# Qutrit States

1. Sep 26, 2009

### Kreizhn

1. The problem statement, all variables and given/known data
Design a procedure (consisting of unitary operators and a measurement) that best distinguishes the following states
$$\frac1{\sqrt3} \left( | 0 \rangle + | 1 \rangle + |2 \rangle \right)$$
$$\frac1{\sqrt3} \left( | 0 \rangle - | 1 \rangle - |2 \rangle \right)$$
$$\frac1{\sqrt3} \left( -| 0 \rangle + | 1 \rangle - |2 \rangle \right)$$
$$\frac1{\sqrt3} \left( -| 0 \rangle - | 1 \rangle + |2 \rangle \right)$$

3. The attempt at a solution
The inner product of all four states is the same, and comes out to $-\frac13$. Geometrically, if we were to consider the linear space spanned by the basis states $|a \rangle$ for $a = 1 ,2 , 3$ then these states would form the vertices of a tetrahedron. I can then find a mapping that puts two of these states onto the same basis state, but I'm unsure if this is optimal. Any ideas?