Pauli problem in dimension 4
The finite-dimensional Pauli problem asks how many measurements in orthonormal bases determine every pure state of a \(d\)-level quantum system up to a global phase. Four bases always suffice, and the answer is known to be three for \(d=2\) and four for \(d=3\) and \(d\geq5\). Dimension four remained open because the embedding argument used in other dimensions fails there: the pure-state space \(\mathbb{CP}^3\) embeds in \(\R^9\), the space of the nine independent probabilities of three bases. We show that three bases do not suffice in \(\C^4\), so exactly four are needed. The same argument…
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