Book contents
- Frontmatter
- Contents
- Contributors
- Preface
- 1 No-Cloning in Categorical Quantum Mechanics
- 2 Classical and Quantum Structuralism
- 3 Generalized Proof-Nets for Compact Categories with Biproducts
- 4 Quantum Lambda Calculus
- 5 The Quantum IO Monad
- 6 Abstract Interpretation Techniques for Quantum Computation
- 7 Extended Measurement Calculus
- 8 Predicate Transformer Semantics of Quantum Programs
- 9 The Structure of Partial Isometries
- 10 Temporal Logics for Reasoning about Quantum Systems
- 11 Specification and Verification of Quantum Protocols
- Index
- References
3 - Generalized Proof-Nets for Compact Categories with Biproducts
Published online by Cambridge University Press: 05 July 2014
By
Book contents
- Frontmatter
- Contents
- Contributors
- Preface
- 1 No-Cloning in Categorical Quantum Mechanics
- 2 Classical and Quantum Structuralism
- 3 Generalized Proof-Nets for Compact Categories with Biproducts
- 4 Quantum Lambda Calculus
- 5 The Quantum IO Monad
- 6 Abstract Interpretation Techniques for Quantum Computation
- 7 Extended Measurement Calculus
- 8 Predicate Transformer Semantics of Quantum Programs
- 9 The Structure of Partial Isometries
- 10 Temporal Logics for Reasoning about Quantum Systems
- 11 Specification and Verification of Quantum Protocols
- Index
- References
Summary
Abstract
Just as conventional functional programs may be understood as proofs in an intuitionistic logic, so quantum processes can also be viewed as proofs in a suitable logic.We describe such a logic, the logic of compact closed categories and biproducts, presented both as a sequent calculus and as a system of proof-nets. This logic captures much of the necessary structure needed to represent quantum processes under classical control, while remaining agnostic to the fine details. We demonstrate how to represent quantum processes as proof-nets and show that the dynamic behavior of a quantum process is captured by the cut-elimination procedure for the logic. We show that the cut-elimination procedure is strongly normalizing: that is, that every legal way of simplifying a proof-net leads to the same, unique, normal form. Finally, taking some initial set of operations as nonlogical axioms, we show that that the resulting category of proof-nets is a representation of the free compact closed category with biproducts generated by those operations.
3.1 Introduction
3.1.1 Logic, Processes, and Categories
Birkhoff and von Neumann initiated the logical study of quantum mechanics in their well-known 1936 paper. They constructed a logic by assigning a proposition letter to each observable property of a given quantum system and studied negations, conjunctions, and disjunctions of these properties. The resulting lattice is nondistributive, and so the heart of what is called “quantum logic” is the study of various kinds of nondistributive lattices.
- Type
- Chapter
- Information
- Semantic Techniques in Quantum Computation , pp. 70 - 134Publisher: Cambridge University PressPrint publication year: 2009
References
, , and (1999) Nuclear and trace ideals in tensored *-categories. Journal of Pure and Applied Algebra 143:3–17.CrossRefGoogle Scholar
, and (2004) A categorical semantics of quantum protocols. In Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science: LICS 2004, pages 415-425. IEEE Computer Society.Google Scholar
, and (2006) A categorical quantum logic. Mathematical Structures in Computer Science 16(3):469–489. Special Issue for the Proceedings of QPL 2004.CrossRefGoogle Scholar
, , and (1996) Interaction categories and the foundations of typed concurrent programming. In , editor, Proceedings of the 1994 Marktoberdorf Summer School on Deductive Program Design, pages 35–113. Springer-Verlag.Google Scholar
, , and (2002) Geometry of interaction and linear combinatory algebras. Mathematical Structures in Computer Science 12:625–665.CrossRefGoogle Scholar
, and (1994) New foundations for the geometry of interaction. Information and Computation 111(1):53–119. Conference version appeared in LiCS '92.CrossRefGoogle Scholar
, and (1995) Higher-dimensional algebra and topological quantum field theory. Journal ofMathematical Physics 36:6073–6105.Google Scholar
(1979) *-Autonomous Categories, volume 752 of Lecture Notes in Mathematics. SpringerVerlag.CrossRefGoogle Scholar
(1991) *-autonomous categories and linear logic. Mathematical Structures in Computer Science 1:159–178.CrossRefGoogle Scholar
, , , , , and (1993) Teleporting an unknown quantum state via dual classical and EPR channels. Physical Review Letters, pages 1895-1899.Google Scholar
, , , and (1996) Natural deduction and coherence for weakly distributive categories. Journal of Pure and Applied Algebra 113:229–296.CrossRefGoogle Scholar
, and (1997) Weakly distributive categories. Journal of Pure and Applied Algebra 113(2):229–296.Google Scholar
