Symmetric quantum joint measurements on multiple qubits

We investigate the generalization of symmetric quantum joint measurements on multiple qubits. We first describe a method for constructing a symmetric joint measurement basis for three qubits by utilizing single-qubit states corresponding to the four vertices of an isohedral tetrahedron on the Bloch sphere. We then demonstrate the expected tetrahedral symmetry of the current measurement basis and discuss its application in a trilocal star-shaped network. This architecture enables us to generalize the two-qubit symmetric joint measurement to an $n$-qubit version, preserving the tetrahedral or hexahedral symmetry.