Mutually Unbiased Bases for Continuous Variables

Abstract

The concept of mutually unbiased bases is studied for N pairs of continuous variables. To find mutually unbiased bases reduces, for specific states related to the Heisenberg-Weyl group, to a problem of symplectic geometry. Given a single pair of continuous variables, three mutually unbiased bases are identified while five such bases are exhibited for two pairs of continuous variables. For N = 2, the golden ratio occurs in the definition of these mutually unbiased bases suggesting the relevance of number theory not only in the finite-dimensional setting.

5 pages, no figures, revised to be identical to published text

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• Periodic discretized continuous observables are neither continuous nor discrete
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• Discrete phase-space structures and Wigner functions for $N$ qubits