# Fisher information from likelihood function for discrete quantum case

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.

• That seems correct to me. Commented Mar 4 at 13:18

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).