Mean of continuous variables observable via measurement on a single qubit

It is shown that the mean value of any observable with a bounded spectrum can be uniquely determined from binary statistics of the measurement performed on a single-qubit ancilla coupled to a given system. The corresponding positive operator-valued measure fully encodes the observable structure. The method is generalized to the case of distant-laboratory paradigm and is considered in the context of entanglement detection with few local measurements. The results are also discussed in the context of quantum programming.