Types of quantum field thories
category with duals (list of them)
dualizable object (what they have)
Central aspects of finite quantum mechanics (with finite-dimensional state space, notably for tensor products of qbit states) and quantum computation follow formally from the formal properties of the category FinHilb of finite-dimensional Hilbert spaces. These properties are axomatized by saying that Hilb is an example of a †-compact category.
Conversely, much of finite quantum mechanics and quantum computation can be formulated in any †-compact category, and general reasoning about †-compact categories themselves yields results about quantum mechanics and quantum computation.
Let be a †-compact category.
We list various concepts in quantum mechanics and their corresponding incarnation in terms of structures in .
An observable in quantum mechanics formulated on a Hilbert space is modeled by a self-adjoint operator, and the classical measurement outcomes of this operator provide, at least under some assumptions, an orthogonal basis on the Hilbert space.
That, more abstractly, the notion of orthogonal basis of an object can be phrased intrinsically inside any suitable -compact category is the point made in (CoeckePavlovicVicary)
In (Vicary) it is shown that in -compact categories with all finite limits over certain “tree-like” diagrams compatible with the -structure, this has the properties that
If furthermore every bounded sequence of measurements in with values in has a least upper bound, then it follows that this field coincides with the complex numbers
Symmetric monoidal categories such as †-compact categories have as internal logic a fragment of linear logic and as type theory a flavor of linear type theory. In this fashion everything that can be formally said about quantum mechanics in terms of †-compact categories has an equivalent expression in formal logic/type theory. It has been argued (Abramsky-Duncan 05, Duncan 06) that this linear logic/linear type theory of quantum mechanics is the correct formalization of “quantum logic”. An exposition of this point of view is in (Baez-Stay 09).
with an expanded version in
A fairly comprehensive account of the underlying theory of string diagrams is in
A pedagogical exposition of the graphical calculus is in
More basic introductions are in
A comprehensive collection of basics and of recent developments is in
The formalization of orthogonal bases in -compact categories is in
The role of complex numbers in general -compact categories is discussed in
Completely positive maps in terms of -categories are discussed in
An exposition along these lines is in