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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.

At $A=\pi /4$, defineThey are normalized and orthogonal. The full state is the pure superposition $(|\uparrow d_{\uparrow }{E}_{\leftarrow }⟩+|\downarrow d_{\downarrow }{E}_{\to }⟩)/\sqrt{2}$. A nonselective von Neumann process 1 selecting the original detector records givesWithout imposing any projection on the full state, tracing out the orthogonal environment states already gives ${\rho }_{SD}={\rho }_{up/down}$ from part (a). These two global descriptions share that reduced state but differ in their global coherence. Quantum decoherence does not make a pure global state literally equal to the corresponding statistical mixture.