Heisenberg's error-disturbance relations: A joint measurement-based experimental test

The original Heisenberg error-disturbance relation was recently shown to be not universally valid and two different approaches to reformulate it were proposed. The first one focuses on how the error and disturbance of two observables A and B depend on a particular quantum state. The second one asks how a joint measurement of A and B affects their eigenstates. Previous experiments focused on the first approach. Here we focus on the second one. First, we propose and implement an extendible method of quantum-walk-based joint measurements of noisy Pauli operators to test the error-disturbance relation for qubits introduced in the work of Busch et al. [Phys. Rev. A 89, 012129 (2014)], where the polarization of the single photon, corresponding to a walker's auxiliary degree of freedom that is commonly known as a coin, undergoes a position-and time-dependent evolution. Then we formulate and experimentally test a universally valid state-dependent relation for three mutually unbiased observables. We therefore establish a method of testing error-disturbance relations.