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arXiv:2105.06244 · Electronic Proceedings in Theoretical Computer Science

A Graphical Calculus for Lagrangian Relations

Cole Comfort, Aleks Kissinger

Published Oct 31, 2022• 10 citations
arXivHTMLPDF

Abstract

Symplectic vector spaces are the phase spaces of linear mechanical systems. The symplectic form describes, for example, the relation between position and momentum as well as current and voltage. The category of linear Lagrangian relations between symplectic vector spaces is a symmetric monoidal subcategory of relations which gives a semantics for the evolution -- and more generally linear constraints on the evolution -- of various physical systems. We give a new presentation of the category of Lagrangian relations over an arbitrary field as a `doubled' category of linear relations. More precisely, we show that it arises as a variation of Selinger's CPM construction applied to linear relations, where the covariant orthogonal complement functor plays the role of conjugation. Furthermore, for linear relations over prime fields, this corresponds exactly to the CPM construction for a suitable choice of dagger. We can furthermore extend this construction by a single affine shift operator to obtain a category of affine Lagrangian relations. Using this new presentation, we prove the equivalence of the prop of affine Lagrangian relations with the prop of qudit stabilizer theory in odd prime dimensions. We hence obtain a unified graphical language for several disparate process theories, including electrical circuits, Spekkens' toy theory, and odd-prime-dimensional stabilizer quantum circuits.

DOI
10.4204/eptcs.372.24

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Symplectic vector spaces are the phase spaces of linear mechanical systems. The symplectic form describes, for example, the relation between position and momentum as well as current and voltage. The category of linear Lagrangian relations between symplectic vector spaces is a symmetric monoidal subcategory of relations which gives a semantics for the evolution -- and more generally linear constraints on the evolution -- of various physical systems. We give a new presentation of the category of Lagrangian relations over an arbitrary field as a `doubled' category of linear relations. More precisely, we show that it arises as a variation of Selinger's CPM construction applied to linear relations, where the covariant orthogonal complement functor plays the role of conjugation. Furthermore, for linear relations over prime fields, this corresponds exactly to the CPM construction for a suitable choice of dagger. We can furthermore extend this construction by a single affine shift operator to obtain a category of affine Lagrangian relations. Using this new presentation, we prove the equivalence of the prop of affine Lagrangian relations with the prop of qudit stabilizer theory in odd prime dimensions. We hence obtain a unified graphical language for several disparate process theories, including electrical circuits, Spekkens' toy theory, and odd-prime-dimensional stabilizer quantum circuits.
We introduce a family of diagrammatical equational theories unifying two research programs: categorical quantum mechanics and graphical linear algebra. We prove their completeness with respect to denotational semantics described in terms of relations between vector spaces equipped with symplectic structure. This provides versatile graphical languages encompassing both affinely constrained classical mechanical systems, as well as odd-prime-dimensional stabiliser and Gaussian quantum circuits. Terms are described by labelled graphs with input and output interfaces, and the languages are equipped with equational theories amenable to standard graph rewriting techniques. In order to reason about large composite systems, we introduce a compact scalable notation where the vertices are themselves labelled by graphs. This notation allows us to state new and powerful rewrite rules which operate on diagrams at a large scale. We also show how this notation neatly captures some important constructions, such as graph states of quantum computing and the impedance and admittance matrices of electrical networks.
We give generators and relations for the hypergraph props of Gaussian relations and positive affine Lagrangian relations. The former extends Gaussian probabilistic processes by completely-uninformative priors, and the latter extends Gaussian quantum mechanics with infinitely-squeezed states. These presentations are given by adding a generator to the presentation of real affine relations and of real affine Lagrangian relations which freely codiscards effects, as well as certain rotations. The presentation of positive affine Lagrangian relations provides a rigorous justification for many common yet informal calculations in the quantum physics literature involving infinite-squeezing. Our presentation naturally extends Menicucci et al.'s graph-theoretic representation of Gaussian quantum states with a representation for Gaussian transformations. Using this graphical calculus, we also give a graphical proof of Braunstein and Kimble's continuous-variable quantum teleportation protocol. We also interpret the LOv-calculus, a diagrammatic calculus for reasoning about passive linear-optical quantum circuits in our graphical calculus. Moreover, we show how our presentation allows for additional optical operations such as active squeezing.
This paper explores the application of category theory to provide a semantic framework for quantum vector spaces, which are fundamental to quantum mechanics and quantum information. We move beyond the traditional algebraic and analytical descriptions of Hilbert spaces to adopt a more abstract, compositional, and graphical approach rooted in monoidal categories. This categorical perspective allows for a unified understanding of quantum phenomena, including superposition, entanglement, and measurement, by representing quantum systems as objects and processes as morphisms. We delve into the concept of compact closed categories and dagger symmetric monoidal categories as the appropriate mathematical structures to capture the essential features of quantum theory, such as duality, adjoints, and the tensor product for composite systems. Furthermore, the paper discusses how graphical calculi, arising from these categorical structures, offer an intuitive and rigorous method for reasoning about quantum protocols and computations, facilitating the design and verification of quantum algorithms. By providing a high-level, axiomatic framework, categorical semantics clarifies the foundational principles of quantum mechanics and opens new avenues for developing quantum programming languages and understanding the interplay between quantum information and computation.
Polar duality is a well-known concept from convex geometry and analysis. In the present paper, we study two symplectically covariant versions of polar duality keeping in mind their applications to quantum mechanics. The first variant makes use of the symplectic form on phase space and allows a precise study of the covariance matrix of a density operator. The latter is a fundamental object in quantum information theory., The second variant is a symplectically covariant version of the usual polar duality highlighting the role played by Lagrangian planes. It allows us to define the notion of "geometric quantum states" with are in bijection with generalized Gaussians.
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 introduce the CP*-construction on a dagger compact closed category as a generalisation of Selinger's CPM-construction. While the latter takes a dagger compact closed category and forms its category of "abstract matrix algebras" and completely positive maps, the CP*-construction forms its category of "abstract C*-algebras" and completely positive maps. This analogy is justified by the case of finite-dimensional Hilbert spaces, where the CP*-construction yields the category of finite-dimensional C*-algebras and completely positive maps. The CP*-construction fully embeds Selinger's CPM-construction in such a way that the objects in the image of the embedding can be thought of as "purely quantum" state spaces. It also embeds the category of classical stochastic maps, whose image consists of "purely classical" state spaces. By allowing classical and quantum data to coexist, this provides elegant abstract notions of preparation, measurement, and more general quantum channels.