nLab quantum information

under construction

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) 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, …

Related entries

• quantum logic, quantum theory

• quantum state preparation, quantum measurement

• quantum gate, quantum circuit

• quantum probability, quantum mechanics

• quantum communication

• quantum computation, quantum cryptography

• quantum sensing

• qbit, quantum circuit model

• quantum teleportation

• quantum error correcting code

• information theory, information geometry

• Kochen-Specker theorem

• MIP-star=RE

• entanglement entropy, holographic entanglement entropy

References

General

Textbook accounts:

• Michael A. Nielsen, Isaac L. Chuang, Quantum computation and quantum information, Cambridge University Press (2000) [doi:10.1017/CBO9780511976667, pdf, pdf]

• Stephen Barnett, Quantum Information, Oxford University Press (2009) [ISBN:9780198527633]

• Benjamin Schumacher, Michael Westmoreland, Quantum Processes, Systems, and Information, Cambridge University Press (2010) [doi:10.1017/CBO9780511814006]

• Mark M. Wilde, Quantum Information Theory, Cambridge University Press (2013) [doi:10.1017/CBO9781139525343, arXiv:1106.1445]

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

• Masahito Hayashi, Quantum information theory - mathematical foundation, Graduate Texts in Physics, Springer (2017) [doi:10.1007/978-3-662-49725-8]

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

• John Watrous, The Theory of Quantum Information, Cambridge University Press (2018) [doi:10.1017/9781316848142, webpage, pdf]

• Joseph M. Renes, Quantum Information Theory (2015) [pdf] De Gruyter (2022) [doi:10.1515/9783110570250]

• Garth Warner: Positivity, University of Washington [pdf, pdf]

Book collection:

• Dirk Bouwmeester, Artur Ekert, Anton Zeilinger (eds.), The Physics of Quantum Information – Quantum Cryptography, Quantum Teleportation, Quantum Computation, Springer (2020) [doi:10.1007/978-3-662-04209-0]

Lecture notes:

• Greg Kuperberg, A concise introduction to quantum probability, quantum mechanics, and quantum computation (2005) [pdf, pdf]

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

• Edward Witten, A Mini-Introduction To Information Theory, Rivista Nuovo Cimento 43 (2020) 187–227 [arXiv:1805.11965, doi:10.1007/s40766-020-00004-5]

• Scott Aaronson, Introduction to Quantum Information Science (2018) [pdf, webpage]

  Introduction to Quantum Information Science II (2022) [pdf]

See also:

• IQOQI Vienna, List of Open Quantum Problems

With continuous variables:

• Samuel L. Braunstein, Peter van Loock, Quantum information with continuous variables, Rev. Mod. Phys. 77 2 (2005) 513 [arXiv:quant-ph/0410100, doi:10.1103/RevModPhys.77.513]

• Samuel L. Braunstein, Arun K. Pati, Quantum Information with Continuous Variables, Springer (2003) [doi:10.1007/978-94-015-1258-9]

In a context of quantum optics:

• Tim Byrnes, Ebubechukwu Ilo-Okeke, Section 11 of: Quantum Atom Optics: Theory and Applications to Quantum Technology, Cambridge University Press 2021 (arXiv:2007.14601, CUP)

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:

• Wikipedia, Quantum information

• Wikipedia, Bures metric

• Quantiki – Quantum Information Portal and Wiki

• Report of the Snowmass 2021 Theory Frontier Topical Group on Quantum Information Science [arXiv:2209.14839]

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:

• Alonso Botero, Quantum Information and the Representation Theory of the Symmetric Group, Rev. colomb. mat. vol.50 no. 2 Bogotá July/Dec. 2016 (doi:10.15446/recolma.v50n2.62210, pdf)

In relation to topological phases of matter:

• Bei Zeng, Xie Chen, Duan-Lu Zhou, Xiao-Gang Wen:

