Tomographically complete sets of orthonormal bases in finite systems
Mohamed Ahmed Shalaby, Apostol Vourdas · Journal of Physics A Mathematical and Theoretical · 2011
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.