- #1
Johny Boy
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- TL;DR
- I want to confirm and discuss my idea of how to derive the Fisher information from the likelihood function of a state sent through a quantum circuit (such as the Mach-Zehnder).
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. Lastly, would I be correct that for the discrete case the Fisher information is given by $$I(\theta):= \sum_{x} \bigg(\frac{\partial }{\partial \theta} \log L(x;\theta)\bigg)^2L(x; \theta),$$
where the summation over ##x## refers to all the different permutations of outcomes {##m_{02},~m_{11},~m_{20}##} such that ##M = m_{02}+m_{11}+m_{20}## holds. Is this this the correct understanding of how to derive the Fisher information for this discrete phase estimation scheme? Many thanks for your time and assistance.
- 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. Lastly, would I be correct that for the discrete case the Fisher information is given by $$I(\theta):= \sum_{x} \bigg(\frac{\partial }{\partial \theta} \log L(x;\theta)\bigg)^2L(x; \theta),$$
where the summation over ##x## refers to all the different permutations of outcomes {##m_{02},~m_{11},~m_{20}##} such that ##M = m_{02}+m_{11}+m_{20}## holds. Is this this the correct understanding of how to derive the Fisher information for this discrete phase estimation scheme? Many thanks for your time and assistance.