(2005a) De-linearizing linearity: Projective quantum axiomatics from strong compact closure. In Proceedings of the 3rd International Workshop on Quantum Programming Languages.Google Scholar
, and (2008) Interacting quantum observables. In , , , , , and , editors, Automata, Languages and Programming, 35th International Colloquium, ICALP 2008, Reykjavik, Iceland, July 7-11, 2008, Proceedings, Part II, volume 5126 of Lecture Notes in Computer Science, pages 298–310. Springer.Google Scholar
, , and (2008a) Bases in diagrammatic quantum protocols. In Proceedings of Mathematical Foundations of Programming Semantics XXIV.Google Scholar
, , and (2008b) A new description of orthogonal bases. Mathematical Structures in Computer Science13pp, to appear, arxiv.org/abs/0810.0812.Google Scholar
, and (1989) The structure of multiplicatives. Archive Mathematical Logic 28(3):181–203.CrossRefGoogle Scholar
(1991) Catégories tannakiennes. In Grothendieck Festschrift, volume 2, pages 111–194. Birkhauser.Google Scholar
, and (2008) A topos foundation for theories of physics I. Formal languages for physics. Journal of Mathematical Physics 49.Google Scholar
(2004) Believe it or not, Bell states are a model of multiplicative linear logic. Technical Report PRG-RR-04-18, Oxford University Computing Laboratory.Google Scholar
(1935) Untersuchungen uber das Schliessen. In , editor, The Collected Papers of Gerhard Gentzen. North-Holland.Google Scholar
(1987b) Multiplicatives. In , editor, Logic and Computer Science: New Trends and Applcations, pages 11–37. Rendiconti del Seminario Mathematico dell'Universita e Politecnico di Torino.Google Scholar
(1996) Proof-nets: the parallel syntax for proof theory. In , and , Logic and Algebra. Volume 180 of Lecture Notes in Pure and Applied Mathematics. Marcel Dekker.Google Scholar
, , and (1992) Normal forms and cut-free proofs as natural transformations. In , editor, Logic from Computer Science, Mathematical Sciences Research Institute Publications 21, pages 217-241. Springer-Verlag.Google Scholar
, , and (2009) A topos for algebraic quantum theory. Communications in Mathematical Physics. To appearGoogle Scholar
(2008) Finite products are biproducts in a compact closed category. Journal of Pure and Applied Algebra 212(2):394–400.CrossRefGoogle Scholar
(1980) The formulaæ-types notion of construction. In , and , editors, To H.B. Curry, Essays on Combinatory Logic, Lambda Calculus and Formalism, pages 479–490. Academic Press.Google Scholar
, and van (2003) Proof nets for unit-free multiplicative-additive linear logic. In Proc. Logic in Computer Science 2003. IEEE.Google Scholar
, and (2003) Gluing and orthogonality for models of linear logic. Theoretical Computer Science 294:183–231.CrossRefGoogle Scholar
(1977) Remarques sur la théories des jeux à deux personnes. Gazette des sciences mathématiques du Québec 1(4).Google Scholar
, and (1991) The geometry of tensor categories i. Advances in Mathematics 88:55-113.CrossRefGoogle Scholar
, , and (1996) Traced monoidal categories. Mathematical Proceedings of the Cambridge Philosophical Society 119:447–468.CrossRefGoogle Scholar
(1972) Many-variable functorial calculus I, volume 281 of Lecture Notes in Mathematics, pages 66–105. Springer.Google Scholar
, and (1980) Coherence for compact closed categories. Journal of Pure and Applied Algebra 19:193–213.CrossRefGoogle Scholar
(1968) Deductive systems and categories I. syntactic calculus and residuated categories. Mathematical Systems Theory 2(4):287–318.Google Scholar
(1969) Deductive systems and categories (ii). In , editor, Category Theory, Homology Theory and their Applications I, volume 87 of Lecture Notes in Mathematics, pages 76–122. Springer.Google Scholar
, and (1986) Introduction to Higher Order Categorical Logic, volume 7 of Cambridge Studies in Advanced Mathematics. Cambridge University Press.Google Scholar
, and Tortora de (2004) Slicing polarized additive normalisation. In , , , and , editors, Linear Logic in Computer Science, volume 316 of London Mathematical Society Lecture Notes, pages 247–282. Cambridge University Press.Google Scholar
(1994) Models of Lambda Calculi and Linear Logic: Structural, Equational and Proof-theoretic Characterisations. Ph.D. thesis, St Hugh's College, Oxford.Google Scholar
(1971) Applications of negative dimensional tensors. In Combinatorial Mathematics and its Applications, pages 221–244. Academic Press.Google Scholar
(2004) Towards a quantum programming language. Mathematical Structures in Computer Science 14(4):527–586.CrossRefGoogle Scholar
(2005) Dagger compact closed categories and completely positive maps. In Proceedings of the 3rd International Workshop on Quantum Programming Languages.Google Scholar
(2001) The Logic of Physical Properties in Static and Dynamic Perspective. Ph.D. thesis, Vrije Universiteit Brussel.Google Scholar
(1987) On natural transformations of distinguished functors and their superpositions in certain closed categories. Journal of Pure and Applied Algebra 47.CrossRefGoogle Scholar
- 3
- Cited by