Does the matrix representation of a logic gate change according to the basis states?
To expand upon my question, allow me to introduce a problem that I'm attempting:
Suppose $\lvert+\rangle = \frac{1}{\sqrt{2}}(\lvert0\rangle + \lvert1\rangle)$ and $\lvert-\rangle = \frac{1}{\sqrt{2}}(\lvert0\rangle - \lvert1\rangle)$. Find the action of the CZ gate on the basis states $\lvert+\rangle\lvert0\rangle$, $\lvert-\rangle\lvert0\rangle$, $\lvert+\rangle\lvert1\rangle$, $\lvert-\rangle\lvert1\rangle$ and $\lvert0\rangle\lvert+\rangle$, $\lvert0\rangle\lvert-\rangle$, $\lvert1\rangle\lvert+\rangle$, $\lvert1\rangle\lvert-\rangle$.
For clarification:
$$ \lvert0\rangle = \begin{pmatrix} 1 \ 0 \end{pmatrix}, \lvert1\rangle = \begin{pmatrix} 0 \ 1 \end{pmatrix}$$
In this question, I am asked to calculate the action of the CZ matrix on the different basis states. This is a straightforward task, but what I am confused about is the mathematical form of the CZ matrix/operator.
The CZ gate is defined to be a gate that applies the Z Pauli matrix on the target qubit when the control qubit is $\lvert1\rangle$. For the basis states $\lvert00\rangle$, $\lvert01\rangle$, $\lvert10\rangle$ and $\lvert11\rangle$, the CZ gate has the following matrix form:
$$CZ = \begin{pmatrix} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0\ 0 & 0 & 1 & 0\ 0 & 0 & 0 & -1 \end{pmatrix} \tag{1} $$
As you can see from the question that I'm working on, the basis states are different. 656
How am I supposed to tackle this question? I have two approaches in mind:
Approach 1: The definition of any logic gate is defined with respect to the basis states $\lvert0\rangle$ and $\lvert1\rangle$. Therefore, the matrix in equation 1 is still valid for my basis states and I should compute $CZ\lvert+\rangle\lvert0\rangle$, $CZ\lvert-\rangle\lvert0\rangle$, $CZ\lvert+\rangle\lvert1\rangle$, etc.
Approach 2: The definition of a logic gate changes according to the basis states. The matrix form in equation 1 is no longer valid. Instead, I should apply the Z Pauli matrix on the target qubits if the control qubit is $\lvert1\rangle$ i.e. I apply the Z Pauli matrix on the target qubits in $\lvert1\rangle\lvert+\rangle$ and $\lvert1\rangle\lvert-\rangle$ as they are the only basis states which have the control qubit $\lvert1\rangle$.
I am new to this topic, so if I have missed out anything important or not explained my point clearly, please comment below so I can update the question. 66