Measuring average of non-Hermitian operator with weak value in a Mach-Zehnder interferometer

Quantum theory allows direct measurement of the average of a non-Hermitian operator using the weak value of the positive semidefinite part of the non-Hermitian operator. Here, we experimentally demonstrate the measurement of the weak value and the average value of non-Hermitian operator by an interferometric technique. Our scheme is unique as we can directly obtain the weak value from the interference visibility and the phase shift in a Mach-Zehnder interferometer without using any weak measurement or postselection. Both the experiments discussed here were performed with laser sources, but the results would be the same with average statistics of single-photon experiments. Thus, the present experiment opens up the possibility of measuring the weak value and the average value of a non-Hermitian operator without weak interaction and postselection, which can have several technological applications.