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The two-path interference argument constructed a state that predicts path and detector probabilities. We now ask a new question:

Given the state now, what rule predicts the state later?

1. What experiment produces a time-update problem?

The previous post defined

\[\Omega:=\{A,B\}, \qquad \mathcal H:=\mathbb C^\Omega :=\{\psi\mid\psi:\Omega\to\mathbb C\}.\]

This note begins with an experiment we want to predict: a two-path interferometer. We will discover what information ordinary probabilities fail to retain, construct a state that retains it, and derive the rule that maps that state to detector probabilities.

Only after the real two-dimensional construction is complete will we ask why the same state is usually written with complex numbers.