  A general quantum measurement on an unknown quantum state enables us to
estimate what the state originally was. Simultaneously, the measurement has a
destructive effect on a measured quantum state which is reflected by the
decrease of the output fidelity. We show for any $d$-level system that quantum
non-demolition (QND) measurement controlled by a suitably prepared ancilla is a
measurement in which the decrease of the output fidelity is minimal. The ratio
between the estimation fidelity and the output fidelity can be continuously
controlled by the preparation of the ancilla. Different measurement strategies
on the ancilla are also discussed. Finally, we propose a feasible scheme of
such a measurement for atomic and optical 2-level systems based on basic
controlled-NOT gate.
