= Solution
<Quantum complementarity> concerns experimental arrangements or observables that cannot be combined into one sharp classical description. For a <qubit>, the <Pauli operators> $X$ and $Z$ have incompatible eigenbases: a definite $Z$ value gives equal probabilities for the two $X$ outcomes. One can measure either observable sharply on a given preparation, but a joint measurement cannot reproduce both sharp projective measurements on every state. Complementarity is not merely ignorance of two simultaneously definite quantum properties.
In a balanced two-path <interferometer>, <which-path information> reduces the <interference> available when the paths recombine. If the path becomes entangled with detector states,
$$
|\Psi\rangle=\frac{|u\rangle|E_u\rangle+e^{i\varphi}|v\rangle|E_v\rangle}{\sqrt2},
$$
the path <reduced density matrix> has off-diagonal magnitude $|\langle E_u|E_v\rangle|/2$. Its <interference visibility> is $V=|\langle E_u|E_v\rangle|$. For equally likely pure marker states, optimal path distinguishability is $D=\sqrt{1-|\langle E_u|E_v\rangle|^2}$, giving $D^2+V^2=1$ in this ideal model. Identical marker states preserve full <interference>; orthogonal ones permit perfect path discrimination and erase the unconditional fringes.
A concrete optical realization puts different <polarizations> on the two arms of a <Mach-Zehnder interferometer>. A detector that ignores <polarization> sees reduced <interference> because <polarization> records the path. Projecting the outgoing light onto a common <polarization> erases that information in the selected subensemble and can recover conditional fringes; summing all outcomes again removes them. This is a physical change in which alternatives remain distinguishable, not a retrospective alteration of earlier detections.
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