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Journal of Mathematical Physics

Arithmetic, mutually unbiased bases and complementary observables

M. D. Sheppeard

Published Feb 1, 2010• 1 citation
Publisher

Abstract

Complementary observables in quantum mechanics may be viewed as Frobenius structures in a dagger monoidal category, such as the category of finite dimensional Hilbert spaces over the complex numbers. On the other hand, their properties crucially depend on the discrete Fourier transform and its associated quantum torus, requiring only the finite fields that underlie mutually unbiased bases. In axiomatic topos theory, the complex numbers are difficult to describe and should not be invoked unnecessarily. This paper surveys some fundamentals of quantum arithmetic using finite field complementary observables, with a view considering more general axiom systems.

DOI
10.1063/1.3271045

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Complementary observables in quantum mechanics may be viewed as Frobenius structures in a dagger monoidal category, such as the category of finite dimensional Hilbert spaces over the complex numbers. On the other hand, their properties crucially depend on the discrete Fourier transform and its associated quantum torus, requiring only the finite fields that underlie mutually unbiased bases. In axiomatic topos theory, the complex numbers are difficult to describe and should not be invoked unnecessarily. This paper surveys some fundamentals of quantum arithmetic using finite field complementary observables, with a view considering more general axiom systems.
Ongoing work in quantum information emphasises the need for a structural understanding of quantum speedups: in this work, we focus on the quantum Fourier transform and the structures in quantum theory that enable it. We elucidate a general connection in any process theory between the Fourier transform and strongly complementary observables, i.e. Hopf algebras in dagger symmetric monoidal categories. We generalise the necessary tools of representation theory from fdHilb to arbitrary dagger symmetric monoidal categories. We define groups, characters and representations, and we prove their relation to strong complementarity. The Fourier transform is then defined in terms of pairs of strongly complementary observables, in both the abelian and non-abelian case. In the abelian case, we draw the connection with Pontryagin duality and provide categorical proofs of the Fourier Inversion Theorem, the Convolution Theory, and Pontryagin duality. Our work finds application in the novel characterisation of the Fourier transform for the category fRel of finite sets and relations. This is a result of interest for the study of categorical quantum algorithms, as the usual construction of the quantum Fourier transform in terms of Fourier matrices is shown to fail in fRel. Despite this, the process theoretic perspective on the Fourier transform is sensible in this setting. Furthermore, our categorical setting provides a generalisation of the abelian Fourier transform from finite-dimensional Hilbert spaces to finite-dimensional modules over arbitrary semirings, as well as a further generalisation to finite non-abelian groups, including a fully categorical generalisation of the Gelfand-Naimark theorem.
A certain class of Frobenius algebras has been used to characterize orthonormal bases and observables on finite-dimensional Hilbert spaces. The presence of units in these algebras means that they can only be realized finite-dimensionally. We seek a suitable generalization, which will allow arbitrary bases and observables to be described within categorical axiomatizations of quantum mechanics. We develop a definition of H*-algebra that can be interpreted in any symmetric monoidal dagger category, reduces to the classical notion from functional analysis in the category of (possibly infinite-dimensional) Hilbert spaces, and hence provides a categorical way to speak about orthonormal bases and quantum observables in arbitrary dimension. Moreover, these algebras reduce to the usual notion of Frobenius algebra in compact categories. We then investigate the relations between nonunital Frobenius algebras and H*-algebras. We give a number of equivalent conditions to characterize when they coincide in the category of Hilbert spaces. We also show that they always coincide in categories of generalized relations and positive matrices.
The basic methods of constructing the sets of mutually unbiased bases in the Hilbert space of an arbitrary finite dimension are reviewed and an emerging link between them is outlined. It is shown that these methods employ a wide range of important mathematical concepts like, e.g., Fourier transforms, Galois fields and rings, finite and related projective geometries, and entanglement, to mention a few. Some applications of the theory to quantum information tasks are also mentioned. KEY WORDS: mutually unbiased bases; d-dimensional Hilbert space; Galois fields and rings; maximally entangled states.
A certain class of Frobenius algebras has been used to characterize orthonormal bases and observables on finite-dimensional Hilbert spaces. The presence of units in these algebras means that they can only be realized finite-dimensionally. We seek a suitable generalization, which will allow arbitrary bases and observables to be described within categorical axiomatizations of quantum mechanics. We develop a definition of H*-algebra that can be interpreted in any symmetric monoidal dagger category, reduces to the classical notion from functional analysis in the category of (possibly infinite-dimensional) Hilbert spaces, and hence provides a categorical way to speak about orthonormal bases and quantum observables in arbitrary dimension. Moreover, these algebras reduce to the usual notion of Frobenius algebra in compact categories. We then investigate the relations between nonunital Frobenius algebras and H*algebras. We give a number of equivalent conditions to characterize when they coincide in the category of Hilbert spaces. We also show that they always coincide in categories of generalized relations and positive matrices.