How to merge beams with POVMs?

[naima]

When in a stern-gerlach electrons are not detected the two beams can merge behind. We just have to add the amplitudes.
this is easy because up> and down> are orthogonal.
Suppose that we have a POVM setup with 3 output channels: F1 F2 and F3 as operators. They sum to Id but have not to be orthogonal.
there are three detectors which click with probabilities P1 P2 and P3 ($Pi = Tr(\rho Fi)$)
Suppose now that we replace detectors 1 and 2 by a merger Have we to add F1 and F2? (add probabilities).
With $F1 = M1^\dagger M1$ and $F2 = M2^\dagger M2$ we would get
$M1^\dagger M1 + M2^\dagger M2$ and not $(M1+M)^\dagger(M+M)$
This is not the same. so what is the rule when we merge beams?

 

[kith]

In theory, it depends on the experimental setup. If you just split the beam and remerge it, you have a coherent process and add amplitudes. If you prepare two different ensembles ρ₁ and ρ₂ and mix them, you have an incoherent process and add probabilities.

In practice, I don't think it makes much difference how you model the situation. Decoherence in the first process makes the outcome look identical to the second.

 

[naima]

Man'ko gave https://www.researchgate.net/publication/222549915_Inner_composition_law_of_pure_states_as_a_purification_of_impure_states .
There is a composition law for density matrices.
look at (eq 17)