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Journal of Physics A Mathematical and Theoretical

Canonical representation of three-qubit states with real amplitudes

Oscar Mario Perdomo

Published Oct 8, 2021• 2 citations
Publisher

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).

DOI
10.1088/1751-8121/ac2e27

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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).
From the Schmidt representation we have that, up to local gates, every 2-qubit state can be written as [ϕ⟩=λ1[00⟩+λ2[11⟩[ϕ\rangle=λ_1 [00\rangle+λ_2 [11\rangle[ϕ⟩=λ1​[00⟩+λ2​[11⟩ with λ1λ_1λ1​ and λ2λ_2λ2​ 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⟩=λ1[000⟩+λ2eiθ[100⟩+λ3[101⟩+λ4[110⟩+λ5[111⟩[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[ϕ⟩=λ1​[000⟩+λ2​eiθ[100⟩+λ3​[101⟩+λ4​[110⟩+λ5​[111⟩ with λi≥0λ_i\ge0λi​≥0 and 0≤θ≤π0\le θ\le π0≤θ≤π. In this paper, we show that no every 3-qubit state with real amplitudes can be transform in the form [ϕ⟩=λ1[000⟩+λ2[100⟩+λ3[101⟩+λ4[110⟩+λ5[111⟩[ϕ\rangle=λ_1 [000\rangle+λ_2 [100\rangle+λ_3[101\rangle+λ_4[110\rangle+λ_5[111\rangle[ϕ⟩=λ1​[000⟩+λ2​[100⟩+λ3​[101⟩+λ4​[110⟩+λ5​[111⟩ by using local gates in the orthogonal group (the group generated by Ry(θ)R_y(θ)Ry​(θ) and XXX gates). We also show that, up to local gates in the orthogonal group, every 3-qubit with real amplitudes can be written as [ϕ⟩=λ1[000⟩+λ2[011⟩+λ3[101⟩+λ4[110⟩+λ5[111⟩[ϕ\rangle=λ_1 [000\rangle+λ_2 [011\rangle+λ_3[101\rangle+λ_4[110\rangle+λ_5[111\rangle[ϕ⟩=λ1​[000⟩+λ2​[011⟩+λ3​[101⟩+λ4​[110⟩+λ5​[111⟩ with the λiλ_iλi​ real numbers. An explanation of this result can be found in the youtube video \url{https://youtu.be/gDN20QHzsoQ}
We present a generalization to three qubits of the standard Bloch sphere representation for a single qubit and of the seven-dimensional sphere representation for two qubits presented in Mosseri et al (Mosseri R and Dandoloff R 2001 J. Phys. A: Math. Gen. 34 10243). The Hilbert space of the three-qubit system is the 15-dimensional sphere S 15 , which allows for a natural (last) Hopf fibration with S 8 as base and S 7 as fibre. A striking feature is, as in the case of one and two qubits, that the map is entanglement sensitive, and the two distinct ways of un-entangling three qubits are naturally related to the Hopf map. We define a quantity that measures the degree of entanglement of the three-qubit state. Conjectures on the possibility of generalizing the construction for higher qubit states are also discussed.
We present a complete set of local unitary invariants for generic multiqubit systems which gives necessary and sufficient conditions for two states being local unitary equivalent. These invariants are canonical polynomial functions in terms of the generalized Bloch representation of the quantum states. In particular, we prove that there are at most 12 polynomial local unitary invariants for two-qubit states and at most 90 polynomials for three-qubit states. Comparison with Makhlin's 18 local unitary invariants is given for two-qubit systems.
Pure states of three-qubit quantum systems are represented by eight-dimensional complex vectors, whereas their unitary transformations are described by 8×88 \times 88×8 complex matrices. Motivated by this correspondence, we extend our algebraic model for two-qubit systems to eight-dimensional spaces endowed with commutative and associative multiplication. To each eight-dimensional number we assign a matrix and prove that the resulting correspondence is a homomorphic bijection. We derive conditions under which the associated matrix is unitary and hence represents a three-qubit quantum gate. This leads to a specific algebraic notion of a unitary state: a normalized state whose multiplication matrix is unitary. We construct 16,384 basis spaces; within each fixed space, multiplication generates a commuting family of quantum gates. For any two unitary states in the class considered, we give an explicit transition-gate construction within one of these algebraic families. For real components, the unitary states in a representative basis space form a union of thirty-two four-dimensional manifolds in R8\mathbb{R}^{8}R8. The construction therefore yields new structured classes of three-qubit unitary operators and invariant state sets, while its extension to larger multiqubit systems remains open.
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.