Quantum Non-Demolition Measurements and Leggett-Garg inequality

Abstract: Quantum non-demolition measurements define a non-invasive protocol to extract information from a quantum system that we aim to monitor. They exploit an additional quantum system that is sequentially coupled to the system. Eventually, by measuring the additional system, we can extract information about temporal correlations developed by the quantum system dynamics with respect to a given observable. This protocol leads to a quasi-probability distribution for the measured observable outcomes, which can be negative. We prove that the presence of these negative regions is a necessary and sufficient condition for the violation of macrorealism. This is a much stronger condition than the violation of the Leggett-Garg inequalities commonly used for the same task. Indeed, we show that there are situations in which Leggett-Garg inequalities are satisfied even if the macrorealism condition is violated. As a consequence, the quantum non-demolition protocol is a privileged tool to identify with certainty the quantum behavior of a system. As such, it has a vast number of applications to different fields from the certification of quantumness to the study of the quantum-to-classical transition.