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:
and with emphasis on quantum computation:
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)
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 -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
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 -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:
Last revised on August 31, 2024 at 19:25:12. See the history of this page for a list of all contributions to it.