Detecting violations of macrorealism when the original Leggett-Garg inequalities are satisfied

Abstract

Macroscopic realism (MR) is the notion that a time-evolving system possesses definite properties, irrespective of past or future measurements. Quantum-mechanical theories can, however, produce violations of MR. Most research to date has focused on a single set of conditions for MR, the Leggett-Garg inequalities (LGIs), and on a single data set, the standard data set, which consists of single-time averages and second-order correlators of a dichotomic variable $Q$ for three times. However, if such conditions are all satisfied, then where is the quantum behavior? In this paper we provide an answer to this question by considering expanded data sets obtained from finer-grained measurements and MR conditions on those sets. We consider three different situations in which there are violations of MR that go undetected by the standard LGIs. First, we explore higher-order LGIs on a data set involving third- and fourth-order correlators, using spin-$\frac{1}{2}$ and spin-1 systems. Second, we explore the pentagon inequalities and a data set consisting of all possible averages and second-order correlators for measurements of $Q$ at five times. Third, we explore the LGIs for a trichotomic variable and measurements made with a trichotomic operator to again identify violations for a spin-1 system beyond those seen with a single dichotomic variable. We also explore the regimes in which combinations of two and three-time LGIs can be satisfied and violated in a spin-1 system, extending recent work. We discuss the possible experimental implementation of all the above results.