# How does the Pauli Y gate act in the $|+\rangle, |-\rangle$ basis?

The X gate in the $$|+\rangle$$, $$|-\rangle$$ basis becomes the Z gate and vice versa.

What is the Pauli Y gate as a matrix transformation in the $$|+\rangle$$, $$|-\rangle$$ basis?

• What do you mean by X gate become Z gate? Does it mean that Z in Hadamard basis transform basis states between each other as X gate in computational basis? If so, please note that Y works similarly in circular basis. May 15 at 7:18

We have,

$$Y=i|+\rangle\langle -| -i|-\rangle\langle +|$$

You can calculate it easily using the fact that,

$$|0\rangle = \frac{1}{\sqrt 2}(|+\rangle + |-\rangle)$$,

$$|1\rangle = \frac{1}{\sqrt 2}(|+\rangle - |-\rangle)$$, and

$$Y = i|1\rangle\langle 0| -i|0\rangle\langle 1|$$

So, it has the same matrix up to a sign change.

Alternative solution is to use what you already mentioned in your question:

The X gate in the $$|+\rangle$$, $$|-\rangle$$ basis becomes the Z gate and vice versa

with the fact, $$Y=iXZ$$, and $$XZ = -ZX$$.