Weak value advantage in overcoming noise on the primary system

Abstract: The weak value exhibits numerous intriguing characteristics, such as values outside the operator spectrum, leading to unexpected phenomena. The measurement protocol used for measuring the weak value has been the subject of an on-going controversy. In particular, the possibility of gaining a metrological advantage using weak measurements was questioned. A rigorous characterization of this advantage is still missing when the primary system is noisy. We thus consider here the challenge of learning an unknown operator under the influence of noise on the primary system. For unital noise channels, we prove that the weak value measurement protocol (WVMP) is quadratically more robust to noise than strong measurements. Since the WVMP makes use both of weak entanglement as well as postselection, one might suspect that the advantage is solely due to the postselection aspect of the WVMP. We refute this by showing that for the amplitude and phase damping noise channel, the WVMP achieves a quadratic advantage even over strong measurement protocols which are allowed to apply postselection. By this we rigorously prove that in certain cases, the WVMP possesses a strict, provable advantage in robustness to noise.