  We calculate the trade-off between the quality of estimating the quantum
state of an ensemble of identically prepared qubits and the minimum level of
disturbance that has to be introduced by this procedure in quantum mechanics.
The trade-off is quantified using two mean fidelities: the operation fidelity
which characterizes the average resemblance of the final qubit state to the
initial one, and the estimation fidelity describing the quality of the obtained
estimate. We analyze properties of quantum operations saturating the
achievability bound for the operation fidelity versus the estimation fidelity,
which allows us to reduce substantially the complexity of the problem of
finding the trade-off curve. The reduced optimization problem has the form of
an eigenvalue problem for a set of tridiagonal matrices, and it can be easily
solved using standard numerical tools.