  Quantum Information Meets Quantum Matter – From Quantum Entanglement to Topological Phases of Many-Body Systems, Quantum Science and Technology (QST), Springer (2019) $[$arXiv:1508.02595, doi:10.1007/978-1-4939-9084-9$]$

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

• Bowen Chen, Bartlomiej Czech, Zi-zhi Wang, Quantum Information in Holographic Duality, Rept. Prog. Phys. 85 (2022) 4, 046001 [arXiv:2108.09188, doi:10.1088/1361-6633/ac51b5]

Quantum resources

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

• Charles H. Bennett, A resource-based view of quantum information, Quantum Information & Computation 4 6 (2004) 460–466 [doi:10.5555/2011593.2011598]

• Igor Devetak; Aram W. Harrow; Andreas J. Winter, A Resource Framework for Quantum Shannon Theory, IEEE Transactions on Information Theory 54 10 (2008) [doi:10.1109/TIT.2008.928980]

• Eric Chitambar, Gilad Gour, Quantum Resource Theories [arXiv:1806.06107]

Quantum information theory via String diagrams

General

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):

• 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, doi:10.1109/LICS.2004.1319636$]$

• Samson Abramsky, Bob Coecke, Abstract Physical Traces, Theory and Applications of Categories, 14 6 (2005) 111-124. [tac:14-06, arXiv:0910.3144]

• Samson Abramsky, Bob Coecke, Categorical quantum mechanics, in Handbook of Quantum Logic and Quantum Structures, Elsevier (2008) $[$arXiv:0808.1023, ISBN:9780080931661, doi:10.1109/LICS.2004.1319636$]$

• Bob Coecke, De-linearizing Linearity: Projective Quantum Axiomatics from Strong Compact Closure, Proceedings of the 3rd International Workshop on Quantum Programming Languages (2005), Electronic Notes in Theoretical Computer Science 170 (2007) 49-72 [doi:10.1016/j.entcs.2006.12.011, arXiv:quant-ph/0506134]

On the relation to quantum logic/linear logic:

• 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$]$

Early exposition with introduction to monoidal category theory:

• Bob Coecke, Kindergarten quantum mechanics $[$arXiv:quant-ph/0510032$]$

• Bob Coecke, Introducing categories to the practicing physicist $[$arXiv:0808.1032$]$

• John Baez, Mike Stay, Physics, topology, logic and computation: a rosetta stone in: New Structures for Physics, Bob Coecke (ed.), Lecture Notes in Physics 813, Springer (2011) 95-174 $[$arxiv/0903.0340$]$

• Bob Coecke, Eric Oliver Paquette, Categories for the practising physicist, in: New Structures for Physics, Lecture Notes in Physics 813, Springer (2010) $[$arXiv:0905.3010, doi:10.1007/978-3-642-12821-9_3$]$

• Bob Coecke, Quantum Picturalism, Contemporary Physics 51 1 (2010) $[$arXiv:0908.1787, doi:10.1080/00107510903257624$]$

Review in contrast to quantum logic:

• Samson Abramsky, Bob Coecke, Physics from Computer Science: a Position Statement, International Journal of Unconventional Computing 3 3 (2007) $[$pdf, ijuc-3-3-p-179-197$]$

and with emphasis on quantum computation:

• Jamie Vicary, The Topology of Quantum Algorithms, (LICS 2013) Proceedings of 28th Annual ACM/IEEE Symposium on Logic in Computer Science (2013) 93-102 $[$arXiv:1209.3917, doi:10.1109/LICS.2013.14$]$

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

• Peter Selinger, Dagger compact closed categories and completely positive maps, Electronic Notes in Theoretical Computer Science 170 (2007) 139-163 $[$doi:10.1016/j.entcs.2006.12.018, web, pdf$]$

• Bob Coecke, Chris Heunen, Pictures of complete positivity in arbitrary dimension, Information and Computation 250 50-58 (2016) $[$arXiv:1110.3055, doi:10.1016/j.ic.2016.02.007$]$

