Heisenberg uncertainty for qubit measurements

Reports on experiments recently performed in Vienna [Erhard et al., Nature Phys. 8, 185 (2012), 10.1038/nphys2194] and Toronto [Rozema et al., Phys. Rev. Lett. 109, 100404 (2012), 10.1103/PhysRevLett.109.100404] include claims of a violation of Heisenberg's error-disturbance relation. In contrast, we have presented and proven a Heisenberg-type relation for joint measurements of position and momentum [Phys. Rev. Lett. 111, 160405 (2013), 10.1103/PhysRevLett.111.160405]. To resolve the apparent conflict, we formulate here a general trade-off relation for errors in qubit measurements, using the same concepts as we did in the position-momentum case. We show that the combined errors in an approximate joint measurement of a pair of ±1-valued observables A ,B are tightly bounded from below by a quantity that measures the degree of incompatibility of A and B. The claim of a violation of Heisenberg is shown to fail because it is based on unsuitable measures of error and disturbance. Finally we show how the experiments mentioned may directly be used to test our error inequality.