nLab quantum information theory via string diagrams -- references

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$]$