A Graphical Calculus for Lagrangian Relations

Symplectic vector spaces are the phase space 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 of 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. Linear Lagrangian relations, or more generally, affine Lagrangian relations provide a rich, compositional setting for modelling the evolutions of various physical systems. For example, certain classes of electrical circuits can be interpreted in terms of Lagrangian relations over the field of real rational functions [7, 8]. On a quite different note, the state preparation and quantum evolution of p-dimensional generalizations of Spekkens’ toy theory [31] and (consequently) odd-prime-dimensional stabilizer quantum theory [22] have semantics in terms of affine Lagrangian relations over Fp. Specifically, the state preparation corresponds to the affine Lagrangian relations from the tensor unit, and the evolution corresponds to affine symplectomorphisms. In this paper we extend this correspondance to the full category of Lagrangian relations, giving these circuits a proper categorical treatment. Formally, the category of Lagrangian relations is the symmetric monoidal subcategory of linear relations where the objects are symplectic vector spaces and the morphisms are linear relations satisfying an extra condition which can be captured graphically as the following, where V ⊥ denotes the orthogonal complement and the grey box denotes the antipode from the graphical theory of linear relations: