A new description of orthogonal bases

Bob Coecke, Dusko Pavlovic, Jamie Vicary

    Research output: Contribution to journalArticleAcademicpeer-review

    44 Citations (Scopus)
    35 Downloads (Pure)

    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.
    Original languageEnglish
    Pages (from-to)555-567
    Number of pages13
    JournalMathematical structures in computer science
    Volume23
    Issue number3
    DOIs
    Publication statusPublished - 9 Nov 2012

    Keywords

    • Orthogonal bases

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