nLab finite quantum mechanics in terms of dagger-compact categories

under construction

Idea

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.

Quantum mechanical concepts 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 $C$.

Classical measurement outcomes

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).

Complex phases

The underlying “algebra of quantum amplitudes” of the corresponding quantum mechanical system is the endomorphism monoid of the tensor unit

$$\mathbb{C}_C = End_C(I) \,.$$

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

$$\mathbb{C}_C = \mathbb{C}$$

and moreover

$$\mathbb{R}_C = \mathbb{R} \,.$$

Completely positive maps

The behaviour of quantum channels and completely positive maps has an elegant categorical description in terms of $\dagger$-compact categories. See (Selinger and Coecke).

Quantum logic

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).

Related concepts

• order-theoretic structure in quantum mechanics

• linear logic, linear type theory, quantum logic, quantum computing

• JBW-algebraic quantum mechanics

• tensor network, string diagram

References

The idea that the natural language of quantum mechanics and quantum computation is that of †-compact categories became popular with the publication

• Samson Abramsky, Bob Coecke, A categorical semantics of quantum protocols , Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS’04). IEEE Computer Science Press (2004) (arXiv:quant-ph/0402130)

with an expanded version in

• Samson Abramsky, Bob Coecke, Categorical quantum mechanics, in Handbook of Quantum Logic and Quantum Structures vol II, Elsevier, 2008 (arXiv:0808.1023)

A fairly comprehensive account of the underlying theory of string diagrams is in

• Peter Selinger, Dagger compact closed categories and completely positive maps (web, pdf)

Textbook account of the underlying category theory:

• Chris Heunen, Jamie Vicary, Categories for Quantum Theory, Oxford University Press 2019 (ISBN:9780198739616)

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

• Bob Coecke, Ross Duncan, Interacting Quantum Observables: Categorical Algebra and Diagrammatics, New J. Phys. 13 (2011) 043016 $[$arXiv:0906.4725, doi:10.1088/1367-2630/13/4/043016$]$

The formalization of orthogonal bases in $\dagger$-compact categories is in

• Bob Coecke, Dusko Pavlovic, Jamie Vicary, A new description of orthogonal bases (arXiv:0810.0812)

The role of complex numbers in general $\dagger$-compact categories is discussed in

• Jamie Vicary, Completeness of $\dagger$-categories and the complex numbers (arXiv:0807.2927)

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, Mathematical Structures in Computer Science 16 3 (2006) $[$arXiv:quant-ph/0512114, doi:10.1017/S0960129506005275$]$

• Ross Duncan, Types for quantum mechanics, 2006 (pdf, slides)

An exposition along these lines is in

• John Baez, Mike Stay, Physics, topology, logic and computation: a rosetta stone, arxiv/0903.0340; in “New Structures for Physics”, ed. Bob Coecke, Lecture Notes in Physics 813, Springer, Berlin, 2011, pp. 95-174