Information gain in quantum continual measurements

Inspired by works on information transmission through quantum channels, we propose the use of a couple of mutual entropies to quantify the efficiency of continual measurement schemes in extracting information on the measured quantum system. Properties of these measures of information are studied and bounds on them are derived. 1 Quantum measurements and entropies We speak of quantum continual measurements when a quantum system is taken under observation with continuity in time and the output is not a single random variable, but rather a stochastic process [1, 2]. The aim of this paper is to quantify, by means of entropic quantities, the effectiveness of a continual measurement in extracting information from the underlying quantum system. Various types of entropies and bounds on informational quantities can be introduced and studied in connection with continual measurements [3–5]. In particular, in Ref. [5] the point of view was the one of information transmission: the quantum system is a channel in which some information is encoded at an initial time; the continual measurement represents the decoding apparatus. In this paper, instead, we consider the quantum system in itself, not as a transmission channel, and we propose and study a couple of mutual entropies giving two indexes of how good is the continual measurement in extracting information about the quantum system. 1.1 Algebras, states, entropies From now on H will be a separable complex Hilbert space, the space where our quantum system lives.