Revealing quantum correlation functions with zero measurement back-action

Exploring the joint probability distribution of a quantum system in sequential measurements is an important issue in quantum thermodynamics. A straightforward way to achieve this task is to let the system interacts with an ancilla qubit and then measure the ancilla qubit with the projection measurements. However, projection measurements disturb the quantum state thus influencing the subsequent measurement statistics, often referred to as the back-action of the quantum measurement. Alternatively, one can experimentally obtain the real and imaginary parts of the correlation function without the back-action but needs an explicit knowledge of the system's dynamics. In this work, we propose and experimentally verify that the quantum correlation functions can be reconstructed from ancilla-assisted measurements without measurement back-action, where the measurement setup is independent of the system dynamics. During the measurement protocol, POVM (positive operator-valued measure) other than projection measurements are performed on the ancilla qubit. We demonstrate this protocol in a trapped ¹⁷¹Yb⁺-¹³⁸Ba⁺ ion system, where ¹⁷¹Yb⁺ ion and ¹³⁸Ba⁺ ion role as the system qubit and the ancilla qubit, respectively. The hybrid system enables us to reconstruct quantum statistics unaffected by the measurement back-action. The Mølmer-Sørensen (M-S) gate is adopted to generate entanglement between two ions. Two-point and three-point correlation functions are experimentally demonstrated, which shows that our protocol can be extended to multi-point correlation measurements straightforwardly.