Optimal discrimination of geometrically uniform qubit and qutrit states

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.

Figure 1

The Bloch sphere representation of the generating state ${\vec{v}}_0$. By selecting the axis of rotation as the $z$ axis, the three states ${\vec{v}}_0,{\vec{v}}_1,\text{and}{\vec{v}}_2$ all lie in the $z=\zeta$ plane, and the angle between any two adjacent vectors in the $z=\zeta$ plane is $2\pi /3$.

Figure 2

(a) The projection of the Bloch vectors ${\vec{v}}_0,{\vec{v}}_1,\text{and}{\vec{v}}_2$ of the qubit states on $z=\zeta$. Since the states are not required to be pure, their Bloch vectors may not touch the border circle $x^2+y^2=1-{\zeta }^2$. The angle between any two adjacent vectors is $2\pi /3$, and their position is completely determined by the length of the generating state ${\vec{v}}_0$ and the azimuthal angle $\varphi$. Only when the states are pure, i.e., the vector ${\vec{v}}_i$ reaches the border circle, the optimal measurement is the square-root measurement. In general, the optimal measurement for discriminating $\{{\vec{v}}_0,{\vec{v}}_1,{\vec{v}}_2\}$ is the square-root measurement obtained for $\{{\vec{v}}_0^{\prime },{\vec{v}}_1^{\prime },{\vec{v}}_2^{\prime }\}$. (b) The optimal measurement on the equator plane. The representing vectors ${\vec{\mu }}_0,{\vec{\mu }}_1,\text{and}{\vec{\mu }}_2$ are all of unit length, and the measurement is completely determined by the azimuthal angle $\varphi$. The Bloch vectors $\{{\vec{v}}_0,{\vec{v}}_1,{\vec{v}}_2\}$ and $\{{\vec{v}}_0^{\prime },{\vec{v}}_1^{\prime },{\vec{v}}_2^{\prime }\}$ both correspond to the optimal measurement represented by the vectors $\{{\vec{\mu }}_0,{\vec{\mu }}_1,{\vec{\mu }}_2\}$.