delay: $\tau$= 1 as.[]{data-label="figure_sat"}](figure4.eps){width="90.00000%"}

Discussion {#discussion .unnumbered}
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Advantages of using weak value amplification (I): when the detector cannot work above a certain photon number. {#advantages-of-using-weak-value-amplification-i-when-the-detector-cannot-work-above-a-certain-photon-number. .unnumbered}
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Let us suppose that we have at hand light detectors that cannot be used with more than $N_0$ photons. Any limitation on the detection time or the signal power would produce such limitation. The technical advantages of using WVA in this scenario has been previously pointed out [@jordan2014]. Here we make this apparent from a quantum estimation point of view, and quantify this advantage.

Fig. 4(a) shows the minimum probability of error as a function of the number of photons ($N$) entering (and leaving) the interferometer. For $N_0=10^6$, inspection of the figure shows that the probability of error is $P_\mathrm{error}=1.3 \times
10^{-1}$. This is the best we can do with this experimental scheme and these particular detectors without resorting to weak value amplification. However, if we project the output signal from the interferometer into a specific polarisation state, and increase the flux of photons, we can decrease the probability of error, without necessarily going to a regime of high depletion of the signal [@torres2012]. For instance, with $\theta=53.2^{\circ}$, and a flux of 