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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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