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### What is the meaning of $\langle e_k|U|e_0\rangle$ when $U$ acts on a larger Hilbert space than that in which $|e_0\rangle$ and $|e_k\rangle$ live?
TL;DR: We can understand the object $E_k=\langle e_k|U|e_0\rangle$ rigorously in two steps. First, think of $\langle e_k|$ and $|e_0\rangle$ as linear functions. Next, treat the implicit operation in \$...