Schmidt representation of 3-qubits with real amplitudes

TL;DR

This paper investigates the Schmidt representation for 3-qubit states with real amplitudes, showing limitations of local orthogonal gates in transforming these states into certain standard forms and proposing alternative canonical forms.

Contribution

It demonstrates that not all real-amplitude 3-qubit states can be transformed into the known form using orthogonal local gates, and introduces a new canonical form achievable with such gates.

Findings

Not all real 3-qubit states can be transformed into the standard form with orthogonal gates.

Every real 3-qubit state can be expressed in a new canonical form using orthogonal local gates.

The paper clarifies the limitations and possibilities of local orthogonal transformations for real 3-qubit states.

Abstract

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}