What quantum gates are needed to get the state $|01\rangle+|10\rangle$ from $|00\rangle$? I was wondering if I start with two qubits in the state
$$|00\rangle$$
If it's possible to apply gates to get it to the state
$$\frac{|01\rangle + |10\rangle}{\sqrt{2}}$$
I have tried applying the Hadamard Gate, Controlled X etc, But I couldn't make this state. So I'm curious if it's possible and I am just missing something very obvious.
 A: Applying Hadamard + CNOT takes you from $|00\rangle$ to $\dfrac{|00\rangle + |11\rangle}{\sqrt{2}}$. Now, just apply the single-qubit $X$ Pauli operator (which swaps $|0\rangle$ with $|1\rangle$ and vice-versa) to either one of the two qubits, and you get the target state $\dfrac{|01\rangle + |10\rangle}{\sqrt{2}}$.
A: Let's consider a space of four states $\{|00\rangle, |01\rangle, |10\rangle, |11\rangle\}$. The question is whether exist any operator for which,
$$
\left(\begin{array}{cccc}
a_{11} & a_{12} & a_{13} & a_{14} \\
a_{21} & a_{22} & a_{23} & a_{24} \\
a_{31} & a_{32} & a_{33} & a_{34} \\
a_{41} & a_{42} & a_{43} & a_{44} \\
\end{array}\right) 
\left(
\begin{array}{c}
1 \\ 0 \\ 0 \\ 0
\end{array}
\right) = \frac{1}{\sqrt{2}} 
\left(
\begin{array}{c}
0 \\ 1 \\ 0 \\ 1
\end{array}
\right)
$$
Of course one can invent such an operator, e.g.
$$
\hat{a} = \frac{1}{\sqrt{2}}\left(\begin{array}{cccc}
0 & a_{12} & a_{13} & a_{14} \\
1 & a_{22} & a_{23} & a_{24} \\
0 & a_{32} & a_{33} & a_{34} \\
1 & a_{42} & a_{43} & a_{44} \\
\end{array}\right),
$$
but I do not know if it corresponds to any known quantum gate. Remaining $a_{ij}$ can be arbitrary.
