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Linearity of the detector's unitary time evolution maps the two input branches separately, giving the quantum premeasurement stateIn the ordered tensor product basis $(|\uparrow d_{\uparrow }⟩,|\uparrow d_{\downarrow }⟩,|\downarrow d_{\uparrow }⟩,|\downarrow d_{\downarrow }⟩)$, its pure density matrix isA nonselective projective measurement in the indicated detector basis applies $\rho \mapsto {\sum }_j(I_S\otimes {\Lambda }_j)\rho (I_S\otimes {\Lambda }_j)$, removing the two off-diagonal terms:This is the usual nonselective meaning of von Neumann process 1. It is not equal to the pure projector $|{\Phi }^C⟩⟨{\Phi }^C|$. If “the detector has clicked” means conditioning on an actual up outcome, the normalized postselected state is instead $|\uparrow d_{\uparrow }⟩$, with probability $1/2$. The correlated superposition describes the coherent premeasurement before that selection.

The problem of explaining which basis represents stable measurement records. A maximally entangled quantum premeasurement state can have several equally weighted Schmidt decompositions. Choosing a different decomposition does not alter that pure state, but declaring a projection in a different basis generally changes the resulting mixture. Environment-induced pointer-basis selection supplies a dynamical account of robust records, while leaving the further question of a single realized outcome distinct.

The projection step in the traditional measurement account. For outcome projectors $P_j$, the nonselective state is ${\sum }_j{P}_j\rho P_j$; conditional on outcome $j$ it is $P_j\rho P_j/\mathrm{Tr}(P_j\rho )$. Distinguishing these from the coherent state produced by quantum premeasurement avoids identifying a pure correlated projector with its dephased mixture.