Schmidt representation of 3-qubits with real amplitudes

Abstract

From the Schmidt representation we have that, up to local gates, every 2-qubit state can be written as $[\varphi ⟩={\lambda }_1[00⟩+{\lambda }_2[11⟩$ 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 $[\varphi ⟩={\lambda }_1[000⟩+{\lambda }_2{e}^{i\theta }[100⟩+{\lambda }_3[101⟩+{\lambda }_4[110⟩+{\lambda }_5[111⟩$ with ${\lambda }_i\ge 0$ 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 $[\varphi ⟩={\lambda }_1[000⟩+{\lambda }_2[100⟩+{\lambda }_3[101⟩+{\lambda }_4[110⟩+{\lambda }_5[111⟩$ 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 $[\varphi ⟩={\lambda }_1[000⟩+{\lambda }_2[011⟩+{\lambda }_3[101⟩+{\lambda }_4[110⟩+{\lambda }_5[111⟩$ with the ${\lambda }_i$ real numbers. An explanation of this result can be found in the youtube video \url{https://youtu.be/gDN20QHzsoQ}

arXiv comment9 pages. A youtube video explainnig the result in this paper: https://youtu.be/gDN20QHzsoQ

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