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- This is a new thread that takes up the question in the title, a question discussed (off topic) in another thread.

For the background of the discussion see my Insight artice ''Quantum Physics via Quantum Tomography'' and the posts #405 and later of the thread ''Nature Physics on quantum foundations''.

The state of system+detector is ##\widehat\rho=\rho_S\otimes\rho_D##. Premise 2 amounts to

(B1) The ##P_k## form a POVM, i.e., they are Hermitian positive semidefinite operators summing to the identity. This leads to response probabilities

$$p_k=\mathrm{tr}_SP_k \rho_S=\mathrm{tr}(P_k\otimes 1)\widehat\rho.$$

Premise 1 and 4 amount to

(B2) The ##\Pi_k## form a complete family of orthogonal projectors to the joint eigenspaces of a vector of commuting operators corresponding to simultaneous measurement results ##a_k##, interpreted according to Born's rule for projective measurements. This leads to the response probabilities

$$p_k=\mathrm{tr}_D\Pi_k \rho_D=\mathrm{tr}(1\otimes \Pi_k)\widehat\rho.$$

Premise 3 says that the probabilities ##p_k## in (B1) and (B2) are the same.

Now please complete the argument that (B1) and (B2) explain the POVM in terms of the projective measurements.

Trying to translate your statements into precise formulas, and using ##\Pi## in place of ##E## (which to me signifies an energy level, not an operator):Morbert said:Consider a microscopic system ##s## being measured, and the pointer ##M## doing the measuring

Premise 1)

The pointer must be describable with quantum mechanics. I.e. There must be a Hilbert space ##\mathcal{H}_s\otimes\mathcal{H}_M## in principle.

Premise 2)

Given some POVM ##P_k##, there must be an associated measure ##E_k## for the pointer positions

Premise 3)

If the pointer really does measure the microscopic system, then it must be the case that rates are given by

$$p_k = \mathrm{tr}_s\rho_{s}P_k = \mathrm{tr}_{s,M}\rho_{s,M} E_k = \mathrm{tr}_{s,M}P^\dagger_k\rho_{s,M}P_k E_k$$ I.e. The rates must be repoducible by both measures ##P_k## and ##E_k##

Premise 4)Since the pointer positions are mutually exclusive, it must be the case that ##E_kE_{k'} = \delta_{k,k'}##

I think premises 1 + 2 would give us an quantum mechanically describable ancilla that must exist and premises 3 + 4 would say the measurement scenario involving this ancilla must also be describable with a projective decomposition. Which of these would you take issue with?

The state of system+detector is ##\widehat\rho=\rho_S\otimes\rho_D##. Premise 2 amounts to

(B1) The ##P_k## form a POVM, i.e., they are Hermitian positive semidefinite operators summing to the identity. This leads to response probabilities

$$p_k=\mathrm{tr}_SP_k \rho_S=\mathrm{tr}(P_k\otimes 1)\widehat\rho.$$

Premise 1 and 4 amount to

(B2) The ##\Pi_k## form a complete family of orthogonal projectors to the joint eigenspaces of a vector of commuting operators corresponding to simultaneous measurement results ##a_k##, interpreted according to Born's rule for projective measurements. This leads to the response probabilities

$$p_k=\mathrm{tr}_D\Pi_k \rho_D=\mathrm{tr}(1\otimes \Pi_k)\widehat\rho.$$

Premise 3 says that the probabilities ##p_k## in (B1) and (B2) are the same.

Now please complete the argument that (B1) and (B2) explain the POVM in terms of the projective measurements.

For the moment, assume that the eigenvalues are discrete, so that this is not yet an issue.A. Neumaier said:In practice, pointers on a continuous scale are readable only approximately. Hence the actual measurements average over some neighborhood, which spoils exact orthogonality

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