  From the Schmidt representation we have that, up to local gates, every
2-qubit state can be written as $[\phi\rangle=\lambda_1 [00\rangle+\lambda_2
[11\rangle$ with $\lambda_1$ and $\lambda_2$ real numbers. For 3-qubits states,
it is known, PRL 2000 85, that up to local gates every 3-qubit state can be
written as $[\phi\rangle=\lambda_1 [000\rangle+\lambda_2 e^{i \theta
}[100\rangle+\lambda_3[101\rangle+\lambda_4[110\rangle+\lambda_5[111\rangle$
with $\lambda_i\ge0$ and $0\le \theta \le \pi$. In this paper, we show that no
every 3-qubit state with real amplitudes can be transform in the form
$[\phi\rangle=\lambda_1 [000\rangle+\lambda_2
[100\rangle+\lambda_3[101\rangle+\lambda_4[110\rangle+\lambda_5[111\rangle$ by
using local gates in the orthogonal group (the group generated by $R_y(\theta)$
and $X$ gates). We also show that, up to local gates in the orthogonal group,
every 3-qubit with real amplitudes can be written as $[\phi\rangle=\lambda_1
[000\rangle+\lambda_2
[011\rangle+\lambda_3[101\rangle+\lambda_4[110\rangle+\lambda_5[111\rangle$
with the $\lambda_i$ real numbers. An explanation of this result can be found
in the youtube video \url{https://youtu.be/gDN20QHzsoQ}
