In a recent Letter by Budroni and Emary [Phys. Rev. Lett. 113, 050401 (2014)], it is shown that the quantum violation of a three-time Leggett-Garg inequality for a dichotomic qutrit system can exceed the Lüders bound. This is obtained by using a degeneracy-breaking projective measurement rule which the authors termed the “von Neumann rule.” Such violation can even approach the algebraic maximum in the asymptotic limit of system size. In this paper, we first examine the reason behind such violation of the Lüders bound in quantum mechanics and its conceptual relevance in the Leggett-Garg scenario. Further, we demonstrate the violation of the Lüders bound of the simplest noncontextual inequality involving three commuting observables. The implication of such violations of the Lüders bound of Leggett-Garg inequalities and noncontextual inequalities is discussed.

• Received 15 February 2018

DOI:https://doi.org/10.1103/PhysRevA.98.042135