Unifying theory of quantum state estimation using past and future information

Abstract

We propose a unifying theory for estimating unknown quantum states using observed data, both before (past) and after (future) the estimation time. State estimation using past-future observed data has been studied even in the classical cases; for example via the state smoothing technique, which has been shown to perform better than the state filtering (using only the past information). In the quantum regime, there are mainly three existing formalisms that take into account the information both before and after the estimation time, i.e., the weak-value formalism [Phys. Rev. Lett. 60 1351 (1988)], the quantum most-likely path [Phys. Rev. A 88 042110 (2013)], and the quantum state smoothing [Phys. Rev. Lett. 115, 180407 (2015)]. Considering a partially observed quantum system, in which there exist both observed and unobserved records from continuous monitoring of the system, we give a common formulation that establishes the connection among three existing formalisms. The state estimators are calculated based on the expected cost minimization, either in the state space or the unknown record space. Our theory not only unifying existing formalisms for quantum state estimation, but also suggest new estimators that can be applied in practical scenarios.