Information gain and measurement disturbance for quantum agents

The traditional formalism of quantum measurement (hereafter ``TQM'') describes processes where some properties of quantum states are extracted and stored as classical information. While TQM is a natural and appropriate description of how humans interact with quantum systems, it is silent on the question of how a more general, quantum, agent would do so. How do we describe the observation of a system by an observer with the ability to store not only classical information but quantum states in its memory? In this paper, we extend the idea of measurement to a more general class of sensors for quantum agents which interact with a system in such a way that the agent's memory stores information (classical or quantum) about the system under study. For appropriate sensory interactions, the quantum agent may ``learn'' more about the system than would be possible under any set of classical measurements -- but as we show, this comes at the cost of additional measurement disturbance. We experimentally demonstrate such a system and characterize the tradeoffs, which can be done by considering the information required to erase the effects of a measurement.