= Solution
With two coherent paths, the <probability amplitude> at a screen is $\psi_1(x)+e^{i\phi}\psi_2(x)$. The <Born rule> gives
$$
p(x)=|\psi_1|^2+|\psi_2|^2+
2\operatorname{Re}\left(e^{i\phi}\psi_1^*\psi_2\right).
$$
The final term shifts when the relative phase changes, producing <interference fringes>. For slit separation $d$ in the far field, the phase difference is approximately $2\pi d\sin\theta/\lambda$; constructive <interference> occurs at $d\sin\theta=n\lambda$. Finite slit widths add a <diffraction> envelope, so an ideal equally bright, infinitely narrow slit model does not predict the full observed envelope.
<single-electron interference pattern buildup> starts with isolated localized detector events at sufficiently low source intensity. Repeated events accumulate the <interference> <probability> distribution. Thus detection is discrete while propagation is described by coherent <probability amplitudes>. The effect cannot be explained by <interference> between different particles if the apparatus is occupied by at most one particle at a time. A motivating realization used a field-emission <electron> source, an <electron> biprism to recombine two paths, <electron> lenses and a position-sensitive <electron> counter; it recorded individual arrivals building up the fringes. This was a biprism realization of two-path <interference>, not a plate with two literal slits.<Demonstration of single-electron buildup of an interference pattern by Akira Tonomura and others>[section III, American Journal of Physics 57, pp. 117–120]{ref}
The crucial comparison is between both coherent paths open and a preparation that supplies which-path information. If the paths become correlated with detector states $|D_1\rangle,|D_2\rangle$, the joint state is $\psi_1|D_1\rangle+e^{i\phi}\psi_2|D_2\rangle$. Ignoring that detector multiplies the <interference> term by $\langle D_1|D_2\rangle$. Orthogonal detector states remove the term completely, while identical detector states preserve it. This is a consequence of <entanglement> and the <partial trace>; a human need not read the detector.
In a real test, check the single-path distributions, detector background, source coherence and stability of the relative phase. Classical phase drift can wash out fringes without supplying which-path information. Conversely, localized detections alone do not establish coherent two-path propagation; the controlled phase dependence of their accumulated distribution provides the discriminating evidence.
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