  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.
