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arXiv:2012.14031

Schmidt representation of 3-qubits with real amplitudes

Oscar Mario Perdomo

Published Dec 27, 2020• 0 citations
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Abstract

From the Schmidt representation we have that, up to local gates, every 2-qubit state can be written as $[ϕ\rangle=λ_1 [00\rangle+λ_2 [11\rangle$ with $λ_1$ and $λ_2$ 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 $[ϕ\rangle=λ_1 [000\rangle+λ_2 e^{i θ}[100\rangle+λ_3[101\rangle+λ_4[110\rangle+λ_5[111\rangle$ with $λ_i\ge0$ and $0\le θ\le π$. In this paper, we show that no every 3-qubit state with real amplitudes can be transform in the form $[ϕ\rangle=λ_1 [000\rangle+λ_2 [100\rangle+λ_3[101\rangle+λ_4[110\rangle+λ_5[111\rangle$ by using local gates in the orthogonal group (the group generated by $R_y(θ)$ 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 $[ϕ\rangle=λ_1 [000\rangle+λ_2 [011\rangle+λ_3[101\rangle+λ_4[110\rangle+λ_5[111\rangle$ with the $λ_i$ real numbers. An explanation of this result can be found in the youtube video \url{https://youtu.be/gDN20QHzsoQ}

DOI
10.48550/arxiv.2012.14031

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Abstract Let us say that a three-qubit state u 000 |000⟩ + u 001 |001⟩ + ⋯ + u 111 |111⟩ is real if all its amplitudes u rst are real numbers. We will prove that for every real three-qubit state | ϕ ⟩, there exist three angles θ 0 , θ 1 and θ 2 such that R y ( θ 2 ) ⊗ R y ( θ 1 ) ⊗ R y ( θ 0 )| ϕ ⟩ is a three-qubit of the form λ 1 |000⟩ + λ 2 |011⟩ + λ 3 |101⟩ + λ 4 |110⟩ + λ 5 |111⟩ with the λ i real numbers. In contrast with the general case, the case of three-qubits with complex amplitudes, we proved that for three qubit states, the dimension of the real entanglement space (the space obtained by identifying real qubit states with local orthogonal gates, instead of local unitary gates) is 4 and in this paper we find four linearly independent polynomial invariants of degree 4 which are not possible to find for the different Schmidt representations of three qubit states. See (Acín et al 2000 Phys. Rev. Lett. 85 1560; Acín et al 2001 J. Phys. A: Math. Gen. 34 6725; Carteret et al 2000 J. Math. Phys. 41 7932; Sudbery 2001 J. Phys. A: Math. Gen. 34 643).
We will call a pure qubit state real if all its amplitudes are real numbers. We show that any real 3-qubit state can be prepared using Ry(θ)R_y(\theta)Ry​(θ) gates and at most four controlled-ZZZ gates, and we conjecture that four is optimal. We also present an algorithm -- different from the 2008 algorithm given by Znidaric, Giraud and Georgeot -- that prepares any 3-qubit state using local gates and at most three controlled-ZZZ gates. Videos showing how our method works for two- and three-qubit states can be found at https://youtu.be/LIdYSs-rE-o and https://youtu.be/Kne0Vq7gyzQ
In this paper, we present a new and algebraically simple algorithm that prepares any pure three-qubit state using only local gates and at most three controlled-[Formula: see text] gates. In 2008, Znidaric et al. already showed that three is the optimal number of controlled-[Formula: see text] gates required for the preparation of three-qubit states if in addition to those gates only local gates are used. If we restrict the local gates to [Formula: see text] gates, our algorithm provides a way to prepare any pure three-qubit state [Formula: see text] with real amplitudes using at most four controlled-[Formula: see text] gates when its hyperdeterminant is negative and three or fewer controlled-[Formula: see text] gates otherwise. We conjecture the existence of three-qubit states with real amplitudes that cannot be prepared using only [Formula: see text] gates and less than four controlled-[Formula: see text] gates.
We prove for any pure three-quantum-bit state the existence of local bases which allow to build a set of five orthogonal product states in terms of which the state can be written in a unique form. This leads to a canonical form which generalizes the two-quantum-bit Schmidt decomposition. It is uniquely characterized by the five entanglement parameters. It leads to a complete classification of the three-quantum-bit states. It shows that the right outcome of an adequate local measurement always erases all entanglement between the other two parties.
Let us denote by [Formula: see text] the Clifford group (the circuit or operations generated by Hadamard, [Formula: see text] phase and the controlled-NOT gates) and by [Formula: see text] the set of qubit states that can be prepared by circuits from the Clifford group. In other words, a state [Formula: see text] if [Formula: see text] where [Formula: see text]. We will refer to states in [Formula: see text] as Clifford states. This paper studies the set of all three-qubit Clifford states. We prove that [Formula: see text] has 8640 states and if we define two states [Formula: see text] and [Formula: see text] in [Formula: see text] to be equivalent if [Formula: see text], with [Formula: see text] a local transformation in [Formula: see text], then the resulting quotient space has five orbits. More exactly, [Formula: see text] where the orbit [Formula: see text] is made up of states with entanglement entropy [Formula: see text]. For example, the first orbit [Formula: see text] contains the state [Formula: see text] and corresponds to the unentangled Clifford states. We say that [Formula: see text] is a real state if all its amplitudes [Formula: see text] are real numbers. We also say that an operator is real if all the entries of its matrix representation with respect to the computational basis are real numbers. In this paper, we also study the set of real Clifford 3 qubits and the way this set splits when we identify two real Clifford states [Formula: see text] and [Formula: see text] to be equivalent if [Formula: see text] where [Formula: see text] is a local real Clifford operator. An interesting aspect that follows from this study of Clifford states is the existence of two real Clifford states [Formula: see text] and [Formula: see text] that can be connected with a Clifford local transformation but they cannot be connected with a real Clifford local transformation. This is, the equation [Formula: see text] for [Formula: see text], does have a solution in the set of local transformations from [Formula: see text] but it does not have a solution among the local transformations from [Formula: see text] that are real. We go a little deeper and show that the equation [Formula: see text] does not have a solution for any local operation (not necessarily Clifford) whose entries are real numbers. Finally, we show how the CNOT gates act on the set of Clifford states and also in the set of real Clifford states.