Journal of Physics A Mathematical and Theoretical

Tomographically complete sets of orthonormal bases in finite systems

Mohamed Ahmed Shalaby, Apostol Vourdas

Published Aug 2, 2011• 11 citations

Abstract

Quantum systems where the position and momentum are in the ring ( d is an odd integer) are considered. A tomographically complete set of bases, i.e. a set of bases such that probabilities from tomography experiments corresponding to these bases, can be used for the calculation of an arbitrary density matrix, is considered. Such a set of bases is constructed by considering a factorization of a system with dimension d = p 1 p 2 (where p 1 , p 2 are prime numbers) in terms of two subsystems with dimensions p 1 and p 2 . Appropriate combination of the mutually unbiased bases in the subsystems, leads to a tomographically complete set of bases in the full system. This set leads to probabilities along all the lines through the origin (0, 0) in the phase space, with exactly d points. It is shown that the number of these lines is ψ( d ) (the Dedekind function). The general theory is exemplified with examples.

DOI: 10.1088/1751-8113/44/34/345303

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