 Solution

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 byTaking the partial trace over the environment gives the system–apparatus reduced density matrixEnvironmental records make $⟨E_1|E_0⟩$ very small for macroscopically distinct alternatives. If independent fragments have overlaps $d_k$, their combined overlap is ${\prod }_k{d}_k$, explaining the rapid suppression. The apparatus interaction also favours robust records, giving environment-induced pointer-basis selection rather than an arbitrary choice of basis.^[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 $|B_0⟩,|B_1⟩$, the global observable $|B_0⟩⟨B_1|+|B_1⟩⟨B_0|$ has expectation $2\mathrm{Re}(a^{\ast }b)$ 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.