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It is my understanding that the non-diagonal terms in the density matrix of a macroscopic system that it is initially in an entangled state go exponentially fast to zero as the system interacts with the environment. The off diagonal elements reach a value of zero asymptotically, in an infinite amount of time, but never become zero.

Does this mean that a macroscopic system is always in superposition, only that one of those states is highly unlikely to be observed in a practical measurement (but in the end, the cat is still alive and dead, only that more of one than of the other)?

Does decoherence theory have a different view on this issue?

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  • $\begingroup$ decoherence has not yet solved the measurement problem $\endgroup$ – user83548 May 28 '16 at 22:22
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The first part of the sentence is right, given some assumptions, the rest is not.

The most precise description of every physical system is in terms of a very general superposition of a priori possible states. There is never any collapse.

But the rest isn't true. An off-diagonal element of the density matrix doesn't correspond to any single state. It corresponds to the information about the relative phase between two different states. If a cat is in the pure state $a|{\rm dead}\rangle+b|{\rm alive}\rangle$, then the off-diagonal element $\rho_{\rm dead, alive}$ is equal to $a^* b$. If some interaction with the environment makes the relative phase between $a,b$ ambiguous, then – even if both $a,b$ are large (significantly nonzero), we have to calculate $\rho_{\rm dead, alive}$ by some averaging and the result is basically zero.

Therefore, decoherence means that the information about the relative phases of the probability amplitudes of the "preferred bases vectors" is disappearing due to the interactions with the environment.

Only the diagonal entries of the density matrix in the preferred basis may be understood as probabilities of states. And none of these probabilities of allowed states goes to zero due to decoherence. All of them are predicted to be nonzero, even with decoherence. Decoherence in no way includes the "collapse" that would make some diagonal entries zero (and only one of them would be basically $1$).

Decoherence only "approximately erases" the off-diagonal entries of the density matrix – those that can't be interpreted as probabilities of any particular states; these off-diagonal elements are exactly those that don't have any counterpart in classical physics at all. But decoherence does not erase any diagonal entries, not even approximately.

Decoherence means that the states of the preferred basis may be analyzed by the classical logic – the quantum interference between them is negligible. But decoherence in no way replaces the collapse or the measurement that induces it. One still needs an observer who observes one option (e.g. whether a cat is alive or dead), and only when the observer observes this information, the density matrix's diagonal entries collapse to $(0,1,0,0)$ or something like that. Decoherence is a phenomenon or a way to organize a calculation within quantum mechanics. Quantum mechanics always needs observers to create the sharp outcomes and this fact obviously cannot be "undone" by any calculation that works within quantum mechanics, and decoherence is one of them.

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