On the violation of L$\ddot{u}$ders bound of macrorealist and noncontextual inequalities

TL;DR

This paper critically examines recent claims of violating L"uders bounds in quantum systems, showing that such violations depend on measurement basis choices and questioning the validity of the measurement rules used.

Contribution

It demonstrates that violations of L"uders bounds are basis-dependent and questions the conceptual relevance of these violations in macrorealist and noncontextual inequalities.

Findings

Violation of L"uders bound depends on measurement basis choice

Violation of non-contextual inequality observed, challenging previous claims

Raises doubts about the validity of the von Neumann measurement rule

Abstract

In a recent Letter [PRL, 113, 050401 (2014)], it is shown that the quantum violation of a three-time Leggett-Garg inequality (LGI) for a dichotomic qutrit system can exceed the L$\ddot{u}$ders bound. This is obtained by using a degeneracy breaking projective measurement rule which the authors termed as von Neumann rule. Such violation can even approach the algebraic maximum in the asymptotic limit of system size. In this paper, we question the implication of such violation of L$\ddot{u}$ders bound and its conceptual relevance in LG scenario. We note an important fact that the basis for implementing the proposed von Neumann rule for a degenerate observable is non-unique and show that the violation of L$\ddot{u}$ders bound is crucially dependent on the choice of basis. Further, we demonstrate the violation of L$\ddot{u}$ders bound of the simplest non-contextual inequality (NCI) which is in contrast to the reasoning provided in the aforementioned Letter. This result further raises the doubts regarding the validity of the proposed rule as a viable projective measurement. We discuss the relevance of such results with respect to the usual quantum violation of LGI and NCI.