Generalized $W$ state of four qubits with exclusively the three-tangle

Sebastian Gartzke and Andreas Osterloh

Phys. Rev. A 98, 052307 – Published 6 November 2018

Abstract

We single out a class of states possessing only the three-tangle but distributed all over four qubits. This is a three-site analog of states from the $W$ class. The latter possess exclusively globally distributed pairwise entanglement as measured by the concurrence. We perform an analysis for four qubits, showing that such a state indeed exists. To this end we analyze specific states of four qubits for which all possible SL invariants vanish, and hence which are part of the $SL$ null cone. Instead, they will possess a certain unitary invariant. In analyzing the three-tangle of rank-two reduced density matrices of these states, we manage to show that in this particular case we reach the convex roof exactly. As an interesting by-product this solution is extended in the rank-two case to a homogeneous polynomial SL-invariant measure of entanglement of degree $2 m$ , if there are two states which correspond to an at most $n$ -fold degenerate solution in the zero polytope for $0 < n < m$ that can be combined with the convexified minimal characteristic curve at an $(2 m - n)$ -fold zero yielding a decomposition of $ρ$ . If more than one such state does exist in the zero polytope, a minimization must be performed. If no decomposition of $ρ$ is obtained in this way, it provides a better lower bound than the lowest convexified curve.

Received 22 December 2017
Revised 10 July 2018

DOI:https://doi.org/10.1103/PhysRevA.98.052307

Physics Subject Headings (PhySH)

Quantum entanglement

Quantum Information, Science & Technology

Authors & Affiliations

Sebastian Gartzke and Andreas Osterloh^*

Institut für Theoretische Physik, Universität Duisburg-Essen, D-47048 Duisburg, Germany

^*andreas.osterloh@uni-due.de

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Vol. 98, Iss. 5 — November 2018

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Figure 1
The Bloch sphere of density matrices made of the orthonormal states $| ψ_{i} 〉, i \in {1, 2}$ , is shown, together with two superpositions $| Z_{i} 〉, i \in {1, 2}$ , of them. These two states give a valid decomposition of the density matrix $ρ$ .
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Figure 2
Characteristic curves for certain values of $φ$ : $φ = 0$ [blue widely dashed curve that interconnects concavely the points at $p = 0$ and 1 with $(p, \sqrt{τ_{3}}) = (1 / 2, 0)]$ to the straight orange dashed line at $φ = π / 2$ up to $φ = π$ (the lowest red curve). The curves are symmetrically distributed around $φ = 0$ and $φ = π$ . The red lowest curve is the minimal characteristic curve and is already convex. Therefore it constitutes a lower bound to $\hat{\sqrt{τ_{3}}}$ . The zero polytope consists of a threefold degenerate zero at the angle $φ = π$ and a single zero at $φ = 0$ .
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Figure 3
Upper bounds to the convex roof $\hat{\sqrt{τ_{3}}}$ of $\sqrt{τ_{3}}$ . The characteristic curves at the angles $φ = 0$ (upper blue widely dashed curve) and $φ = π$ (lowest red curve) are shown together with valid decompositions of the density matrix, namely a convex combination of (a) one of the two eigenstates and $ρ_{0} = \frac{1}{2} (| Ψ (\frac{1}{2}, 0) 〉 〈 Ψ (\frac{1}{2}, 0) | + | Ψ (\frac{1}{2}, π) 〉 〈 Ψ (\frac{1}{2}, π) |)$ (black dotted lines) being a convex sum of states from the zero polytope $Ψ (\frac{1}{2}, 0)$ and $Ψ (\frac{1}{2}, π)$ , (b) $| Ψ (p, 0) 〉 〈 Ψ (p, 0) |$ and $| Ψ (p, π) 〉 〈 Ψ (p, π) |$ (magenta dashed line), (c) the state $| Ψ (\frac{1}{2}, 0) 〉 〈 Ψ (\frac{1}{2}, 0) |$ from the zero polytope and the state $| Ψ (p_{2} (1 / 2, p), π) 〉 〈 Ψ (p_{2} (1 / 2, p), π) |$ such that the line connecting both states intersects the center line of the Bloch sphere at $(2 p - 1)$ (orange curve below the dotted line) as shown in Fig. 1, and (d) the same as in case (c) but with $| Ψ (\frac{1}{2}, π) 〉 〈 Ψ (\frac{1}{2}, π) |$ from the zero polytope and the corresponding state $| Ψ (p_{2} (1 / 2, p), 0) 〉 〈 Ψ (p_{2} (1 / 2, p), 0) |$ (green dash-dotted curve above the dashed and dotted line). The orange curve coincides with the convex roof.
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Figure 4
Some characteristic curves are shown besides those at the angles $φ = 0$ (blue dashed curve that concavely drops to zero at $p$ about 0.83), $φ = π / 2$ (orange widely dashed curve), and $φ = π$ (red dashed curve that convexly drops to zero at $p$ about 0.17), together with several valid decompositions of the density matrix and hence upper bounds to the convex roof $\hat{\sqrt{τ_{3}}}$ of $\sqrt{τ_{3}}$ , namely a convex combination of (a) one of the two eigenstates and $ρ_{0} = \frac{1}{2} (| Ψ (p_{0; 1}, 0) 〉 〈 Ψ (p_{0; 1}, 0) | + | Ψ (p_{0, 2}, π) 〉 〈 Ψ (p_{0, 2}, π) |)$ (black dotted lines) of the states taken from the zero polytope given by $Ψ (p_{0; 1}, 0)$ and a threefold solution in $Ψ (p_{0, 2}, π)$ , (b) the state $| Ψ (p_{0; 1}, 0) 〉 〈 Ψ (p_{0; 1}, 0) |$ from the zero polytope and a state $| Ψ (p_{2} (p_{0; 1}, p), π) 〉 〈 Ψ (p_{2} (p_{0; 1}, p), π) |$ such that the line connecting both states intersects the center line of the Bloch sphere at $(2 p - 1)$ (solid orange curve below the dotted black line), (c) the same as in case (b) but with $| Ψ (p_{0; 2}, π) 〉 〈 Ψ (p_{0; 2}, π) |$ from the zero polytope and the corresponding state $| Ψ (p_{2} (p_{0; 2}, p), 0) 〉 〈 Ψ (p_{2} (p_{0; 2}, p), 0) |$ (dash-dotted green curve above the dotted black line). The orange curve is already convex and therefore coincides with the convex roof. We also show the convexified characteristic curve (red dotted line attached to the red dashed curve corresponding to the characteristic curve at the angle $π$ ). It however does not lead to a diminishing of the effective three-tangle (dotted orange curve above the convex roof) as one would expect. This is due to the linear increase of weight with decreasing value of $p$ . It therefore approaches to the convexified characteristic curve for $p \to 1$ .
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