Partition of 3-qubits using local gates

It is well known that local gates have smaller error than non-local gates. For this reason it is natural to make two states equivalent if they differ by a local gate. Since two states that differ by a local gate have the same entanglement entropy, then the entanglement entropy defines a function in the quotient space. In this paper we study this equivalence relation on (i) the set $\mathbb{R}\hbox{Q}(3)$ of 3-qubit states with real amplitudes, (ii) the set $Q\mathcal{C}$ of 3-qubit states that can be prepared with gates on the Clifford group, and (iii) the set $Q\mathbb{R}\mathcal{C}$ of 3-qubit states in $Q\mathcal{C}$ with real amplitudes. We show that the set $Q\mathcal{C}$ has 8460 states and the quotient space has 5 elements. We have $\frac{Q\mathcal{C}}{\sim}=\{S_{0},S_{2/3,1},S_{2/3,2},S_{2/3,3},S_{1}\}$. As usual, we will call the elements in the quotient space, orbits. We have that the orbit $S_0$ contains all the states that differ by a local gate with the state $|000\rangle$. There are 1728 states in $S_0$ and as expected, they have zero entanglement entropy. All the states in the orbits $S_{2/3,1},S_{2/3,2},S_{2/3,3}$ have entanglement entropy $2/3$ and each one of these orbits has 1152 states. Finally, the orbit $S_{1}$ has 3456 elements and all its states have maximum entanglement entropy equal to one. We also study how the controlled not gates $CNOT(1,2)$ and $CNOT(2,3)$ act on these orbits. For example, we show that when we apply a $CNOT(1,2)$ to all the states in $S_0$, then 960 states go back to the same orbit $S_0$ and 768 states go to the orbit $S_{2/3,1}$. Similar results are obtained for $\mathbb{R}Q\mathcal{C}$. We also show that the entanglement entropy function reaches its maximum value 1 in more than one point when acting on $\frac{\hbox{$\mathbb{R}$Q(3)} }{\sim}$.