  Large weak values have been used to amplify the sensitivity of a linear
response signal for detecting changes in a small parameter, which has also
enabled a simple method for precise parameter estimation. However, producing a
large weak value requires a low postselection probability for an ancilla degree
of freedom, which limits the utility of the technique. We propose an
improvement to this method that uses entanglement to increase the efficiency.
We show that by entangling and postselecting $n$ ancillas, the postselection
probability can be increased by a factor of $n$ while keeping the weak value
fixed (compared to $n$ uncorrelated attempts with one ancilla), which is the
optimal scaling with $n$ that is expected from quantum metrology. Furthermore,
we show the surprising result that the quantum Fisher information about the
detected parameter can be almost entirely preserved in the postselected state,
which allows the sensitive estimation to approximately saturate the optimal
quantum Cram\'{e}r-Rao bound. To illustrate this protocol we provide simple
quantum circuits that can be implemented using current experimental
realizations of three entangled qubits.
