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As far as I understood decoherence theory, it explains, why we are not able to measure superposition of macroscopic objects in some specific basis, which turns out to be the position-basis in most cases. As I'm currently trying to understand the multiple worlds interpretation (MWI), very often I find texts that say MWI needs decoherence theory as a completion.

What I don't understand is $(1)$ why MWI needs decoherence and $(2)$ how MWI can be combined with decoherence.

I'm even not sure if this question is simply too coarse-grained, or if there is one simple thought, which I have overlooked yet.

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The MWI is basically a reinterpretation of the von Neumann measurement scheme. The latter tries to include an idealized measurement apparatus into full quantum description of some external observer.

Let's try to measure the observable $\hat{A}=\sum_{k} A_k\hat{\mathcal{P}}_k$ where $\hat{\mathcal{P}}_k=|\psi_k\rangle\langle\psi_k|$. The probabilities are given by the Born rule and the consequent measurement deal with the transformed wavefunction obtained with the projection postulate which is commonly known as a wavefunction collapse, \begin{equation} P(A=A_k)=\langle\psi|\hat{\mathcal{P}_k}|\psi\rangle,\quad |\psi\rangle\mapsto |\psi_k\rangle \end{equation} the interaction of this ideal measurement apparatus with the measured object is more or less by definition looks like this, \begin{equation} |\Psi(t_0)\rangle=|0\rangle_{app}\otimes\sum_{k}\alpha_k|\psi_{k}\rangle_{obj}\mapsto|\Psi(t_{meas})\rangle=\sum_{k}\alpha_k|A_k\rangle_{app}\otimes|\psi_{k}\rangle_{obj} \end{equation} So the measurement apparatus becomes entangled with the object in such a way that its registered value is perfectly correlated with the initial observable $\hat{A}$. Now, the von Neumann used this simply to show that if such ideal measurement apparatus is included as extra layer of the quantum description then if observer actually measures the apparatus i.e. the observable $\hat{A}_{app}\equiv \sum_k|A_k\rangle_{app}\langle A_k|_{app}$, not the object itself then the probabilities of the outcomes and projection rules remain consistent with the description without the measurement apparatus. Please note that the whole scheme assumes that a measurement apparatus can be described by the pure quantum state therefore neglecting its interaction with the environment completely (the flaw that was already understood back then)

The MWI appears when you

  • Replace the sole measurement apparatus with the whole macroscopic environment with all macroscopic objects entangled between each other in such a way that they are all perfectly correlated. As simplification if you were able to assign some quantum state not only to the measurement apparatus but also to the scientist Alice, scientist Bob, their cat and Alice's chair you should replace $|A_k\rangle_{app}$ with, \begin{equation} |A_k\rangle_{env}=|A_k\rangle_{app}|A_k\rangle_{Alice}|A_k\rangle_{Bob}|A_k\rangle_{cat}|A_k\rangle_{chair} \end{equation} as result the $|\Psi\rangle$ is interpreted as an unobservable quantum state of the whole universe that assumed to evolve as a closed quantum system. The step fundamental for MWI is that after the measurement the universal state becomes a superposition of the branches that because of the quantum evolution linearity evolve independently of each other.

  • Consider then a series of the von Neumann-like measurements with device and environment making records. So if we assume that after the measurement the first observable eigenstates transforms into, \begin{equation} |\psi_k\rangle\mapsto \sum_l \beta_{kl}|\phi\rangle_l \end{equation} where $|\phi\rangle_l$ are eigenstates of some second observable $\hat{B}$ then the two subsequent von Neumann measurements yield, \begin{aligned} |0\rangle_{env}\sum_k\alpha_k|\psi_k\rangle\mapsto\sum_k\alpha_k|A_k\rangle_{env}|\psi_k\rangle\mapsto\\ \sum_k\alpha_k|A_k\rangle_{env}\sum_l\beta_{kl}|\phi_l\rangle \mapsto\sum_{k,l}\alpha_k\beta_{kl}|A_k,B_l\rangle_{env}|\phi\rangle_l \end{aligned} Then the branching structure appears in the coefficients. You get $\alpha_k\beta_{kl}$ but not some non-decomposable $\gamma_{kl}$. Because of that if you look on the branches like $|A_k,\ldots\rangle$ they doesn't care whether you omitted all branches with different values of $A$ after the first measurement. So this is the attempt to explain the projection postulate - the observer as a part of such quantum universe percieve as if "collapse" happened whereas the evolution of the whole universe is unitary. Everett and his followers attempted to derive the Born rule i.e. that $|\alpha_k\beta_{kl}|^2$ actually gives a probability of the branch. However all those derivations are circular and only show that if you assume such probability for the branch this is consistent with the predicted statistics appearing in many measurements.

The issue is that both von Neumann measurement scheme and MWI were formulated at extremely heuristic level without any demonstration how this sort of thing appears in actual systems.

The MWI in its original formulation didn't explain why all the macroscopic objects are correlated in such a perfect way. It didn't explain why this entanglement occurs in such a way that we get a semiclassical macroscopic world with some preference for localization in both coordinate and momentum spaces. Because of this lack of detail many people developed a lot of extremely naive ideas. For example that the universal quantum state is a superposition of a strictly classical "worlds".

The decoherence can explain how actually the universal quantum state may develop MWI-like structure. If you don't care about purely philosophical discussions and trying to get rid of the Born rule then the consistent histories (or decoherent histories) approach is "the way to do MWI right" i.e. the way to show that vague ideas of the Copenhagen school about the macroscopic world can actually get solid justification if we assume that the quantum theory can be applied on the cosmological scales.

With enough coarse-graining you may get decoherence being extremely strong. The universal quantum state in such an idealized limit indeed can be seen as experiencing MWI-like branching. That is the probabilities for the coarse-grained histories will be the same as if you calculated them on the branched state. But you should understand that it's only a simplified picture. The distinct classical worlds are not actually predicted by the decoherence, that's why more refined versions of MWI talk more about "many minds" rather than "many worlds" (though the distinct "mind" can only exist as a coarse grained approximation too)

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