Since (as is known) no Hilbert space description is possible for two independent quantum systems (theorem 1), we investigate what kind of complete atomistic ortho-lattice (cao lattice) ${\\cal L}$ is suitable to model the physical properties of the total system. This leads us to impose five conditions P1 to P5 on ${\\cal L}$ . We show that conditions P1 to P4 are physically necessary and prove that if ${\\cal L}$ satisfies conditions P1 to P5, then ${\\cal L}$ is the separated product proposed by Aerts in 1982 to describe independent quantum systems (theorem 2). Finally, we explain why the separated product may not be the right candidate and therefore ask whether there exists a cao-lattice not isomorphic to the separated product that satisfies conditions P1 to P4.Since (as is known) no Hilbert space description is possible for two independent quantum systems (theorem 1), we investigate what kind of complete atomistic ortho-lattice (cao lattice) ${\\cal L}$ is suitable to model the physical properties of the total system. This leads us to impose five conditions P1 to P5 on ${\\cal L}$. We show that conditions P1 to P4 are physically necessary and prove that if ${\\cal L}$ satisfies conditions P1 to P5, then ${\\cal L}$ is the separated product proposed by Aerts in 1982 to describe independent quantum systems (theorem 2). Finally, we explain why the separated product may not be the right candidate and therefore ask whether there exists a cao-lattice not isomorphic to the separated product that satisfies conditions P1 to P4.

Apr 17, 20025 citations

The property lattice of spatially separated quantum systems

Nicolas Gisin

Reports on Mathematical PhysicsJun 1, 19862 citations

Independence of Quantum Systems

Jan Hamhalter

Springer eBooksJan 1, 20030 citations

A Presentation of Quantum Logic Based on an and then Connective

Daniel J. Lehmann

When a physicist performs a quantic measurement, new information about the system at hand is gathered. This article studies the logical properties of how this new information is combined with previous information. It presents Quantum Logic as a propositional logic under two connectives: negation and the and then operation that combines old and new information. The and then connective is neither commutative nor associative. Many properties of this logic are exhibited, and some small elegant subset is shown to imply all the properties considered. No independence or completeness result is claimed. Classical physical systems are exactly characterized by the commutativity, the associativity, or the monotonicity of the and then connective. Entailment is defined in this logic and can be proved to be a partial order. In orthomodular lattices, the operation proposed by Finch in [3] satisfies all the properties studied in this article. All properties satisfied by Finch's; operation in modular lattices are valid in Quantum Logic. It is not known whether all properties of Quantum Logic are satisfied by Finch's; operation in modular lattices. Non-commutative, non-associative algebraic structures generalizing Boolean algebras are defined, ideals are characterized and a homomorphism theorem is proved.When a physicist performs a quantic measurement, new information about the system at hand is gathered. This article studies the logical properties of how this new information is combined with previous information. It presents Quantum Logic as a propositional logic under two connectives: negation and the and then operation that combines old and new information. The and then connective is neither commutative nor associative. Many properties of this logic are exhibited, and some small elegant subset is shown to imply all the properties considered. No independence or completeness result is claimed. Classical physical systems are exactly characterized by the commutativity, the associativity, or the monotonicity of the and then connective. Entailment is defined in this logic and can be proved to be a partial order. In orthomodular lattices, the operation proposed by Finch in [3] satisfies all the properties studied in this article. All properties satisfied by Finch's; operation in modular lattices are valid in Quantum Logic. It is not known whether all properties of Quantum Logic are satisfied by Finch's; operation in modular lattices. Non-commutative, non-associative algebraic structures generalizing Boolean algebras are defined, ideals are characterized and a homomorphism theorem is proved.

arXiv:quant-ph/0701113Journal of Logic and ComputationAug 21, 200716 citations

Intrinsic Properties of Quantum Systems

Petr Hájíček, J. Tolar

arXiv:0806.4437Foundations of PhysicsMar 13, 200911 citations

Linear independence of nearest-neighbor valence bond states in several two-dimensional lattices

Julia Wildeboer, Alexander Seidel

We show for several two-dimensional lattices that the nearest-neighbor valence bond states are linearly independent. To do so, we utilize and generalize a method that was recently introduced and applied to the kagome lattice by one of the authors. This method relies on the choice of an appropriate cell for the respective lattice, for which a certain local linear independence property can be demonstrated. Whenever this is achieved, linear independence follows for arbitrarily large lattices that can be covered by such cells, for both open and periodic boundary conditions. We report that this method is applicable to the kagome, honeycomb, square, squagome, two types of pentagonal, square-octagonal, star, two types of Archimedean, three types of ``martini'', and fullerene-type lattices, e.g., the well known `buckyball.'' Applications of the linear independence property, such as the derivation of effective quantum dimer models or the construction of new solvable spin- $1/2$ models, are discussed.We show for several two-dimensional lattices that the nearest-neighbor valence bond states are linearly independent. To do so, we utilize and generalize a method that was recently introduced and applied to the kagome lattice by one of the authors. This method relies on the choice of an appropriate cell for the respective lattice, for which a certain local linear independence property can be demonstrated. Whenever this is achieved, linear independence follows for arbitrarily large lattices that can be covered by such cells, for both open and periodic boundary conditions. We report that this method is applicable to the kagome, honeycomb, square, squagome, two types of pentagonal, square-octagonal, star, two types of Archimedean, three types of ``martini'', and fullerene-type lattices, e.g., the well known `buckyball.'' Applications of the linear independence property, such as the derivation of effective quantum dimer models or the construction of new solvable spin-$1/2$ models, are discussed.

arXiv:1101.1621Physical Review BMay 26, 201115 citations

Since (as is known) no Hilbert space description is possible for two independent quantum systems (theorem 1), we investigate what kind of complete atomistic ortho-lattice (cao lattice)

{\\cal L}

is suitable to model the physical properties of the total system. This leads us to impose five conditions P1 to P5 on

{\\cal L}

. We show that conditions P1 to P4 are physically necessary and prove that if

{\\cal L}

satisfies conditions P1 to P5, then

{\\cal L}

is the separated product proposed by Aerts in 1982 to describe independent quantum systems (theorem 2). Finally, we explain why the separated product may not be the right candidate and therefore ask whether there exists a cao-lattice not isomorphic to the separated product that satisfies conditions P1 to P4.

1/2

models, are discussed.