Direct quantum process tomography via measuring sequential weak values of incompatible observables

The weak value concept has enabled fundamental studies of quantum measurement and, recently, found potential applications in quantum and classical metrology. However, most weak value experiments reported to date do not require quantum mechanical descriptions, as they only exploit the classical wave nature of the physical systems. In this work, we demonstrate measurement of the sequential weak value of two incompatible observables by making use of two-photon quantum interference so that the results can only be explained quantum physically. We then demonstrate that the sequential weak value measurement can be used to perform direct quantum process tomography of a qubit channel. Our work not only demonstrates the quantum nature of weak values but also presents potential new applications of weak values in analyzing quantum channels and operations.