nLab quantum information

Surveys, textbooks and lecture notes

under construction

Contents

Idea

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…

Categorical 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) to a discussion of quantum information.

The linear adjoint $\left(-{\right)}^{†}$ 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 $\left(-{\right)}^{†}$ from ${\mathrm{Hilb}}^{\mathrm{op}}$ to $\mathrm{Hilb}$. Furthermore, the category FHilb of finite dimensional Hilbert spaces forms a †-compact closed category, or a compact closed category such that ${A}_{*}$ := $\left({A}^{*}{\right)}^{†}=\left({A}^{†}{\right)}^{*}$ and $\left({\eta }_{A}{\right)}^{†}={ϵ}_{{A}^{*}}$.

Graphical notation

Aleks Kissinger: Much of this could probably be incorporated into the page on string diagrams.

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:

(1)$\left({f}_{1}\otimes {f}_{2}\right)\circ \left({g}_{1}\otimes {g}_{2}\right)=\left({f}_{1}\circ {g}_{1}\right)\otimes \left({f}_{2}\circ {g}_{2}\right)$(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 $A,B$ as directed strings and arrows $f:A\to B$ as boxes.

We represent the tensor product as juxtaposition:

and composition as graph composition:

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.

(2)$\left(f\otimes {1}_{D}\right)\circ \left({1}_{A}\otimes g\right)=\left(f\circ {1}_{A}\right)\otimes \left({1}_{D}\circ g\right)=\left({1}_{B}\circ f\right)\otimes \left(g\circ {1}_{C}\right)=\left({1}_{B}\otimes g\right)\circ \left(f\otimes {1}_{C}\right)$(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.

Extensions

CPM, classical structures, …

Revised on April 19, 2013 01:07:03 by Urs Schreiber (82.169.65.155)