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Let $|\psi_{SA}\rangle$ be the state of a system and an apparatus, for example an electron spin and a Stern-Gerlach apparatus.

If $|\psi_S\rangle=\alpha|\uparrow\rangle+\beta |\downarrow\rangle$ prior to the measurement, the combined state after measurement is said to be

$$|\psi_{SA}\rangle=|\uparrow\rangle|up\rangle + |\downarrow\rangle|down\rangle$$

where the up/down states represent the state of the apparatus after measurement.

Then the reduced density matrix of the system, to good approximation, is diagonal:

$$\rho_S = |\alpha|^2 |\uparrow\rangle\langle\uparrow| + |\beta|^2 |\downarrow\rangle\langle\downarrow|$$

When speaking about decoherence in the context of measurement, it is often emphasized that since $\rho_S$ is diagonal, the system is "just in a classical mixture", as if it had lost its quantum properties and just represents a lack of knowledge of the state.

This situation seems completely analogous to that of (for example) the singlet state, wherein the reduced density matrix is also completely diagonal. But there is clearly nothing "classical" about the singlet state: it is a maximally entangled state which violates the Bell Inequality.

So, in what sense does the diagonal density matrix after measurement signify a "classical" ensemble any more than the singlet state?

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That is the whole point

There is no way to distinguish the mixed state corresponding to the statistical ensemble in the classical sense and part of the larger system in the entangled state. The description in quantum theory is the same and all measurements will give the same probability distribution of the results.

The Bell inequalities appears only in correlations between measurements performed on different subsystems. To see such non-classical correlations you should actually keep measuring the whole environment. In that case you would be able to recover the entangled nature of the full system. Of course it's practically impossible.

Please note that the environmental decoherence is NOT synonymous with the measurement. By itself it describes the object of interest coupled to the environment you ignore completely. For example there is some measurement apparatus that interacts with your system but however its results are ignored. When you don't ignore it's results rather you should combine it with the von Neumann measurement scheme. What this combination can give you is that all such macroscopic measurement apparatuses do for all practical purposes agree with each other (which is a foundation for the consistent histories interpretation)

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It doesn't directly; any density matrix is diagonal in some basis because it's Hermitian, so it's not intrinsically meaningful to call out that one is diagonal.

However in the density matrix understanding the expectation values are given by $$\langle A\rangle=\operatorname{Tr}\left(\rho\hat A\right),$$ and so if $\rho = p~\rho_1 + (1-p)~\rho_2$ then the resulting density matrix shows a classical-probabilistic superposition for all observables $A$ simultaneously, $\langle A\rangle = p \langle A\rangle_1 +(1-p) \langle A\rangle_2.$

So, if one has a basis already in mind, as opposed to letting the density matrix choose one's basis, then discovering that the density matrix happens to be diagonal in that basis means that there is a nice interpretation of the density matrix as a classical superposition of those states.

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