Quantum Measurement Tomography
One can imagine a situation in which an apparatus performs some measurement on quantum systems, and determining what particular measurement is desired. The strategy is to send in systems of various known states, and use these states to estimate the outcomes of the unknown measurement. Also known as "quantum estimation", tomography techniques are increasingly important including those for quantum measurement tomography and the very similar quantum state tomography. Since a measurement can always be characterized by a set of POVM's, the goal is to reconstruct the characterizing POVM's . The simplest approach is linear inversion. Similar to in quantum state observation, use
- .
Exploiting linearity as above, this can be inverted to solve for the .
Not surprisingly, this suffers from the same pitfalls as in quantum state tomography. Namely, non-physical results, in particular negative probabilities. Here the will not be valid POVM's, as they will not be positive. Bayesian methods as well as Maximum likelihood estimation of the density matrix can be used to restrict the operators to valid physical results.
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