I am simulating (using Mathematica) a Mach-Zehnder interferometer, with photon counting measurements at the end (based on the setup described in the recent post) for the input state $|\psi\rangle:=|0,2\rangle$ (for the different arms of the Mach-Zehnder I consider angles $\theta/2$ and $-\theta/2$). I set some true value of $\theta$ (say $\theta_{true} = 1.5$).
Using a random number generator, I carry out $\nu = 300$ independent measurements, from these measurement results I construct the Likelihood function $$L(\theta):= P(0,2)^{m_{02}}P(1,1)^{m_{11}}P(2,0)^{m_{20}},$$ (as in the post but without the multinomial coefficients since I retain each measurement result) and then numerically (Mathematica) find the $\theta$ that maximizes the Likelihood function. By doing so I numerically find the Maximum Likelihood Estimator (MLE). I do this many times about $10^5$ times, so that I have $10^5$ MLE estimators of $\theta$. I am trying to saturate the well-known Cramer-Rao bound, which states $(\Delta \theta)^2 \geq 1/(\nu\cdot FI)$, where $FI$ denotes the classical Fisher information (in this case the Fisher Information is analytically found to be $FI=2$). I would expect that for large enough trials $\gtrapprox 10^5$ as stated, I would approach equality in the stated Cramer-Rao bound (for $\nu = 300$ this is $\frac{1}{(300\cdot 2)} = 0.00167$). What I find is that the mean of the estimators seem to stabilize about 1.496 with the variance of the MLEs stabilizing at $(\Delta \theta)^2 \approx 0.006$. This is about a factor 3 larger than what is required to saturate the Cramer-Rao bound, which would be $(\Delta \theta)^2 \approx 0.00167.$
Query: There seems to be an inherent under-estimation (regardless of the number of iterations), in the sense that the mean of the MLEs are always around $1.496$ but never greater than $1.5$, for large enough iterations. Could this be a result of possible MLE bias for this particular setup? Or is this more likely a numerical imprecision lost during the coding process? The estimation is close but not close enough. Thanks for any ideas and assistance.
