Informationally complete measurements allow the estimation of expectation
values of any operator on a quantum system, by changing only the
data-processing of the measurement outcomes. In particular, an informationally
complete measurement can be used to perform quantum tomography, namely to
estimate the density matrix of the quantum state. The data-processing is
generally nonunique, and can be optimized according to a given criterion. In
this paper we provide the solution of the optimization problem which minimizes
the variance in the estimation. We then consider informationally complete
measurements performed over bipartite quantum systems focusing attention on
universally covariant measurements, and compare their statistical efficiency
when performed either locally or globally on the two systems. Among global
measurements we consider the special case of Bell measurements, which allow to
estimate the expectation of a restricted class of operators. We compare the
variance in the three cases: local, Bell, and unrestricted global--and derive
conditions for the operators to be estimated such that one type of measurement
is more efficient than the other. In particular, we find that for factorized
operators and Bell projectors the Bell measurement always performs better than
the unrestricted global measurement, which in turn outperforms the local one.
For estimation of the matrix elements of the density operator, the relative
performances depend on the basis on which the state is represented, and on the
matrix element being diagonal or off-diagonal, however, with the global
unrestricted measurement generally performing better than the local one.Comment: 8 pages, no figure
