Violation of Leggett-Garg inequalities implies information erasure

The Leggett-Garg inequalities were originally introduced for experimentally testing a possible break of the quantum evolution in mesoscopic systems. In this paper, we take a different point of view by focusing on faithful classical simulations of sequential quantum measurements. In this context, the violation of Leggett-Garg inequalities implies that classically simulated quantum measurements induce perturbations into the subsequent evolution of the classical variables. We show that the implication is even stronger and a measurement erases previous information by performing a partial reset on the classical state. Thus, the measuring device acts as a low-temperature bath absorbing entropy from the measured system. Information erasure is a form of preparation contextuality. Our proof is straightforward if one assumes that maximal ignorance of the quantum state is compatible with maximal ignorance of the classical state. We also employ a weaker hypothesis.