Informationally complete measurements on bipartite quantum systems: comparing local with global measurements

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 us 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.