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

Question

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?

Answer 1

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.

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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 $XZ = -ZX$.