- #1

naima

Gold Member

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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?

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?

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