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Applications of Mathematics

Hilbert-space-valued states on quantum logics

Jan Hamhalter, Pavel Pták

Published Jan 1, 1992• 3 citations
Publisher

Abstract

We analyze finitely additive orthogonal states whose values lie in a real Hilbert space. We call them $h$-states. We first consider the important case of $h$-states on a standard Hilbert logic $L(H)$ of projectors in $H$-we describe the $h$-states $s$: $L(H_1) \rightarrow H_2$, where $\text {dim } H_2 \leq$ \text {dim} H_1 < \infty$. In particular, we show that, up to a unitary mapping, every $h$-state $s$: $L(H)\rightarrow H(3\leq \text {dim } H < \infty)$ has to be concentrated on a one-dimensional projection. We also study the $h$-states $s$: $L(H_1)\rightarrow H_2$ for the case of $\text {dim } H_1 = \infty$. The results of the first part complement the papers [10] and [13]. In the second part we investigate $h$-states on general logics. Being motivated by the quantum axiomatics, the main question we ask here is as follow: Given a Hilbert space $H$ with $\text {dim } H < \infty$, what is the class of such logics $L$ that, for any Boolean subalgebra $B$ of $L$, every $h$-states $s$: $B \rightarrow H$ extends over $L$? We answer this question by finding a simple condition characterizing the class (Theorem 3.4]. It turns out that the class is considerably large-it contains e.g. all concrete logics-but, on the other hand, it does not contain all finite logics (we construct a counterexample in the appendix).

