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 mappingsWe 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

Jan 1, 20024 citations

THE LINEARITY OF QUANTUM MECHANICS AT STAKE: THE DESCRIPTION OF SEPARATED QUANTUM ENTITIES

Diederik Aerts, FRANK VALCKENBORGH

We consider the situation of a physical entity that is the compound entity consisting of two 'separated' quantum entities. In earlier work it has been proven by one of the authors that such a physical entity cannot be described by standard quantum mechanics. More precisely, it was shown that two of the axioms of traditional quantum axiomatics are at the origin of the impossibility for standard quantum mechanics to describe this type of compound entity. One of these axioms is equivalent with the superposition principle, which means that separated quantum entities put the linearity of quantum mechanics at stake. We analyze the conceptual steps that are involved in this proof, and expose the necessary material of quantum axiomatics to be able to understand the argumentWe consider the situation of a physical entity that is the compound entity consisting of two 'separated' quantum entities. In earlier work it has been proven by one of the authors that such a physical entity cannot be described by standard quantum mechanics. More precisely, it was shown that two of the axioms of traditional quantum axiomatics are at the origin of the impossibility for standard quantum mechanics to describe this type of compound entity. One of these axioms is equivalent with the superposition principle, which means that separated quantum entities put the linearity of quantum mechanics at stake. We analyze the conceptual steps that are involved in this proof, and expose the necessary material of quantum axiomatics to be able to understand the argument

arXiv:quant-ph/0205162Jun 1, 20023 citations

Orthocomplementation and compound systems

Boris Ischi

Abstract. In their 1936 founding paper on quantum logic, Birkhoff and von Neumann postulated that the lattice describing the experimental propositions concerning a quantum system is orthocomplemented. We prove that this postulate fails for the lattice Lsep describing a compound system consisting of so called separated quantum systems. By separated we mean two systems prepared in different “rooms ” of the lab, and before any interaction takes place. In that case the state of the compound system is necessarily a product state. As a consequence, Dirac’s superposition principle fails, and therefore Lsep cannot satisfy all Piron’s axioms. In previous works, assuming that Lsep is orthocomplemented, it was argued that Lsep is not orthomodular and fails to have the covering property. Here we prove that Lsep cannot admit and orthocomplementation. Moreover, we propose a natural model for Lsep which has the covering property. PACS numbers: 03.65.Ta, 03.65.Ca 1.Abstract. In their 1936 founding paper on quantum logic, Birkhoff and von Neumann postulated that the lattice describing the experimental propositions concerning a quantum system is orthocomplemented. We prove that this postulate fails for the lattice Lsep describing a compound system consisting of so called separated quantum systems. By separated we mean two systems prepared in different “rooms ” of the lab, and before any interaction takes place. In that case the state of the compound system is necessarily a product state. As a consequence, Dirac’s superposition principle fails, and therefore Lsep cannot satisfy all Piron’s axioms. In previous works, assuming that Lsep is orthocomplemented, it was argued that Lsep is not orthomodular and fails to have the covering property. Here we prove that Lsep cannot admit and orthocomplementation. Moreover, we propose a natural model for Lsep which has the covering property. PACS numbers: 03.65.Ta, 03.65.Ca 1.

Oct 12, 20044 citations

Locality and orthomodular structure of compound systems

Teppei Watanabe

A P-lattice is defined as a σ-complete, orthomodular atomic lattice ℒ which is formed by the set of propositions of a physical system. A composition of physical systems in the framework of P-lattices is considered and some notions of locality are given. It is shown that the following statements about compound systems are equivalent. (a) All atoms of a compound system are reducible to those of its subsystems. (b) All pure states of a compound system are separable into those of its subsystems. (c) A compound system has statistical property independence. (d) At least one of the subsystems is classical. (e) Bell-type inequalities hold.A P-lattice is defined as a σ-complete, orthomodular atomic lattice ℒ which is formed by the set of propositions of a physical system. A composition of physical systems in the framework of P-lattices is considered and some notions of locality are given. It is shown that the following statements about compound systems are equivalent. (a) All atoms of a compound system are reducible to those of its subsystems. (b) All pure states of a compound system are separable into those of its subsystems. (c) A compound system has statistical property independence. (d) At least one of the subsystems is classical. (e) Bell-type inequalities hold.

