Condition for macroscopic realism beyond the Leggett-Garg inequalities

Abstract

In 1985, Leggett and Garg put forward the concept of macroscopic realism (macrorealism) and, in analogy to Bell's theorem, derived a necessary condition in terms of inequalities, which are now known as the Leggett-Garg inequalities. In this paper, we discuss another necessary condition called no-signaling in time. It solely bases on comparing the probability distribution for a macrovariable at some time for the cases where previously a measurement has or has not been performed. Although the concept is analogous to the no-signaling condition in the case of Bell tests, it can be violated according to quantum mechanical predictions even in situations where no violation of Leggett-Garg inequalities is possible.

Wigner's form of the Leggett-Garg inequality, the no-signaling-in-time condition, and unsharp measurements

“…Here we may note that Leggett [10,11] has argued that NIM can be regarded to be a 'natural' corollary of the condition of macrorealism per se in the context of what is known as an ideal negative result measurement (applicable for testing both LGI and WLGI) that can, too, be invoked for testing NSIT, as has been noted by Kofler and Brukner [21]. By investigating the robustness of the QM violation of NSIT against unsharpness of measurement, a striking result is obtained that, no matter whatever be the value of the sharpness parameter characterizing precision of the relevant measurements, QM violates NSIT.…”

“…As already mentioned in the introductory section, an alternative necessary condition for the validity of macrorealism has recently been proposed [21] by assuming that the outcome statistics of a measurement would remain unaffected by any prior measurement. This condition, referred to as 'No-Signalling in Time' (NSIT), is the statistical version of NIM used in deriving LGI and can be viewed as the temporal analogue of the no-signalling condition for the spacelike separated measurements used in the EPR-Bohm scenario, with the difference that while any violation of the latter would violate special relativity, the violation of NSIT does not imply any such inconsistency.…”

“…Furthermore, we consider the other proposed necessary condition for macro-realism known as 'No-Signaling in Time' (NSIT) suggested by Kofler and Brukner [21] which stipulates that the statistics of the outcomes of measurements at any instant should not show dependence on whether any prior measurement has been performed; in other words, NSIT is based on applying the condition of non-invasive measurability (NIM) at the statistical level whose violation would imply violation of NIM at the individual level. Here we may note that Leggett [10,11] has argued that NIM can be regarded to be a 'natural' corollary of the condition of macrorealism per se in the context of what is known as an ideal negative result measurement (applicable for testing both LGI and WLGI) that can, too, be invoked for testing NSIT, as has been noted by Kofler and Brukner [21].…”

Violation of noninvasive macrorealism by a superconducting qubit: Implementation of a Leggett-Garg test that addresses the clumsiness loophole

“…Some recent papers have stressed the role of equalities rather than inequalities in testing macrorealism [29,30]. One might consider reframing our Leggett-Garg program in the future by directly checking the equality (1) rather than inserting it into the inequality (2).…”

Temporal steering inequality

“…In addition, the Leggett-Garg (LG) inequality can also be applied to microscopic systems [13][14][15][16], as a tool to examine the quantum coherent dynamics therein. For two-level systems, there is a one-to-one correspondence between the LG inequality and the Bell-CHSH inequality [18], a fundamental consequence of the Choi-Jamiolkowski isomorphism [19].…”