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Cambridge University Press eBooks

Compatibility Relation

Mark Pankov

Published Jan 6, 2020• 0 citations
Publisher

Abstract

Two projections commute if and only if their images are compatible. Using combinatorial methods we describe compatibility preserving bijective transformations of Grassmannians. In some cases, these transformations form a class greater than the class of transformations induced by unitary and anti-unitary operators.
DOI
10.1017/9781108800327.007

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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.
Let HHH be an infinite-dimensional complex Hilbert space and let L(H){\mathcal L}(H)L(H) be the logic formed by all closed subspaces of HHH. For every natural kkk we denote by Gk(H){\mathcal G}_{k}(H)Gk​(H) the Grassmannian consisting of kkk-dimensional subspaces. An orthogonal apartment of Gk(H){\mathcal G}_{k}(H)Gk​(H) is the set consisting of all kkk-dimensional subspaces spanned by subsets of a certain orthogonal base of HHH. Orthogonal apartments can be characterized as maximal sets of mutually compatible elements of Gk(H){\mathcal G}_{k}(H)Gk​(H). We show that every bijective transformation fff of Gk(H){\mathcal G}_{k}(H)Gk​(H) such that fff and f−1f^{-1}f−1 send orthogonal apartments to orthogonal apartments (in other words, fff preserves the compatibility relation in both directions) can be uniquely extended to an automorphism of L(H){\mathcal L}(H)L(H).
There is a natural equivalence relation on representations of the states of a given quantum system in a Hilbert space, two representations being equivalent iff they are related by a unitary transformation. There are two equivalence classes, with members of opposite classes being related by a conjugate-unitary (anti-unitary) transformation. These two conjugacy classes are related in much the same way as are the two imaginary units of a complex field, and there is a priori no basis on which to prefer one over the other in any individual case. This is potentially problematic in that the choice of conjugacy class of a representation determines the sign of energy and other quantities defined as generators of continuous symmetries of the system in question, so that it would appear that principles like conservation of energy for a compound system may hold or fail depending on relative choices of conjugacy class of representations of its subsystems. We show that for any finite set of quantum systems there are exactly two ways of choosing conjugacy classes of representations consistent with the usual tensor-product construction for representing the compound system composed of these. Each is obtained from the other by reversing the conjugacy of all the representations at once. The relation of unitary equivalence for representations of a single system is therefore uniquely extendible to representations of all systems that can interact with it.
It is well known that transformations of $ \mathbb{C}^n $ preserving the standard inner product are unitary transformations. In this paper, all bijective transformations of isotropic sets of $ \mathbb{C}P^{n} $ preserving $ H $-orthogonality in both directions, called $ H $-orthogonal transformations, have been determined. This is a generalization of Uhlhorn's version of Wigner's unitary-antiunitary theorem. The group of $ H $-orthogonal transformations on some other sets of $ \mathbb{C}P^{n} $ were also determined.