We construct the tensor product for the property lattices of two entities. We give a physical interpretation for this tensor product. We show that the tensor product is never an orthocomplemented lattice if both entities are nontrivial. It also never satisfies the covering law. This is the reason why this tensor product does not exist in quantum mechanics and does not exist in quantum logic. We analyze the relation of the tensor product with the usual description of two entities in quantum mechanics. At the end, we give a mathematical way of constructing this tensor product.
We construct the tensor product for the property lattices of two entities. We give a physical interpretation for this tensor product. We show that the tensor product is never an orthocomplemented lattice if both entities are nontrivial. It also never satisfies the covering law. This is the reason why this tensor product does not exist in quantum mechanics and does not exist in quantum logic. We analyze the relation of the tensor product with the usual description of two entities in quantum mechanics. At the end, we give a mathematical way of constructing this tensor product.
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.
A quantum logic is the couple (L,M) where L is an orthomodular σ-lattice and M is a strong set of states on L. The Jauch–Piron property in the σ-form is also supposed for any state of M. A ‘‘tensor product’’ of quantum logics is defined. This definition is compared with the definition of a free orthodistributive product of orthomodular σ-lattices. The existence and uniqueness of the tensor product in special cases of Hilbert space quantum logics and one quantum and one classical logic are studied.
In this work we present an intuitive construction of the quantum logical axiomatic system provided by George Mackey. The goal of this work is a detailed discussion of the results from the paper 'Physical justification for using the tensor product to describe two quantum systems as one joint system' [1] published by Diederik Aerts and Ingrid Daubechies. This means that we want to show how certain composed physical systems from classical and quantum mechanics should be described logically. To reach this goal, we will, like in [1], discuss a special class of axiomatically defined composed physical systems. With the help of certain results from lattice and c-morphism theory (see [2] and [23]), we will present a detailed proof of the statement, that in the quantum mechanical case, a composed physical system must be described via a tensor product space.
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
The concept of an operational logic has been developed by Randall and Foulis ([l]-[4], [10], [11]) as a part of a larger effort to obtain a formalism suitable for expressing, comparing, and evaluating various approaches to empirical science, statistics, and in particular, quantum mechanics. The structure of these logics is similar to that of an orthomodular partially ordered set which is often used as a model for quantum logics. However, the operational logic is a more general structure which, among other features, allows for the creation of a tensor product of logics to represent the coupling of physical systems. Randall and Foulis have shown that, given certain reasonable physical constraints, such a product is not possible within the category of orthomodular posets [12].
Up to now we have limited ourselves to states of a single particle. In the present chapter we shall introduce the description of two-particle states. Once this case is understood, it will be easy to generalize to any number of particles. States of two (or more) particles lead to very rich configurations called entangled states. A remarkable feature is that two entangled quantum particles, even at arbitrarily large spatial separations, continue to form a single entity and no classical probabilistic model is able to reproduce the correlation between particles. In the first section we shall present the essential mathematical formalism, that of the tensor product. This will permit us in Section 6.2 to describe quantum mixtures using the state operator formalism. Section 6.3 is devoted to the study of important physical consequences like the Bell inequalities and interference experiments involving entangled states, which will lead us to a deeper understanding of quantum physics. Finally, in the last section we shall briefly review applications to measurement theory and quantum information theory. The latter is undergoing rapid development at present and has applications to quantum computing, cryptography, and teleportation. The tensor product of two vector spaces Definition and properties of the tensor product We wish to construct the space of states of two physical systems which we assume initially to be completely independent. Let ℌ N 1 and ℌ M 2 be the spaces of states of the two systems, of dimension N and M , respectively.