When making the Pauli Matrix Measurement on an interferometer in the Quantum state , we get with probability and with probability , i.e.

Depending on the value of we can get more/less information per experiment, based on the derivatives of and . Since the underlying value of is uniform on the interval , we can attempt to maximise the information obtained by applying different phase shifts before applying the Measurement. As such, we will consider our observations to be the set , obtained with the underlying probabilities

Our goal is to find the most likely given our observations.


We will apply the Maximum Likelihood Estimation framework. Up to a constant, the probability of our observations equals . Since is Monotonic, , allowing us to simplify the latter expression:

In order to maximise we need to have

While finding the closed formula solution for the equation above might not be possible, finding the roots can be done relatively quickly via the Newton-Raphson method.

Fisher Information

We can calculate the Fisher Information for this setup by:

  • Replace mentions of above with .

Open Questions

Some questions remain. Help me ❓

  1. Is the root unique, up to translations of ?
  2. Can we find the Variance of the estimated value ?
  3. .
    1. Could this cause problems in the equality above due to an absurd sensitivity to “edge” cases?
    2. Could we get rid of this infinite information explosion by adding a possibly normally distributed error term to the measurement of ?