Journal of Mathematical PhysicsApr 16, 20045 citations

Sublattices and Δ-blocks of orthomodular posets

Ivan Chajda, Helmut Länger

Abstract States of quantum systems correspond to vectors in a Hilbert space and observations to closed subspaces. Hence, this logic corresponds to the algebra of closed subspaces of a Hilbert space. This can be considered as a complete lattice with orthocomplementation, but it is not distributive. It satisfies a weaker condition, the so-called orthomodularity. Later on, it was recognized that joins in this structure need not exist provided the subspaces are not orthogonal. Hence, the resulting structure need not be a lattice but a so-called orthomodular poset, more generally an orthoposet only. For orthoposets, we introduce a binary relation $\mathrel \Delta$ and a binary operator $d(x,y)$ that are generalizations of the binary relation $\textrm{C}$ and the commutator $c(x,y)$ , respectively, known for orthomodular lattices. We characterize orthomodular posets among orthogonal posets. Moreover, we describe connections between the relations $\mathrel \Delta$ and $\leftrightarrow$ (the latter was introduced by P. Pták and S. Pulmannová) and the operator $d(x,y)$ . In addition, we investigate certain orthomodular posets of subsets of a finite set. In particular, we describe maximal orthomodular sublattices and Boolean subalgebras of such orthomodular posets. Finally, we study properties of $\Delta$ -blocks with respect to Boolean subalgebras and distributive subposets they include.Abstract States of quantum systems correspond to vectors in a Hilbert space and observations to closed subspaces. Hence, this logic corresponds to the algebra of closed subspaces of a Hilbert space. This can be considered as a complete lattice with orthocomplementation, but it is not distributive. It satisfies a weaker condition, the so-called orthomodularity. Later on, it was recognized that joins in this structure need not exist provided the subspaces are not orthogonal. Hence, the resulting structure need not be a lattice but a so-called orthomodular poset, more generally an orthoposet only. For orthoposets, we introduce a binary relation $\mathrel \Delta$ and a binary operator $d(x,y)$ that are generalizations of the binary relation $\textrm{C}$ and the commutator $c(x,y)$, respectively, known for orthomodular lattices. We characterize orthomodular posets among orthogonal posets. Moreover, we describe connections between the relations $\mathrel \Delta$ and $\leftrightarrow$ (the latter was introduced by P. Pták and S. Pulmannová) and the operator $d(x,y)$. In addition, we investigate certain orthomodular posets of subsets of a finite set. In particular, we describe maximal orthomodular sublattices and Boolean subalgebras of such orthomodular posets. Finally, we study properties of $\Delta$-blocks with respect to Boolean subalgebras and distributive subposets they include.

arXiv:2004.01962Journal of Logic and ComputationJun 25, 20200 citations

THE SEMILATTICE, BOX, AND LATTICE-TENSOR PRODUCTS IN QUANTUM LOGIC

Boris Ischi

Abstract. Given two complete atomistic lattices L1 and L2, we define a set S = S(L1, L2) of complete atomistic lattices by means of three axioms (natural regarding the description of quantum compound systems), or in terms of a universal property with respect to a given class of bimorphisms. We prove that S is a complete lattice. The bottom element L1 ∧○L2 is the separated product of Aerts. For atomistic lattices with 1 (not complete), L1 ∧○L2 ∼ = L1□L2 the box product of Grätzer and Wehrung, and, in case L1 and L2 are moreover coatomistic, L1 ∧○L2 ∼ = L1 ⊠ L2 the lattice tensor product. The top element L1 ∨○L2 is the (complete) join-semilattice tensor product of Fraser, which is isomorphic to the tensor products of Chu and Shmuely. With some additional hypotheses on L1 and L2 (true if L1 and L2 are moreover orthomodular with the covering property), we prove that S is a singleton if and only if L1 or L2 is distributive, if and only if L1 ∨○L2 has the covering property. Our main result reads: L ∈ S admits an orthocomplementation if and only if L = L1 ∧○L2. For L1 and L2 moreover irreducible, we characterize the automorphisms of each L ∈ S in terms of those of L1 and L2. At the end, we construct an example L1 ⇓○L2 in S which has the covering property. 1.Abstract. Given two complete atomistic lattices L1 and L2, we define a set S = S(L1, L2) of complete atomistic lattices by means of three axioms (natural regarding the description of quantum compound systems), or in terms of a universal property with respect to a given class of bimorphisms. We prove that S is a complete lattice. The bottom element L1 ∧○L2 is the separated product of Aerts. For atomistic lattices with 1 (not complete), L1 ∧○L2 ∼ = L1□L2 the box product of Grätzer and Wehrung, and, in case L1 and L2 are moreover coatomistic, L1 ∧○L2 ∼ = L1 ⊠ L2 the lattice tensor product. The top element L1 ∨○L2 is the (complete) join-semilattice tensor product of Fraser, which is isomorphic to the tensor products of Chu and Shmuely. With some additional hypotheses on L1 and L2 (true if L1 and L2 are moreover orthomodular with the covering property), we prove that S is a singleton if and only if L1 or L2 is distributive, if and only if L1 ∨○L2 has the covering property. Our main result reads: L ∈ S admits an orthocomplementation if and only if L = L1 ∧○L2. For L1 and L2 moreover irreducible, we characterize the automorphisms of each L ∈ S in terms of those of L1 and L2. At the end, we construct an example L1 ⇓○L2 in S which has the covering property. 1.

arXiv:math/0304350Jan 1, 20053 citations

\mathrel \Delta

and a binary operator

d(x,y)

that are generalizations of the binary relation

\textrm{C}

and the commutator

c(x,y)

, respectively, known for orthomodular lattices. We characterize orthomodular posets among orthogonal posets. Moreover, we describe connections between the relations

\mathrel \Delta

and

\leftrightarrow

(the latter was introduced by P. Pták and S. Pulmannová) and the operator

d(x,y)

. In addition, we investigate certain orthomodular posets of subsets of a finite set. In particular, we describe maximal orthomodular sublattices and Boolean subalgebras of such orthomodular posets. Finally, we study properties of

\Delta

-blocks with respect to Boolean subalgebras and distributive subposets they include.