2026/07/17 by Jef Pauwels, Nicolas Gisin
#quant-ph
We give a closed-form construction of the n-qubit Elegant Joint Measurement (EJM) proposed in [PRL 136, 190201 (2026)] and show that it is part of a tunable family of measurements with tetrahedrally arranged Bloch vectors. The construction is based on the interference pattern implied by a single phase polynomial built from the elementary symmetric functions. It realises a regular tetrahedral measurement for every n, and the corresponding measurement unitary lies at level n+1 of the Clifford hierarchy. Starting from this measurement, we ask whether the size of the local tetrahedron -- and hence the entanglement of the basis -- can be varied while preserving its symmetry. For every even n the answer is yes, and remarkably the size follows the same one-parameter law that governs the known two-qubit family, interpolating down to a 1-uniform basis. For n=3 the EJM is locally isolated, while for odd n≥5 we do not know an analogous closed-form family. We also give an analogous construction, valid for every n ≥3, with square local geometry.