Bayes cost of parameter estimation for a quantum system interacting with an environment

Abstract

The Bayes cost of parameter estimation is studied for a quantum system which is influenced by an external environment, where the cost function is assumed to be a quadratic function of a difference between true and estimated values. When the reduced time evolution of a quantum system is determined by the time-dependent Lindblad equation, it is found how the Bayes cost changes with time. The Bayes cost increases monotonously with time for the Markovian environment, while it shows an oscillatory behavior for the non-Markovian environment due to the memory effect. Furthermore, in order to investigate how initial correlation between quantum system and environment, an analytic expression of the Bayes cost is derived for a qubit-oscillator system. It is found for both Markovian and non-Markovian environments that the Bayes cost can take a value smaller than the initial one in the presence of the initial correlation. The decrease in the Bayes cost is due to the backflow of information that is included in the initially correlated part.