Optimal discrimination of geometrically uniform qubit and qutrit states

Abstract

We consider discrimination of geometrically uniform states, and analytically solve this problem for qubits. For qutrits, we obtain the exact solution to the optimal measurement when the defining unitary matrix for the geometrically uniform states is degenerate. We also show that if the unitary is nondegenerate then the optimal measurement for discriminating geometrically uniform qubits or qutrits can always be expressed as a rank-1 operator, which converts the original discrimination problem into an optimization of a sum of trigonometric functions with two real variables in the case of qutrits. Additionally, a geometrical interpretation of the optimal measurement for discriminating geometrically uniform qubits is given via the Bloch sphere representation.