DOI
10.21136/am.1992.104491

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Crítica (México D F En línea)•Nov 9, 1979•0 citations
We analyze finitely additive orthogonal states whose values lie in a real Hilbert space. We call them hhh-states. We first consider the important case of hhh-states on a standard Hilbert logic L(H)L(H)L(H) of projectors in HHH-we describe the hhh-states sss: L(H1)→H2L(H_1) \rightarrow H_2L(H1​)→H2​, where dim H2≤\text {dim } H_2 \leqdim H2​≤ \text {dim} H_1 < \infty.Inparticular,weshowthat,uptoaunitarymapping,every. In particular, we show that, up to a unitary mapping, every .Inparticular,weshowthat,uptoaunitarymapping,everyh−state-state −states:: :L(H)\rightarrow H(3\leq \text {dim } H < \infty)hastobeconcentratedonaone−dimensionalprojection.Wealsostudythehas to be concentrated on a one-dimensional projection. We also study thehastobeconcentratedonaone−dimensionalprojection.Wealsostudytheh−states-states −statess:: :L(H_1)\rightarrow H_2forthecaseoffor the case offorthecaseof\text {dim } H_1 = \infty.Theresultsofthefirstpartcomplementthepapers[10]and[13].Inthesecondpartweinvestigate. The results of the first part complement the papers [10] and [13]. In the second part we investigate .Theresultsofthefirstpartcomplementthepapers[10]and[13].Inthesecondpartweinvestigateh−statesongenerallogics.Beingmotivatedbythequantumaxiomatics,themainquestionweaskhereisasfollow:GivenaHilbertspace-states on general logics. Being motivated by the quantum axiomatics, the main question we ask here is as follow: Given a Hilbert space −statesongenerallogics.Beingmotivatedbythequantumaxiomatics,themainquestionweaskhereisasfollow:GivenaHilbertspaceHwithwithwith\text {dim } H < \infty,whatistheclassofsuchlogics, what is the class of such logics ,whatistheclassofsuchlogicsLthat,foranyBooleansubalgebrathat, for any Boolean subalgebrathat,foranyBooleansubalgebraBofofofL,every, every ,everyh−states-states −statess:: :B \rightarrow Hextendsoverextends overextendsoverL$? We answer this question by finding a simple condition characterizing the class (Theorem 3.4]. It turns out that the class is considerably large-it contains e.g. all concrete logics-but, on the other hand, it does not contain all finite logics (we construct a counterexample in the appendix).
We consider the standard quantum logic L(H){\mathcal L}(H)L(H) associated to a complex Hilbert space HHH, i.e. the lattice of closed subspaces of HHH together with the orthogonal complementation. The orthogonality and compatibility relations are defined for any logic. In the standard quantum logic, they have a simple interpretation in terms of operator theory. For example, two closed subspaces (propositions in the logic L(H){\mathcal L}(H)L(H)) are compatible if and only if the projections on these subspaces commute. We present both classical and more resent results on transformations of L(H){\mathcal L}(H)L(H) and the associated Grassmannians which preserve the orthogonality or compatibility relation. The first result in this direction was classical Wigner's theorem.
Quantum logic combines two sharp propositions by one join, which exists whether or not the propositions can be tested together and which keeps no record of that difference. We define a quantum hyperstructure to be a multivalued composition on the orthogonal projectors of a complex Hilbert space. Given two projectors, the composition returns every projector that commutes with both and whose range lies in the sum of their ranges. Its value is the family of sharp propositions still jointly compatible with the pair and supported on the subspace the pair generates. Associativity fails in general, already for a commuting triple, and holds on every Boolean algebra of mutually commuting projectors. In finite dimension the family is classified by the joint eigenspaces of the two projectors and by the principal angles between their ranges. The same composition determines the projector order and the compatibility relation. It is not a hyper effect sum, and the poset of its commuting joins is not a quantum suplattice. The construction makes the indeterminacy of incompatible measurement a law of composition, rather than only a failure of distributivity.
In this paper, we study linear operators on real and complex Euclidean spaces which are real-orthogonal projections. It is a generalization of such standard (complex) orthogonal projections for which only the real part of scalar product vanishes. We note the difference between properties of real-orthogonal projections on real and on complex spaces. We can compare some partial order properties of orthogonal and of real-orthogonal projections. We prove that the set of all real-orthogonal projections in a finite-dimensional complex or real space is a quantum pseudo-logic.
The complete and atomic orthomodular lattice structure which is formed by the projectors (or equivalently, by the closed subspaces) of a complex and separable Hilbert space will be indicated in this paper by LG (H), where H is the Hilbert space. (We take operations and distinct elements for granted.) LG (H) is, classically considered, the “typical” logic of quantum mechanics, in the following sense: if H is the space associated with physical system S, then H’s rays, the ultramaximal operators, etc., represent typical properties of S, such as observable states and magnitudes. LG (H) is simply an element in model sets of more general theories such as the orthomodular lattice set, which we formalize in § 2, using a second-order logic. This generalization, which we call “formal” because it consists merely of giving more permissive axioms that facilitate the apparition of new structures other than LG (H), can yet be extended. In § 2 we use infinitarian logics, where a numerable number of conjunctions and quantificators is permitted, to express the theory of orthomodular partially ordered sets (posets). That formulation uses essentially the logic called (omega-1, omega-1), and a conjecture is raised about proving it cannot be realized within a weaker logic. The way to prove this, it is suggested, should be analogous to the way the indefinability of well-ordering is proved. Such generalizations are analogous to those obtained when we pass, e.g., from the field of real numbers to any commutative, Archimedian, etc., field; then to a commutative ring with and unity, and so on. They are, therefore, formal, in the sense that (1) they do not “say” much about the initial structure which motivated the generalization and that (2) in generalizing, we allow the “entrance” of new structures that have little to do with the original motivation. For instance, in the theory of orthomodular lattices, besides Hilbert-space logics (or those based on other topological vectorial spaces), new models appear, such as finite or orthomodular lattices, for which the name “quantum logic” has little sense. In § 3 we study the modifications, and eventual generalizations, of the basic structure that generates quantum logic, i.e., the infinite-dimensional separable complex Hilbert space structure. A first way seems to be the generalizing of the field, and it is analyzed in § 3.1. In principle, it seems natural to admit that the field must contain some copy of the field of real numbers, or have at least an expressive force as great as theirs. This is why the use of real fields —and for a greater certainty, of complex fields— in the H spaces of quantum mechanics seems to be fully justified. They study of quaternions, carried on by Emch and others, but not clearly completed yet, could lead to an equivalent quantum mechanics, notwithstanding the non-commutativity of this field. Anyway, real, complex and quaternions seem to be only fields with a “natural” right to sustentate these spaces. The incompleteness of rational fields, and their undesirable properties as to dimension and measurability, cause them to be rejected. Other proposals are studied. Leaving aside Hensel’s bodies, that seem to be wholly different in nature, we are left with p-adics Qp and Galois’ fields as the only reasonable candidates. Know algebraic results are then invoked, according to which an extension or completion of these bodies would not lead to the desired results. Quite simply, we would not, save on trivial instances, have any logic at all. It is a well-known fact that by completing p-adic fields (let K be such a completion) we arrive at space (V, K) upon that field, which does not allow to define hermitian forms for dimensions greater than 4. Actually, no involutory anti-automorphism of the field (such as complex conjugation) permits to associate any quadratic form for a dimension greater than or equal to 5. Things are even worse in the case of Galois’ fields where limitative results begin to appear already in dimensions greater than 2. In § 3.2 we choose to accept the complex numbers body C as a canonic body, and see whether it is possible to generalize space into a class that need not be Hilbert’s. There are various possibilities: to consider Banach spaces, or else more or less general local convex spaces, and particularly, to focus analysis on Mackey spaces. For Banach spaces there appear Maczynski’s limitative results; for Mackey spaces, “analogous” results from Wilbur. (Cf. references at the end of the essay.) As a matter of fact, only infinite-dimensional spaces can be of any interest. In this case, if a Mackey space has an “acceptable” topology, i.e., a stronger one than the so-called weak convergence topology, and its LG (M) is really a logic, then M is once again a Hilbert space. Several of these results to the conclusion that we can hardly obtain a different logic (a more general or anyway more informative one) by taking the class LG (E) of some non-Hilbert space E. Rayski’s proposition about substituting Schwartz space for Hilbert space is also considered, but although such a substitution would modify actual work “within” the space in the interests of formalization, it is natural to expect the apparition of a logic LG (S), equivalent to former logics. Maybe the part special logics play in quantum mechanics will be entirely unnecessary when the axiomatizing of a physical system can be done in terms of some kind of spaces that leaves no doubt about its usefulness. [C.L.]