nLab quantum information



Quantum systems

quantum logic

quantum physics

quantum probability theoryobservables and states

quantum information

quantum computation


quantum algorithms:

quantum sensing

quantum communication


under construction



Quantum information refers to data that can be physically stored in a quantum system.

Quantum information theory is the study of how such information can be encoded, measured, and manipulated. A notable sub-field is quantum computation, a term often used synonymously with quantum information theory, which studies protocols and algorithms that use quantum systems to perform computations.

Categorical quantum information refers to a program in which the cogent aspects of Hilbert space-based quantum information theory are abstracted to the level of symmetric monoidal categories.

Quantum protocols and algorithms

Brief synopsis of teleportation, entanglement swapping, BB84, E91, Deutsch-Jozsa, Shor should go here…

Category-theoretic formulation

There is a formulation of (aspects of) quantum mechanics in terms of dagger-compact categories. This lends itself to (and is in fact motivated by) a discussion of quantum information.

The linear adjoint () (-)^\dagger gives Hilbert spaces the structure of a †-category. The category of Hilb of Hilbert spaces forms a †-symmetric monoidal category, that is, a symmetric monoidal category equipped with a symmetric monoidal functor () (-)^\dagger from Hilb opHilb^{op} to HilbHilb. Furthermore, the category FHilb of finite dimensional Hilbert spaces forms a †-compact closed category, or a compact closed category such that A *A_* := (A *) =(A ) *(A^*)^\dagger = (A^\dagger)^* and (η A) =ϵ A *(\eta_A)^\dagger = \epsilon_{A^*}.

Graphical notation

Graphical notation via Penrose notation/string diagrams/tensor networks:

Morphisms in a monoidal category (and 2-categories in general) are inherently two dimensional, where \circ is vertical composition and \otimes is horizontal composition. These satisfy an interchange law:

(f 1f 2)(g 1g 2)=(f 1g 1)(f 2g 2) (f_1 \otimes f_2) \circ (g_1 \otimes g_2) = (f_1 \circ g_1) \otimes (f_2 \circ g_2)

So, if we think of these four morphisms as occupying a spot in 2 dimensional space:

Aleks Kissinger: TODO: figure

we realize that the bracketing from above is essentially meaningless syntax. This notion is the guiding concept for the graphical notation of monoidal categories, or string diagrams. In this notation, we represent objects A,BA,B as directed strings and arrows f:ABf : A \rightarrow B as boxes.

A B f

We represent the tensor product as juxtaposition:

A B f C D g A B f C D g =

and composition as graph composition:

A B f B C g = A B C g f

That is, we perform a pushout along the common edge in the category of typed graphs with boundaries. Consider the interchange law from above, but replacing some of the arrows with identities.

(f1 D)(1 Ag)=(f1 A)(1 Dg)=(1 Bf)(g1 C)=(1 Bg)(f1 C) (f \otimes 1_D) \circ (1_A \otimes g) = (f \circ 1_A) \otimes (1_D \circ g) = (1_B \circ f) \otimes (g \circ 1_C) = (1_B \otimes g) \circ (f \otimes 1_C)

Graphically, this means we can “slide boxes” past each other.

A B D g f C = A B D g f C


CPM, classical structures, …



Textbook accounts:

Book collection:

Lecture notes:

See also:

With continuous variables:

In a context of quantum optics:

Status update:

  • John Preskill, The Physics of Quantum Information, talk at

    The Physics of Quantum Information, 28th Solvay Conference on Physics (2022) (arXiv:2208.08064)

See also:

Further original articles:

  • Carmen Maria Constantin, Sheaf-theoretic methods in quantum mechanics and quantum information theory, PhD thesis, Oxford 2015 arxiv/1510.02561

  • Samson Abramsky, Adam Brandenburger, The sheaf-theoretic structure of nonlocality and contextuality, arxiv/1102.0264

  • Dominik Šafránek, Simple expression for the quantum Fisher information matrix), Phys. Rev. A97 (2018) doi

  • Roman Orus, Entanglement, quantum phase transitions and quantum algorithms (arXiv:quant-ph/0608013)

In Chapter 1 we consider the irreversibility of renormalization group flows from a quantum information perspective by using majorization theory and conformal field theory.

Quantum information in relation to the representation theory of the symmetric group:

In relation to topological phases of matter:

In relation to the AdS-CFT correspondence via holographic entanglement entropy:

Quantum resources

On (entangled) quantum states as resources, not unlike the idea of resources in linear logic:

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:

Last revised on September 22, 2023 at 13:08:54. See the history of this page for a list of all contributions to it.