photons of $N=10^7$, so that after projection $N_{\mathrm{out}}=10^6$ photons reach the detector, the probability of error is decreased to $P_{\mathrm{error}}=9.3 \times 10^{-5}$, effectively enhancing the sensitivity of the experimental scheme (see Fig. 4(b)). The probability of error can be further decreased, also for other projections, at the expense of further increasing the input signal $N$.

In general, the minimum quantum overlap achievable between the states without any projection is $$|\langle \Phi_0|\Phi_1\rangle|^2=\exp \left[-N_0 \left( 1-\gamma
\cos \omega_0 \tau\right) \right],$$ while making use of projection in a weak value amplification scheme is $$|\langle \Phi_0|\Phi_1\rangle|^2=\exp
\left[-\frac{2N_0 \left(1-\gamma \cos \omega_0 \tau
\right)}{1+\gamma \cos \left(\omega_0 \tau-\Gamma-\pi/2 \right)}
\right]. \label{enhancement}$$ Eq. (\[enhancement\]) shows that when the number of photons that the detection scheme can handle is limited ($N_0$), projection into a particular polarisation state, at the expense of increasing the signal level, is advantageous. From a quantum estimation point of view, WVA increases the minimum probability of error reachable, since the projection makes possible to use the maximum number of photons available ($N_0$) with a corresponding enhanced mode overlap. Notice that the effect of using different polarisation projections can be beautifully understood as reshaping of the balance between signal level and mode overlap.

Advantages of using 