Canonical representation of three-qubit states with real amplitudes

Abstract

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).