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under construction
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.
A transparent string diagram calculus in †-compact categories as exposed in (Coecke, Kindergarten quantum mechanics) provides an intuitive and powerful tool for reasoning in $\dagger$-compact categories.
Let $(C,\otimes,I, \dagger)$ be a †-compact category.
We list various concepts in quantum mechanics and their corresponding incarnation in terms of structures in $C$.
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 $\dagger$-compact category is the point made in (CoeckePavlovicVicary).
The underlying “algebra of quantum amplitudes” of the corresponding quantum mechanical system is the endomorphism monoid of the tensor unit
In (Vicary) it is shown that in $\dagger$-compact categories with all finite limits over certain “tree-like” diagrams compatible with the $\dagger$-structure, this $\mathbb{C}_C$ has the properties that
it is a field of characteristic 0 with involution $\dagger$;
the subfield $\mathbb{R}_C$ fixed under $\dagger$ is orderable.
If furthermore every bounded sequence of measurements in $C$ with values in $\mathbb{R}_C$ has a least upper bound, then it follows that this field coincides with the complex numbers
and moreover
The behaviour of quantum channels and completely positive maps has an elegant categorical description in terms of $\dagger$-compact categories. See (Selinger and Coecke).
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).
The idea that the natural language of quantum mechanics and quantum computation is that of †-compact categories became popular with the publication
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
Bob Coecke, Kindergarten quantum mechanics (arXiv:quant-ph/0510032)
Bob Coecke, Quantum Picturalism (arXiv:0908.1787)
More basic introductions are in
Bob Coecke, Introducing categories to the practicing physicist (arXiv:0808.1032)
Bob Coecke, Categories for the practising physicist (arXiv:0905.3010)
Chris Heunen, Jamie Vicary, Lectures on categorical quantum mechanics, 2012 (pdf)
A comprehensive collection of basics and of recent developments is in
The formalization of orthogonal bases in $\dagger$-compact categories is in
The role of complex numbers in general $\dagger$-compact categories is discussed in
Completely positive maps in terms of $\dagger$-categories are discussed in
Peter Selinger, Dagger compact closed categories and completely positive maps (web, pdf)
Bob Coecke, Complete positivity without compactness (pdf)
The relation to quantum logic/linear logic has been expolred in
Samson Abramsky, Ross Duncan, A Categorical Quantum Logic (arXiv:quant-ph/0512114)
Ross Duncan, Types for quantum mechanics, 2006 (pdf, slides)
An exposition along these lines is in
Last revised on September 25, 2017 at 13:35:55. See the history of this page for a list of all contributions to it.