Reading about superdense coding I came upon a calculation I can not understand.We have an EPR entangled pair of qubits $\frac{1}{\sqrt2}(|00\rangle + |11\rangle)$ and we want to apply a Pauli X gate to the first of the entangled qubits. On wikipedia, there is an example showing that such an operation would lead to the state $\frac{1}{\sqrt2}(|10\rangle + |01\rangle)$. I can't see an intuition behind it. I know that the Pauli X gate operates as a quantum NOT, but at the same time I do not know how to write it algebraically.How can a single qubit from an entangled pair be represented? Is it $\frac{1}{\sqrt2}(|0\rangle + |1\rangle)$ for the pair above? And if so, why does the Pauli X gate change it at all (if we wrote the operation using matrices, the qubit would be represented as $\left[\frac{1}{\sqrt2} \frac{1}{\sqrt2}\right]$, which multiplied by the Pauli X gate matrix would not change at all...)
How to apply a 1-qubit gate to a single qubit from an entangled pair?
quantum computing
I came to a conclusion that we do not actually take away a single qubit from an entangled pair and operate it, but operate on both qubits with Identity operator on the one we do not want to change.So basically to apply Pauli X gate to the first qubit from an entangled pair, we apply X tensored with $I$ to the entangled pair to get the result.That is, to operate a not on the first qubit, we evolve the system by$$ \sigma_x \otimes I = \begin{pmatrix}0 & 1 \\1 & 0 \end{pmatrix}\otimes\begin{pmatrix}1 & 0 \\0 & 1 \end{pmatrix} =\begin{pmatrix}0 & 0 & 1 & 0\\0 & 0 & 0 & 1\\1 & 0 & 0 & 0\\0 & 1 & 0&  0\end{pmatrix} $$Intuitively it is like applying a single operation to the first qubit and identity to the second one (effectively not changing its state).Thus if we start with the entangled state $|\psi\rangle \frac{1}{\sqrt2}(|00\rangle + |11\rangle)$ we end up with$$ (\sigma_x \otimes I)|\psi\rangle = \begin{pmatrix}0 & 0 & 1 & 0\\0 & 0 & 0 & 1\\1 & 0 & 0 & 0\\0 & 1 & 0&  0\end{pmatrix} \cdot \frac1{\sqrt{2}}\begin{pmatrix}1 \\0 \\0 \\1 \end{pmatrix} = \frac1{\sqrt{2}}\begin{pmatrix}0 \\1 \\1 \\0 \end{pmatrix} = \frac1{\sqrt{2}}(|01\rangle+|10\rangle)$$