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arXiv:2103.05191 · Electronic Proceedings in Theoretical Computer Science

Exponential Modalities and Complementarity (extended abstract)

Robin Cockett, Priyaa Varshinee Srinivasan

Published Oct 31, 2022• 0 citations
arXivHTMLPDF

Abstract

The exponential modalities of linear logic have been used by various authors to model infinite-dimensional quantum systems. This paper explains how these modalities can also give rise to the complementarity principle of quantum mechanics. The paper uses a formulation of quantum systems based on dagger-linear logic, whose categorical semantics lies in mixed unitary categories, and a formulation of measurement therein. The main result exhibits a complementary system as the result of measurements on free exponential modalities. Recalling that, in linear logic, exponential modalities have two distinct but dual components, ! and ?, this shows how these components under measurement become "compacted" into the usual notion of complementary Frobenius algebras from categorical quantum mechanics.

DOI
10.4204/eptcs.372.15

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The exponential modalities of linear logic have been used by various authors to model infinite-dimensional quantum systems. This paper explains how these modalities can also give rise to the complementarity principle of quantum mechanics. The paper uses a formulation of quantum systems based on dagger-linear logic, whose categorical semantics lies in mixed unitary categories, and a formulation of measurement therein. The main result exhibits a complementary system as the result of measurements on free exponential modalities. Recalling that, in linear logic, exponential modalities have two distinct but dual components, ! and ?, this shows how these components under measurement become "compacted" into the usual notion of complementary Frobenius algebras from categorical quantum mechanics.
This thesis develops the categorical proof theory for the non-compact multiplicative dagger linear logic, and investigates its applications to Categorical Quantum Mechanics (CQM). The existing frameworks of CQM are categorical proof theories of compact dagger linear logic, and are motivated by the interpretation of quantum systems in the category of finite dimensional Hilbert spaces. This thesis describes a new non-compact framework called Mixed Unitary Categories which can accommodate infinite dimensional systems, and develops models for the framework. To this end, it builds on linearly distributive categories, and ∗*∗-autonomous categories which are categorical proof theories of (non-compact) multiplicative linear logic. The proof theory of non-compact dagger-linear logic is obtained from the basic setting of an LDC by adding a dagger functor satisfying appropriate coherences to give a dagger-LDC. From every (isomix) dagger-LDC one can extract a canonical "unitary core" which up to equivalence is the traditional CQM framework of dagger-monoidal categories. This leads to the framework of Mixed Unitary Categories (MUCs): every MUC contains a (compact) unitary core which is extended by a (non-compact) isomix dagger-LDC. Various models of MUCs based on Finiteness Spaces, Chu spaces, Hopf modules, etc., are developed in this thesis. This thesis also generalizes the key algebraic structures of CQM, such as observables, measurement, and complementarity, to MUC framework. Furthermore, using the MUC framework, this thesis establishes a connection between the complementary observables of quantum mechanics and the exponential modalities of linear logic.
Categorical quantum mechanics exploits the dagger compact closed structure of finite dimensional Hilbert spaces, and uses the graphical calculus of string diagrams to facilitate reasoning about finite dimensional processes. A significant portion of quantum physics, however, involves reasoning about infinite dimensional processes, and it is well-known that the category of all Hilbert spaces is not compact closed. Thus, a limitation of using dagger compact closed categories is that one cannot directly accommodate reasoning about infinite dimensional processes. A natural categorical generalization of compact closed categories, in which infinite dimensional spaces can be modelled, is *-autonomous categories and, more generally, linearly distributive categories. This article starts the development of this direction of generalizing categorical quantum mechanics. An important first step is to establish the behaviour of the dagger in these more general settings. Thus, these notes simultaneously develop the categorical semantics of multiplicative dagger linear logic. The notes end with the definition of a mixed unitary category. It is this structure which is subsequently used to extend the key features of categorical quantum mechanics.
We relate notions of complementarity in three layers of quantum mechanics: (i) von Neumann algebras, (ii) Hilbert spaces, and (iii) orthomodular lattices. Taking a more general categorical perspective of which the above are instances, we consider dagger monoidal kernel categories for (ii), so that (i) become (sub)endohomsets and (iii) become subobject lattices. By developing a ‘point-free’ definition of copyability we link (i) commutative von Neumann subalgebras, (ii) classical structures, and (iii) Boolean subalgebras.
Categorical quantum mechanics studies quantum theory in the framework of dagger-compact closed categories. Using this framework, we establish a tight relationship between two key quantum theoretical notions: non-locality and complementarity. In particular, we establish a direct connection between Mermin-type non-locality scenarios, which we generalise to an arbitrary number of parties, using systems of arbitrary dimension, and performing arbitrary measurements, and a new stronger notion of complementarity which we introduce here. Our derivation of the fact that strong complementarity is a necessary condition for a Mermin scenario provides a crisp operational interpretation for strong complementarity. We also provide a complete classification of strongly complementary observables for quantum theory, something which has not yet been achieved for ordinary complementarity. Since our main results are expressed in the (diagrammatic) language of dagger-compact categories, they can be applied outside of quantum theory, in any setting which supports the purely algebraic notion of strongly complementary observables. We have therefore introduced a method for discussing non-locality in a wide variety of models in addition to quantum theory. The diagrammatic calculus substantially simplifies (and sometimes even trivialises) many of the derivations, and provides new insights. In particular, the diagrammatic computation of correlations clearly shows how local measurements interact to yield a global overall effect. In other words, we depict non-locality.
A certain class of Frobenius algebras has been used to characterize orthonormal bases and observables on finite-dimensional Hilbert spaces. The presence of units in these algebras means that they can only be realized finite-dimensionally. We seek a suitable generalization, which will allow arbitrary bases and observables to be described within categorical axiomatizations of quantum mechanics. We develop a definition of H*-algebra that can be interpreted in any symmetric monoidal dagger category, reduces to the classical notion from functional analysis in the category of (possibly infinite-dimensional) Hilbert spaces, and hence provides a categorical way to speak about orthonormal bases and quantum observables in arbitrary dimension. Moreover, these algebras reduce to the usual notion of Frobenius algebra in compact categories. We then investigate the relations between nonunital Frobenius algebras and H*-algebras. We give a number of equivalent conditions to characterize when they coincide in the category of Hilbert spaces. We also show that they always coincide in categories of generalized relations and positive matrices.
A certain class of Frobenius algebras has been used to characterize orthonormal bases and observables on finite-dimensional Hilbert spaces. The presence of units in these algebras means that they can only be realized finite-dimensionally. We seek a suitable generalization, which will allow arbitrary bases and observables to be described within categorical axiomatizations of quantum mechanics. We develop a definition of H*-algebra that can be interpreted in any symmetric monoidal dagger category, reduces to the classical notion from functional analysis in the category of (possibly infinite-dimensional) Hilbert spaces, and hence provides a categorical way to speak about orthonormal bases and quantum observables in arbitrary dimension. Moreover, these algebras reduce to the usual notion of Frobenius algebra in compact categories. We then investigate the relations between nonunital Frobenius algebras and H*algebras. We give a number of equivalent conditions to characterize when they coincide in the category of Hilbert spaces. We also show that they always coincide in categories of generalized relations and positive matrices.