Indefinite Time Directed Quantum Metrology

Abstract

We explore the performance of the metrology scheme by employing a quantum time flip during encoding, a specific case of processes with indefinite time direction, which we refer to as indefinite time directed metrology ($ITDM$). In the case of single parameter estimation of a unitary, we demonstrate that our protocol can achieve Heisenberg scaling $(1/N)$ with product probe states, surpassing the standard quantum limit $(1/{\sqrt{N}})$, where $N$ is the number of particles in the probe. We establish this by computing the quantum Fisher information ($QFI$) which is a lower bound on the root mean square error occurred during parameter estimation. Although we analytically prove the optimality of the symmetric product probe state in $ITDM$, entangled probe states produce a higher $QFI$ than optimal product probes without enhancing scaling, highlighting the non-essentiality of entanglement. For phase estimation, we propose a single-qubit measurement on the control qubit that accomplishes near-optimal Fisher information and eventually reaches Heisenberg scaling. Our findings reveal the best orientation of product probe states in every pertinent situation, emphasizing its independence from the parameter to be estimated in the limiting case. Furthermore, we illustrate the benefits of $ITDM$ in noisy metrology, outperforming existing techniques in some situations.