Anomalous weak values and contextuality: robustness‚ tightness‚ and imaginary parts

Abstract

It has been shown that observations of weak values outside the eigenvalue range of the corresponding operator defy classical explanation in the sense of requiring contextuality [M. F. Pusey, Phys. Rev. Lett. 113, 200401, arXiv:1409.1535]. Here we elaborate on and extend that result in several directions. Firstly, we provide “robust” extensions that account for the failure of realistic postselections to be exactly projective and also allow for the weak measurement to be carried out with a qubit pointer in place of the traditional continuous system. Secondly, we slightly tighten the relevant noncontextuality inequalities and show that (a) no single inequality is tighter and (b) all the operational constraints required by the argument are indeed necessary – if any one of them is dropped, the anomaly can be reproduced classically. Finally, whereas the original result required the real part of the weak value to be anomalous, we also give a version for any weak value with an imaginary part. In short, we provide tight inequalities for witnessing nonclassicality using experimentally realistic measurements of any weak value that can be considered anomalous and clarify the debate surrounding them.It has been shown that observations of weak values outside the eigenvalue range of the corresponding operator defy classical explanation in the sense of requiring contextuality [M. F. Pusey, Phys. Rev. Lett. 113, 200401, arXiv:1409.1535]. Here we elaborate on and extend that result in several directions. Firstly, we provide “robust” extensions that account for the failure of realistic postselections to be exactly projective and also allow for the weak measurement to be carried out with a qubit pointer in place of the traditional continuous system. Secondly, we slightly tighten the relevant noncontextuality inequalities and show that (a) no single inequality is tighter and (b) all the operational constraints required by the argument are indeed necessary – if any one of them is dropped, the anomaly can be reproduced classically. Finally, whereas the original result required the real part of the weak value to be anomalous, we also give a version for any weak value with an imaginary part. In short, we provide tight inequalities for witnessing nonclassicality using experimentally realistic measurements of any weak value that can be considered anomalous and clarify the debate surrounding them.