Revealing multi-time quantum statistics without measurement back-action

Due to the principle of quantum mechanics that measurement disturbs a quantum state, it is a major challenge in obtaining quantum statistics, such as quantum correlation function and quasi-probability distribution. We propose an ancilla-assisted measurement scheme to detour the measurement-induced back-action, which can be applied for any multi-time measurements. With performing sequential positive operator-valued measurements at each time-point, the quantum statistics are reconstructed by weighting their outcomes differently. We employ ¹⁷¹Yb-¹³⁸Ba ions as the system-ancilla system to demonstrate the multi-time measurements without disturbing the system state, and the two- and three-time quantum correlation functions and quasi-probability distributions are reconstructed from the ancilla measurement outcomes. The marginal distribution is verified to be unaffected by the measurement at each time-point, and the nonclassical feature of the reconstructed distribution is identified by the negativity in the joint quasi-probability distribution and the violation of the Leggett-Garg inequality.