CANONICAL REPRESENTATION OF 3-QUBIT STATES WITH REAL AMPLITUDES

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 gates 3-qubit [000\rangle+\lambda_2 e^{i \theta }[100\rangle+\lambda_3[101\rangle+\lambda_4[110\rangle+\lambda_5[111\rangle$ $\lambda_i\ge0$ $0\le \le \pi$. In this paper, show no amplitudes transform in form [100\rangle+\lambda_3[101\rangle+\lambda_4[110\rangle+\lambda_5[111\rangle$ by using orthogonal group (the generated $R_y(\theta)$ $X$ gates). We also group, [011\rangle+\lambda_3[101\rangle+\lambda_4[110\rangle+\lambda_5[111\rangle$ $\lambda_i$ An explanation of result found youtube video \url{https://youtu.be/gDN20QHzsoQ}