  How well can multiple noncommutative observables be implemented by a single
measurement? This is a fundamental problem in quantum mechanics and determines
the optimal performances of many tasks in quantum information science. While
existing studies have been mostly focusing on the approximation of two
observables with a single measurement, in practice multiple observables are
often encountered, for which the errors of the approximations are little
understood. Here we provide an approach to study the approximation of an
arbitrary finite number of observables with a single measurement. With this
approach, we obtain analytical bounds on the errors of the approximations for
an arbitrary number of observables, which significantly improves our
understanding of a fundamental problem. We also provide a tighter bound in
terms of the semi-definite programming, which, in the case of two observables,
can lead to an analytical bound that is tighter than existing bounds. We then
demonstrate the power of the approach by quantifying the tradeoff of the
precisions for the estimation of multiple parameters in quantum metrology,
which is of both fundamental and practical interest.
