Designing a Procedure to Distinguish 4 States

[Kreizhn]

Homework Statement

Design a procedure (consisting of unitary operators and a measurement) that best distinguishes the following states
$$\frac1{\sqrt3} \left( | 0 \rangle + | 1 \rangle + |2 \rangle \right)$$
$$\frac1{\sqrt3} \left( | 0 \rangle - | 1 \rangle - |2 \rangle \right)$$
$$\frac1{\sqrt3} \left( -| 0 \rangle + | 1 \rangle - |2 \rangle \right)$$
$$\frac1{\sqrt3} \left( -| 0 \rangle - | 1 \rangle + |2 \rangle \right)$$

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?

 

[Jhero]

Thank you for your question. In order to distinguish between these states, we can use a measurement known as the Hadamard measurement. This measurement involves applying the Hadamard gate to each state and then measuring the state in the computational basis. This measurement will result in the states being projected onto one of the basis states with equal probability, allowing us to distinguish between them.

To break down the procedure, we can apply the Hadamard gate to each state to get:

\frac1{\sqrt3} \left( \frac{| 0 \rangle + | 1 \rangle}{\sqrt2} + \frac{| 0 \rangle - | 1 \rangle}{\sqrt2} + |2 \rangle \right)
\frac1{\sqrt3} \left( \frac{| 0 \rangle - | 1 \rangle}{\sqrt2} - \frac{| 0 \rangle + | 1 \rangle}{\sqrt2} - |2 \rangle \right)
\frac1{\sqrt3} \left( \frac{-| 0 \rangle + | 1 \rangle}{\sqrt2} + \frac{| 0 \rangle - | 1 \rangle}{\sqrt2} - |2 \rangle \right)
\frac1{\sqrt3} \left( \frac{-| 0 \rangle - | 1 \rangle}{\sqrt2} - \frac{| 0 \rangle + | 1 \rangle}{\sqrt2} + |2 \rangle \right)

We can see that the coefficients of the first term in each state are the same, and similarly for the second and third terms. This means that after applying the Hadamard gate, all four states will have the same probability of being measured in the computational basis, making it a fair measurement to distinguish between them.

I hope this helps answer your question. Please let me know if you have any further questions or concerns.