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Property lattices for independent quantum systems

Boris Ischi

Published Apr 17, 2002• 5 citations
OpenAlex

Abstract

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.

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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) calL{\\cal L}calL is suitable to model the physical properties of the total system. This leads us to impose five conditions P1 to P5 on calL{\\cal L}calL. We show that conditions P1 to P4 are physically necessary and prove that if calL{\\cal L}calL satisfies conditions P1 to P5, then calL{\\cal L}calL 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.
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.
Given a complete orthocomplemented lattice L and a set S of nonnegative real functions on L, sufficient conditions are established that S should fulfill in order that L be atomic. The conditions are investigated under which L may be represented by the lattice of all closed subspaces of a separable Hilbert space. (As is well known, the atomicity of L plays an important role here.) Some unsolved problems are pointed out. In axiomatic quantum mechanics, the lattice L may represent the set of propositions whereas the set of functions S represents the set of physical states. The conditions imposed on the pair (L,S) then have a simple and plausible physical interpretation; an important condition imposed on (L,S) is the existence of the ``maximal'' (i.e., maximally determined) states which appear in the theory as limit constructions.
In the course of the completion of this thesis we have addressed several problems concerning quantum many-body systems and one-dimensional lattice models. Before getting into details, let me briefly outline some of them together with our main contributions, and sketch out some central concepts we shall develop later. To begin with, we have studied the entanglement properties of the ground state of several one-dimensional systems (see Refs. [P1]–[P4]), and investigated several exact spectral properties of lattice models in one dimension, obtaining an analytic expression for the partition function of certain models with long-range interactions (see [P4] and [P5] for a detailed description of the models involved). We have completely characterized the spectrum of the so-called long range t-J model for the first time and, in particular, rigorously proved a conjecture proposed by Saiga and Kuramoto in the nineties for the spin 1/2 case...
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 Δ\mathrel \DeltaΔ and a binary operator d(x,y)d(x,y)d(x,y) that are generalizations of the binary relation C\textrm{C}C and the commutator c(x,y)c(x,y)c(x,y), respectively, known for orthomodular lattices. We characterize orthomodular posets among orthogonal posets. Moreover, we describe connections between the relations Δ\mathrel \DeltaΔ and ↔\leftrightarrow↔ (the latter was introduced by P. Pták and S. Pulmannová) and the operator d(x,y)d(x,y)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 Δ\DeltaΔ-blocks with respect to Boolean subalgebras and distributive subposets they include.
Logical analysis of the (non-)orthogonality relation in quantum physics has a long history, and results in many well-known representation theorems for lattices arising from the study of quantum logic via Kripke frames, such as the result in [2] about ortholattices and orthoframes and the result in [3] about property lattices and state spaces. In my talk, I’m going to focus on Piron lattices, also called irreducible propositional systems, since they play an important role in the study of foundations of quantum theory. To represent Piron lattices, I will introduce the notion of quantum Kripke frame (QKF). A QKF is a Kripke frame in which elements of the set are interpreted as states of a quantum system and the binary relation as the non-orthogonality relation between states. With this interpretation, it is dened to satisfy four conditions which are mathematically simple and physically sensible. For example, one of the conditions is a direct translation of the superposition principle in quantum physics into our relational setting. Then, I will prove that the collection of all bi-orthogonally closed subsets of the set in a QKF forms a Piron lattice, and every Piron lattice is isomorphic to such a collection in some QKF. Therefore, we obtain a representation theorem for Piron lattices via QKFs. The technique used here is inspired by [1], in which Baltag and Smets give a representation theorem for Piron lattices (satisfying Mayet’s condition) via quantum dynamic frames. Moreover, I will show that, by replacing the condition corresponding to the superposition principle with a mathematically simple one, one can obtain Kripke frames which transparently represent the states and properties of classical systems. Connections between QKFs and other mathematical structures like Hilbert geometries and quantum dynamic frames will also be discussed. Finally, I will talk about some directions for future work, for example nding a complete axiomatization of QKFs in modal or hybrid logic and extending QKFs to probabilistic QKFs.
Recently we proved that there are two non-isomorphic models of the calculus of quantum logic corresponding to an infinite-dimensional Hilbert space representation: an orthomodular lattice and a weakly orthomodular lattice. We also discovered that there are two non-isomorphic models of the calculus of classical logic: a distributive lattice (Boolean algebra) and a weakly distributive lattice. In this work we consider implications of these results to a quantum simulator which should mimic quantum systems by giving precise instructions on how to produce input state, how to evolve them, and how to read off the final states. We analyze which conditions quantum states of a quantum computer currently obey and which they should obey in order to enable full quantum computing, i.e., proper quantum mathematics. In particular we find several new conditions which lattices of Hilbert space subspaces must satisfy.