Mutually Unbiased Bases for Continuous Variables

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. PACS: 03.65.-w,03.67.-a,03.65.Ta Mutually unbiased (MU) bases of Hilbert spaces with finite dimension d (as defined by Eq. (1) below) are a useful tool. If you want to experimentally determine the state of a quantum system, given only a limited supply of identical copies, the optimal strategy is to perform measurements with respect to MU bases [1]. To pass a secret message to a second party, you could use quantum cryptography to establish a shared key, a procedure which relies on MU bases in the space C [2, 3] or C [4]. Sending a physical system carrying a spin through a noisy environment, the effect of the interactions on the state of the spin might be modelled by a specific quantum channel, conveniently described in terms of MU bases [5]. Finally, if you happen to be captured by a mean king, you might be able to meet his challenge by knowing about entangled states and MU bases [6]. Many of the ideas which underlie physical concepts defined for discrete variables, that is, in a Hilbert space of finite dimension, survive the transition from spin operators to position and momentum operators. Quantum key distribution [7] and quantum teleportation [8], for example, possess counterparts for continuous variables [9]