state space, on lattice of subspaces of an infinite dimensional Hilbert space, and on infinite dimensional Hilbert space equations as well as on connections between them. In particular we obtain an nvariable generalized orthoarguesian equation whi ..."
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Cited by 1 (0 self)
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Abstract. We provide several new results on quantum state space, on lattice of subspaces of an infinite dimensional Hilbert space, and on infinite dimensional Hilbert space equations as well as on connections between them. In particular we obtain an nvariable generalized orthoarguesian equation which holds in any infinite dimensional Hilbert space. Then we strengthen Godowski’s result by showing that in an ortholattice on which strong states are defined Godowski’s equations as well as the orthomodularity hold. We also prove that all 6 and 4variable orthoarguesian equations presented in the literature can be reduced to new 4 and 3variable ones, respectively and that Mayet’s examples follow from Godowski’s equations. To make a breakthrough in testing these massive equations we designed several novel algorithms for generating Greechie diagrams with an arbitrary number of blocks and atoms (currently testing with up to 50) and for automated checking of equations on them. A way of obtaining complex infinite dimensional Hilbert space from the Hilbert lattice equipped with several additional conditions and without invoking the notion of state is presented. Possible repercussions of the results to quantum computing problems are discussed.