# nLab quantum information

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…

## 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) to 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 $Hilb^{op}$ to $Hilb$. Furthermore, the category FHilb of finite dimensional Hilbert spaces forms a †-compact closed category, or a compact closed category such that $A_*$ := $(A^*)^\dagger = (A^\dagger)^*$ and $(\eta_A)^\dagger = \epsilon_{A^*}$.

### 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:

$(f_1 \otimes f_2) \circ (g_1 \otimes g_2) = (f_1 \circ g_1) \otimes (f_2 \circ g_2)$

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 $f : A \rightarrow B$ 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.

$(f \otimes 1_D) \circ (1_A \otimes g) = (f \circ 1_A) \otimes (1_D \circ g) = (1_B \circ f) \otimes (g \circ 1_C) = (1_B \otimes g) \circ (f \otimes 1_C)$

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

## Extensions

CPM, classical structures, …

## References

Textbook accounts:

• Masahito Hayashi, Quantum information theory - mathematical foundation Graduate Texts in Physics (2017)

• Nielsen, Chuang, Quantum computation and quantum information, Cambridge Univeristy Press, ch1 pdf

• Sumeet Khatri, Mark M. Wilde, Principles of Quantum Communication Theory: A Modern Approach (arXiv:2011.04672)

• Giuliano Benenti, Giulio Casati, Davide Rossini, Principles of Quantum Computation and Information, World Scientific 2018 (doi:10.1142/10909, 2004 pdf)

In a context of quantum optics:

Lecture notes:

• Reinhard Werner, Mathematical methods of quantum information theory, 18 lecture course (2017) video playlist yt

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 the AdS-CFT correspondence via holographic entanglement entropy:

• Bowen Chen, Bartlomiej Czech, Zi-zhi Wang, Quantum Information in Holographic Duality (arXiv:2108.09188)