nLab quantum information theory via dagger-compact categories




physics, mathematical physics, philosophy of physics

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theory (physics), model (physics)

experiment, measurement, computable physics

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Central aspects of quantum mechanics with finite-dimensional spaces of quantum states — such as tensor products of qbit states of relevance in quantum information theory and quantum computation — follow from the formal properties of the dagger-compact category of finite-dimensional Hilbert spaces.

Conversely, much of quantum information theory and quantum computation can thus be formulated internal to any dagger-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 provides an intuitive and powerful tool for reasoning in \dagger-compact categories.

Quantum mechanical concepts in \dagger-compact categories

Let (C,,I,)(C,\otimes,I, \dagger) be a †-compact category.

We list various concepts in quantum mechanics and their corresponding incarnation in terms of structures in CC.

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

C=End C(I). \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 C\mathbb{C}_C has the properties that

  • it is a field of characteristic 0 with involution \dagger;

  • the subfield C\mathbb{R}_C fixed under \dagger is orderable.

If furthermore every bounded sequence of measurements in CC with values in C\mathbb{R}_C has a least upper bound, then it follows that this field coincides with the complex numbers

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

and moreover

C=. \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).


Quantum information theory via String diagrams


The observation that a natural language for quantum information theory and quantum computation, specifically for quantum circuit diagrams, is that of string diagrams in †-compact categories (see quantum information theory via dagger-compact categories):

On the relation to quantum logic/linear logic:

Early exposition with introduction to monoidal category theory:

Review in contrast to quantum logic:

and with emphasis on quantum computation:

Generalization to quantum operations on mixed states (completely positive maps of density matrices):

Textbook accounts (with background on relevant monoidal category theory):

Measurement & Classical structures

Formalization of quantum measurement via Frobenius algebra-structures (“classical structures”):

and the evolution of the “classical structures”-monad into the “spider”-diagrams (terminology for special Frobenius normal form, originating in Coecke & Paquette 2008, p. 6, Coecke & Duncan 2008, Thm. 1) of the ZX-calculus:


Evolution of the “classical structures”-Frobenius algebra (above) into the “spider”-ingredient of the ZX-calculus for specific control of quantum circuit-diagrams:

Relating the ZX-calculus to braided fusion categories for anyon braiding:

The role of complex numbers in general \dagger-compact categories:

Last revised on September 1, 2023 at 09:51:48. See the history of this page for a list of all contributions to it.