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A new description of orthogonal bases

Published online by Cambridge University Press:  09 November 2012

BOB COECKE ,
DUSKO PAVLOVIC  and
JAMIE VICARY
BOB COECKE
Affiliation:
Oxford University Computing Laboratory, Department of Computer Science, University of Oxford, Parks Road, Oxford OX 1 3QD, United Kingdom Email: bob.coecke@cs.ox.ac.uk; Dusko.Pavlovic@rhul.ac.uk; jamie.vicary@cs.ox.ac.uk
DUSKO PAVLOVIC
Affiliation:
Oxford University Computing Laboratory, Department of Computer Science, University of Oxford, Parks Road, Oxford OX 1 3QD, United Kingdom Email: bob.coecke@cs.ox.ac.uk; Dusko.Pavlovic@rhul.ac.uk; jamie.vicary@cs.ox.ac.uk
JAMIE VICARY
Affiliation:
Oxford University Computing Laboratory, Department of Computer Science, University of Oxford, Parks Road, Oxford OX 1 3QD, United Kingdom Email: bob.coecke@cs.ox.ac.uk; Dusko.Pavlovic@rhul.ac.uk; jamie.vicary@cs.ox.ac.uk
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Abstract

We show that an orthogonal basis for a finite-dimensional Hilbert space can be equivalently characterised as a commutative †-Frobenius monoid in the category FdHilb, which has finite-dimensional Hilbert spaces as objects and continuous linear maps as morphisms, and tensor product for the monoidal structure. The basis is normalised exactly when the corresponding commutative †-Frobenius monoid is special. Hence, both orthogonal and orthonormal bases are characterised without mentioning vectors, but just in terms of the categorical structure: composition of operations, tensor product and the †-functor. Moreover, this characterisation can be interpreted operationally, since the †-Frobenius structure allows the cloning and deletion of basis vectors. That is, we capture the basis vectors by relying on their ability to be cloned and deleted. Since this ability distinguishes classical data from quantum data, our result has important implications for categorical quantum mechanics.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 23 , Issue 3 , June 2013 , pp. 555 - 567
Copyright
Copyright © Cambridge University Press 2012

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References

Abramsky, S. and Coecke, B. (2004) A categorical semantics of quantum protocols. In: Proceedings of 19th IEEE conference on Logic in Computer Science 415–425. (Available at arXiv:quant-ph/0402130 & arXiv:0808.1023.)CrossRefGoogle Scholar
Aguiar, M. (2000) A note on strongly separable algebras. In: Special issue in honor of Orlando Villamayor. Boletín de la Academia Nacional de Ciencias (Córdoba, Argentina) 65 51–60.Google Scholar
Carboni, A. and Walters, R. F. C. (1987) Cartesian bicategories I. Journal of Pure and Applied Algebra 49 11–32.CrossRefGoogle Scholar
Coecke, B. and Duncan, R. W. (2008) Interacting quantum observables In: Proceedings of the 35th International Colloquium on Automata, Languages and Programming. Springer-Verlag Lecture Notes in Computer Science 5126 298–310.CrossRefGoogle Scholar
Coecke, B. and Pavlovic, D. (2007) Quantum measurements without sums. In: Chen, G., Kauffman, L. and Lamonaco, S. (eds.) Mathematics of Quantum Computing and Technology, Taylor and Francis 567–604. (Available at arXiv:quant-ph/0608035.)Google Scholar
Coecke, B., Paquette, E. O. and Pavlovic, D. (2008) Classical and quantum structuralism. In: Mackie, I. and Gay, S. (eds.) Semantic Techniques for Quantum Computation, Cambridge University Press 29–69. (Available at arXiv:0904.1997.)Google Scholar
Dieks, D. G. B. J. (1982) Communication by EPR devices. Physics Letters A 92 271–272.CrossRefGoogle Scholar
Joyal, A. and Street, R. (1991) The geometry of tensor calculus I. Advances in Mathematics 88 55–112.CrossRefGoogle Scholar
Kelly, G. M. (1974) On clubs and doctrines. Springer-Verlag Lecture Notes in Mathematics 420 181–256.CrossRefGoogle Scholar
Kock, J. (2004) Frobenius Algebras and 2D Topological Quantum Field Theories, Cambridge University Press.Google Scholar
Lawvere, F. W. (1969) Ordinal sums and equational doctrines. In: Seminar on Triples and Categorical Homology Theory. Springer-Verlag Lecture Notes in Mathematics 80 141–155.CrossRefGoogle Scholar
Murphy, G. J. (1990) C*-Algebras and Operator Theory, Academic Press.Google Scholar
Pati, A. K. and Braunstein, S. L. (2000) Impossibility of deleting an unknown quantum state. Nature 404 164–165. (Available at arXiv:quant-ph/9911090.)CrossRefGoogle ScholarPubMed
Penrose, R. (2005) The Road to Reality: A Complete Guide to the Laws of the Universe, Alfred A. Knopf (New York).Google Scholar
Selinger, P. (2007) Dagger compact categories and completely positive maps. Electronic Notes in Theoretical Computer Science 170 139–163.CrossRefGoogle Scholar
Selinger, P. (2010) A survey of graphical languages for monoidal categories. In: Coecke, B. (ed.) New Structures for Physics. Springer-Verlag Lecture Notes in Physics 813 289–356. (Available at arXiv:0908.3347.)CrossRefGoogle Scholar
Vicary, J. (2008) Categorical formulation of finite-dimensional quantum algebras. (Available at arXiv:0805.0432.)Google Scholar
Wootters, W. and Zurek, W. (1982) A single quantum cannot be cloned. Nature 299 802–803.CrossRefGoogle Scholar