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A general question is posed to the quantum community. Partial results areformulated in a self-contained way. In particular, the title question isanswered affirmatorily in two cases: 1) The case of spin/ angular momentum of apartcle; 2) A general symmetry situation under certain technical assumptions.

When is a set of questions to nature together with sharp answers to those questions in one-to-one correspondence with a set of quantum states?