Pauli problem in dimension 4

Abstract — v1

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 shows that no ten vectors in \C^4 do phase retrieval. Since eleven vectors are known to suffice, the smallest phase-retrieval frame and the smallest rank-one POVM distinguishing all pure states in \C^4 both have eleven elements. All three lower bounds follow from one statement: every six-dimensional real space of traceless Hermitian $4\times4$ matrices with a common isotropic vector, that is, a nonzero $e$ with $e^*Te=0$ for all $T$ in the space, contains a nonzero matrix of rank at most two. We prove it with complex $K$-theory: otherwise, an odd unitary map on the five-sphere would have a $K^1$-class that is nonzero by antipodal symmetry but vanishes because of the isotropic vector.