Motivation

The Hilbert space framework for quantum mechanics, initially proposed by Von Neumann, is not ideal as a model for Quantum information and computation. In this formalism, stating and checking algorithms involves matrix multiplications, which quickly become complicated, when there are more than a few qubits involved. Also, in Computer Science, the fact that there exists a transition from a certain state A, to another state B, is often the only thing that is of interest. For many theoretical purposes, it does not matter how this transition is manifested. As a consequence, parts of the ordinary theory of quantum mechanics, such as the Schroedingen equation are redundant for quantum information theory.
Category theory turns out to be a language for quantum theory that captures enough of the structure of quantum mechanics for the purpose of computation, and is easy to work with at the same time.

The interaction between classical and quantum information plays an important role in many quantum protocols (think of measurements, controlled operations etc.)
In the Hilbert space formalism, and in CPM, classical information can only be described by assigning a basis to a state. This has been the motivation to propose new models which contain the notion of classicality in their structure.

Different Models

Symmetric monoidal compact closed categories

Symmetric monoidal compact closed categories were the first categorical model for quantum mechanis, introduced by Samson Abramsky and Bob Coecke [1]. These categories are a generalisation of Hilbert spaces. It allows for abstract notions of scalars, vectors, entanglement, classical structures, pure states, measurements, no-cloning, and the Born rule. Furthermore, the corresponding diagrammatic language is sound and complete.

CPM

Subsequently, the category CPM was proposed by Peter Selinger, which is the abstract category of mixed states and density operators. The biproduct completion of this category may be used to model classical and quantum information separately[2].

Another category proposed by Peter Selinger that describes classical and quantum information separately is the category of dagger splittings [3].

Special dagger Frobenius algebras

The compact category CP* of special dagger Frobenius algebras. This is a generalisation of C* algebras. Commutative special dagger Frobenius algebras model classical data, non commutative special dagger Frobenius algebras model quantum data, completely positive maps model computational processes.[4],[5]

2-categories

Weak 2-categories are a generalisation of the category 2Hilb. These form a link between quantum information and representation theory. Objects model classical data, 1-morphisms model quantum data, 2-morphisms model computational processes. The resulting symmetric monoidal 2-category provides universal syntactic models that can encode entire procedures as single equations [6]

Applications in computational linguistics

Symmetric monoidal compact categories have turned out to be a useful tool for the processing of natural language.[reference?]

Current Research

Modelling Classical and Quantum information

The 2-categorical semantics for quantum protocols can be combined with the theory of dagger Frobenius algebras in one 2-categorical model. using the construction 2(-), which turns a monoidal category in a 2-category. This construction preserves daggers, compactness and some notion of biproducts.
The category 2(CP*(C)) allows a unified description of quantum
teleportation and classical encryption in a single 2-category, as well as a universal security proof applicable simultaneously to both scenarios. At the moment, 2(CP*(C)) is not yet well understood. [7]

There is an ongoing discussion whether or not research in this direction is meaningful for quantum information theory. The uncertainty arises from the question how to interpret dagger Frobenius algebras as physical phenomena.

Categorical models from an axiomatic approach

Recently it was shown by Giulio Chibirella that we can recover a categorical description of pure states, by an axiomatic approach, starting from a probabilistic-operational model of physics.
[8]

Future directions

Loose ends and concrete open questions:

Papers


An introduction to categorical quantum mechanics
Symmetric monoidal compact closed categories
Dagger Compact Closed Categories and Completely Positive Maps
The category of special dagger Frobenius algebras CP*()
The category of special dagger Frobenius algebras compared to other categories
2-categorical semantics for quantum theory
The 2-category of special dagger Frobenius Algebras 2(CP*())

Talks

Slides

Recovering pure states in categorical quantum mechanics from axioms