Quantum Walk as a Generalized Measuring Device

We show that a one-dimensional discrete time quantum walk can be used to implement a generalized measurement in terms of a positive operator value measure (POVM) on a single qubit. More precisely, we show that for a single qubit any set of rank 1 and rank 2 POVM elements can be generated by a properly engineered quantum walk. In such a scenario the measurement of a particle at a position x = i corresponds to a measurement of a POVM element E-i on a qubit. Since the idea of quantum walks originates from the von Neumann model of measurement, in which the change of the position of the pointer depends on the state of the system that is being measured, we argue that von Neumann measurements can be naturally extended to POVMs if one includes the internal evolution of the system in the model.