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In the context of a single phase estimation problem of a quantum photonics experiment. For example consider a 3-photon quantum circuit (such as the Mach-Zehnder which depends on some phase shift operator which encodes a parameter $\theta$) with a photon counting measurement (two detectors) at the end of the circuit with measurement probabilities:

  • P(0,2): 0 photons detected in Detector 1, 2 photons detected in Detector 2.
  • P(1,1): 1 photon detected in each detector.
  • P(2,0): 2 photons detected in Detector 1, and none in Detector 2.

Consider that at a given time we carry out $M$ total measurements. We will get some set of measurement outcomes {$m_{02},~m_{11},~m_{20}$}, where $M = m_{02}+m_{11}+m_{20}$. Am I correct that we can define the corresponding likelihood function $L(\theta)$ (which we intend to use to evaluate the Fisher information) by: $$L(\theta):= \frac{M!}{(m_{02}! m_{11}! m_{20}!)} P(0,2)^{m_{02}}P(1,1)^{m_{11}}P(2,0)^{m_{20}},$$ where the multinomial coefficients account for the different permutations.

Can anyone advise if this is the correct way to construct the likelihood function for the described experimental scenario.

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    $\begingroup$ That seems correct to me. $\endgroup$
    – narip
    Mar 4 at 13:18

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This is correct. If you retained each of the measurement results as opposed to the total number of times each result occured, you would neglect the multinomial factor.

In addition, it is worth explicitly including the dependence on $\theta$ in the probabilities; the point is that those are not dictated by the experimental results at all. Then, maximizing the likelihood with respect to $\theta$ is equivalent to minimizing the Kullback-Leibler divergence between your measured distribution and a theoretical distribution from a given value of $\theta$ (see wiki or my answer).

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  • $\begingroup$ Many thanks for you assistance. If you have a chance please give your opinion on the following related post. $\endgroup$
    – John Doe
    Mar 9 at 3:01

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