Erratum to “Specker's parable of the over-protective seer: A road to contextuality, nonlocality and complementarity” [Phys. Rep. 506 (2011) 1–39](S0370157311001517)(10.1016/j.physrep.2011.05.001)

In our article  [1], we claim that the inequality in our Eq. (F.3), namely,(1)η≤1N∑X1…XN|m→X1…XN|2∑X1…XN|m→X1…XN|, where m→X1…XN≡∑k=1NXknˆk, is a necessary condition for joint measurability of noisy spin observables along axes {nˆk:k∈{1,…,N}}, but this claim is incorrect. Our mistake was in the paragraph between Eqs. (F.11) and (F.12). The vector described by Eq. (F.12) does not, in fact, maximize the scalar product we consider because we optimized over vectors having fixed 1-norm rather than fixed 2-norm. The bound of Eq. (F.3) is a sufficient condition for joint measurability because it coincides with the sufficient condition we proved in our article, namely the inequality of Eq. (F.4), (2)η≤2N∑X1…XN|m→X1…XN|. The equivalence of the two bounds can be established by noting that ∑X1…XN|m→X1…XN|2=N2N, which is derived with some simple algebra. For the specific examples of joint measurability that we considered in Sec. 7 of our article, the bound of (F.3) is, in fact, necessary and sufficient, so this error does not affect our analysis of those cases. The necessity of η≤12 (respectively η≤13) for the joint measurability of noisy spin observables along a pair (respectively triple) of orthogonal axes (our proposition 8) is, as we noted there, proven in Ref.  [2]. The necessity of η≤2/3 for the joint measurability of noisy spin observables along a triple of axes forming a trine (our Proposition 10) was subsequently established by Kunjwal and Ghosh  [3] (Appendix B), which can be seen as a special case of the necessary condition that they provided in Eq. (B3). In general, however, explicit counterexamples to the necessity of the bound can be found. A trivial example¹ Thanks to Ravi Kunjwal for devising this example. arises for spin axes nˆ1=zˆ, nˆ2=nˆ3=−zˆ, for which the triple of spin observables can be jointly measured even if they are noiseless, so that the correct bound in this case is η≤1, while (F.3) implies η≤2/3. Nontrivial examples can also be found.² T. Heinosaari and R. Uola, private communication; J. Bavaresco, private communication. Proposition 11 of our article is also incorrect. It claims to identify the highest degree of anti-correlation that can be achieved in a joint measurement of three noisy spin observables oriented along trine axes with purity parameter η=2/3. But higher degrees of anti-correlation can in fact be found, implying that our generalized noncontextuality inequality, Eq. (113), can in fact be violated for such measurements, as noted in Refs.  [3] and  [4]. Our mistake here was a conceptual one. We identified a POVM that implemented a joint measurement of our three noisy spin observables and assumed, incorrectly, that it was the only POVM (up to fine-graining) that could do so. Such an inference is unproblematic in the case of projective measurements where if two measurements are jointly implementable then there is a unique measurement (up to fine-graining) that achieves this, namely, the one associated to the projectors onto their joint eigenspaces. In the case of POVMs, however, this is not the case, as is discussed in detail in Ref.  [5].