= Solution
\b[<Quantum decoherence> explains the practical stability of records and suppression of observable <interference>; it does not alone select one outcome.] The <measurement problem> arises when universal <unitary time evolution>, a complete <wave function> description and a unique definite outcome are all demanded. A theory must explain which commitment is changed or supplemented, not merely replace a pure composite description by a reduced one.
Model an ideal <quantum measurement> and its environment by
$$
(a|0\rangle+b|1\rangle)|A_*\rangle|E_*\rangle
\longmapsto
|\Psi\rangle=a|0,A_0,E_0\rangle+b|1,A_1,E_1\rangle,
\qquad |a|^2+|b|^2=1.
$$
Taking the <partial trace> over the environment gives the system–apparatus <reduced density matrix>
$$
\rho_{SA}=|a|^2|0,A_0\rangle\langle0,A_0|
+|b|^2|1,A_1\rangle\langle1,A_1|
+ab^*\langle E_1|E_0\rangle|0,A_0\rangle\langle1,A_1|+\text{adjoint}.
$$
Environmental records make $\langle E_1|E_0\rangle$ very small for macroscopically distinct alternatives. If independent fragments have overlaps $d_k$, their combined overlap is $\prod_kd_k$, explaining the rapid suppression. The apparatus interaction also favours robust records, giving <environment-induced pointer-basis selection> rather than an arbitrary choice of basis.<Pointer basis of quantum apparatus: Into what mixture does the wave packet collapse? by W. H. Zurek>[1981 paper, apparatus–environment coupling and stable pointer records]{ref}
What follows is approximate agreement, for local apparatus <observables>, with a classical probabilistic mixture. What does not follow is that either term has ceased to exist. Even with orthogonal environmental records, the global <quantum state> is a <pure state> with <quantum entanglement>. For orthogonal branches $|B_0\rangle,|B_1\rangle$, the global <observable> $|B_0\rangle\langle B_1|+|B_1\rangle\langle B_0|$ has expectation $2\operatorname{Re}(a^*b)$ in the superposition and zero in the corresponding incoherent mixture. Its practical inaccessibility is not its mathematical disappearance. The reduced state is an <improper mixed state>, not proof that an unknown single branch was selected.
Different interpretations use this result differently. In <Bohmian mechanics>, disjoint environmental configurations make one occupied branch an effective description of the actual apparatus. In the <many-worlds interpretation>, robust branches supply approximately autonomous relative records, without a unique global outcome. In the <GRW model>, a separate stochastic change of the <wave function> selects a dominant component; environmental <decoherence> helps explain ordinary records but is not the collapse law. These are different completions of the argument.
Nor does diagonality derive the <Born rule>. Interpreting the diagonal entries as experienced outcome <probabilities> requires a probability rule or an interpretive argument. Extremely small overlaps also leave approximations and possible <quantum recoherence>; a finite controlled environment can in principle be manipulated to reveal the global coherence. The appropriate verdict is that <decoherence> resolves much of the preferred-record-basis and effective-classicality problem while leaving the unique-outcome and probability problems dependent on further theory. It is an essential physical mechanism in many accounts, not a self-contained solution by relabelling a <partial trace> as collapse.
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