Quantum simulations of macrorealism violation via the QNDM protocol

The Leggett-Garg inequalities have been proposed to identify the quantum behaviour of a system; specifically, the violation of macrorealism. They are usually implemented by performing two sequential measurements on quantum systems, calculating the correlators of such measurements and then combining them arriving at Leggett-Garg inequalities. However, this approach only provides sufficient conditions for the violation of macrorealism. Recently, it was proposed an alternative approach that uses non-demolition measurements and gives both a necessary and sufficient condition for the violation of macrorealism. By storing the information in a quantum detector, it is possible to construct a quasi-probability distribution whose negative regions unequivocally identify the quantum behaviour of the system. Here, we perform a detailed comparison between these two approaches. The use of the IBM quantum simulators allows us to evaluate the performance in real-case situations and to include both the statistical and environmental noise. We find that the non-demolition approach is not only able to always identify the quantum features but it requires fewer resources than the standard Leggett-Garg inequalities. In addition, while the efficiency of the latter is strongly affected by the presence of the noise, the non-demolition approach results incredibly robust and its efficiency remains unchanged by the noise. These results make the non-demolition approach a viable alternative to the Leggett-Garg inequalities to identify the violation of macrorealism.