• Bob Coecke, Chris Heunen, Aleks Kissinger,

  Categories of Quantum and Classical Channels, EPTCS 158 (2014) 1-14 $[$arXiv:1408.0049, doi:10.4204/EPTCS.158.1$]$

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

• Bob Coecke, Aleks Kissinger, Picturing Quantum Processes – A First Course in Quantum Theory and Diagrammatic Reasoning, Cambridge University Press (2017) $[$ISBN:9781107104228$]$

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

  based on:

  Chris Heunen, Jamie Vicary, Lectures on categorical quantum mechanics (2012) [pdf, pdf]

• Bob Coecke, Stefano Gogioso, Quantum in Pictures, Quantinuum Publications (2023) $[$ISBN 978-1739214715, Quantinuum blog$]$

  (focus on ZX-calculus)

Measurement & Classical structures

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

• Bob Coecke, Duško Pavlović, Quantum measurements without sums, in Louis Kauffman, Samuel Lomonaco (eds.), Mathematics of Quantum Computation and Quantum Technology, Taylor & Francis (2008) 559-596 $[$arXiv:quant-ph/0608035, doi:10.1201/9781584889007$]$

• Bob Coecke, Eric Oliver Paquette, POVMs and Naimark’s theorem without sums, Electronic Notes in Theoretical Computer Science 210 (2008) 15-31 $[$arXiv:quant-ph/0608072, doi:10.1016/j.entcs.2008.04.015$]$

• Bob Coecke, Eric Oliver Paquette, Duško Pavlović, Classical and quantum structuralism, in: Semantic Techniques in Quantum Computation, Cambridge University Press (2009) 29-69 $[$arXiv:0904.1997, doi:10.1017/CBO9781139193313.003$]$

• Bob Coecke, Duško Pavlović, Jamie Vicary, A new description of orthogonal bases, Mathematical Structures in Computer Science 23 3 (2012) 555- 567 $[$arXiv:0810.0812, doi:10.1017/S0960129512000047$]$

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:

• Bob Coecke, Ross Duncan, §3 in: Interacting Quantum Observables, in Automata, Languages and Programming. ICALP 2008, Lecture Notes in Computer Science 5126, Springer (2008) $[$doi:10.1007/978-3-540-70583-3_25$]$

• Aleks Kissinger, §§2 in: Graph Rewrite Systems for Classical Structures in $\dagger$-Symmetric Monoidal Categories, MSc thesis, Oxford (2008) $[$pdf, pdf$]$

• Aleks Kissinger, §4 in: Exploring a Quantum Theory with Graph Rewriting and Computer Algebra, in: Intelligent Computer Mathematics. CICM 2009, Lecture Notes in Computer Science 5625 (2009) 90-105 $[$doi:10.1007/978-3-642-02614-0_12$]$

• Bob Coecke, Ross Duncan, Def. 6.4 in: Interacting Quantum Observables: Categorical Algebra and Diagrammatics, New J. Phys. 13 (2011) 043016 $[$arXiv:0906.4725, doi:10.1088/1367-2630/13/4/043016$]$

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:

• Bob Coecke, Ross Duncan, §3 in: Interacting Quantum Observables, in Automata, Languages and Programming. ICALP 2008, Lecture Notes in Computer Science 5126, Springer (2008) $[$doi:10.1007/978-3-540-70583-3_25$]$

• Aleks Kissinger, Graph Rewrite Systems for Classical Structures in $\dagger$-Symmetric Monoidal Categories, MSc thesis, Oxford (2008) $[$pdf, pdf$]$

• Aleks Kissinger, Exploring a Quantum Theory with Graph Rewriting and Computer Algebra, in: Intelligent Computer Mathematics. CICM 2009, Lecture Notes in Computer Science 5625 (2009) 90-105 $[$doi:10.1007/978-3-642-02614-0_12$]$

• 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$]$

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

• Fatimah Rita Ahmadi, Aleks Kissinger, Topological Quantum Computation Through the Lens of Categorical Quantum Mechanics $[$arXiv:2211.03855$]$