Completeness of Graphical Languages for Mixed States Quantum Mechanics

TL;DR

This paper extends graphical languages for quantum mechanics to include mixed states and general quantum operations using a new discard construction, achieving completeness for a broader class of quantum processes.

Contribution

It introduces the discard construction to extend categorical graphical languages to mixed states, enabling reasoning about general quantum operations.

Findings

The discard construction makes any dagger-SMC into a symmetric monoidal category with discard maps.

Several graphical languages are extended and shown to be complete for mixed states.

The construction fails for some cases like Clifford+T quantum mechanics due to lack of enough isometries.

Abstract

There exist several graphical languages for quantum information processing, like quantum circuits, ZX-Calculus, ZW-Calculus, etc. Each of these languages forms a dagger-symmetric monoidal category (dagger-SMC) and comes with an interpretation functor to the dagger-SMC of (finite dimension) Hilbert spaces. In the recent years, one of the main achievements of the categorical approach to quantum mechanics has been to provide several equational theories for most of these graphical languages, making them complete for various fragments of pure quantum mechanics. We address the question of the extension of these languages beyond pure quantum mechanics, in order to reason on mixed states and general quantum operations, i.e. completely positive maps. Intuitively, such an extension relies on the axiomatisation of a discard map which allows one to get rid of a quantum system, operation which is not allowed in pure quantum mechanics. We introduce a new construction, the discard construction, which transforms any dagger-symmetric monoidal category into a symmetric monoidal category equipped with a discard map. Roughly speaking this construction consists in making any isometry causal. Using this construction we provide an extension for several graphical languages that we prove to be complete for general quantum operations. However this construction fails for some fringe cases like the Clifford+T quantum mechanics, as the category does not have enough isometries.