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arXiv:1712.08595 · Physical Review A

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

Sebastian Gartzke, Andreas Osterloh

Published Nov 6, 2018• 8 citations
arXivHTMLPDF

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 $\mathrm{SL}$ 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 $2m$, 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 $(2m\ensuremath{-}n)$-fold zero yielding a decomposition of $\ensuremath{\rho}$. If more than one such state does exist in the zero polytope, a minimization must be performed. If no decomposition of $\ensuremath{\rho}$ is obtained in this way, it provides a better lower bound than the lowest convexified curve.

DOI
10.1103/physreva.98.052307

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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 WWW 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\mathrm{SL}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 2m2m2m, if there are two states which correspond to an at most nnn-fold degenerate solution in the zero polytope for 0<n<m0<n<m0<n<m that can be combined with the convexified minimal characteristic curve at an (2m\ensuremath−n)(2m\ensuremath{-}n)(2m\ensuremath−n)-fold zero yielding a decomposition of \ensuremathρ\ensuremath{\rho}\ensuremathρ. If more than one such state does exist in the zero polytope, a minimization must be performed. If no decomposition of \ensuremathρ\ensuremath{\rho}\ensuremathρ is obtained in this way, it provides a better lower bound than the lowest convexified curve.
I calculate the mixed three-tangle ${\ensuremath{\tau}}_{3}[\ensuremath{\rho}]$ for the reduced density matrices of the four-qubit representant states found in Phys. Rev. A 65, 052112 (2002). In most of the cases, the convex roof is obtained, except for one class, where I provide with an upper bound, which is assumed to be very close to the convex roof. I compare with results published in Phys. Rev. Lett. 113, 110501 (2014). Since the method applied there usually results in higher values for the upper bound, in certain cases it can be understood that the convex roof is obtained exactly, namely, when the zero-polytope where ${\ensuremath{\tau}}_{3}$ vanishes shrinks to a single point.
We provide a complete analysis of mixed three-qubit states composed of a Greenberger-Horne-Zeilinger state and a W state orthogonal to the former. We present optimal decompositions and convex roofs for the three-tangle. Further, we provide an analytical method to decide whether or not an arbitrary rank-2 state of three qubits has vanishing three-tangle. These results highlight intriguing differences compared to the properties of two-qubit mixed states, and may serve as a quantitative reference for future studies of entanglement in multipartite mixed states. By studying the Coffman-Kundu-Wootters inequality we find that, while the amounts of inequivalent entanglement types strictly add up for pure states, this "monogamy" can be lifted for mixed states by virtue of vanishing tangle measures.
Here I present a method for how intersections of a certain density matrix of rank 2 with the zero polytope can be calculated exactly. This is a purely geometrical procedure which thereby is applicable to obtaining the zeros of SL- and SU-invariant entanglement measures of arbitrary polynomial degree. I explain this method in detail for a recently unsolved problem. In particular, I show how a three-dimensional view, namely, in terms of the Bloch-sphere analogy, solves this problem immediately. To this end, I determine the zero polytope of the three-tangle, which is an exact result up to computer accuracy, and calculate upper bounds to its convex roof which are below the linearized upper bound. The zeros of the three-tangle (in this case) induced by the zero polytope (zero simplex) are exact values. I apply this procedure to a superposition of the four-qubit Greenberger-Horne-Zeilinger and WWW state. It can, however, be applied to every case one has under consideration, including an arbitrary polynomial convex-roof measure of entanglement and for arbitrary local dimension.
An algorithm is proposed that serves to handle full-rank density matrices when coming from a lower-rank method to compute the convex roof. This is in order to calculate an upper bound for any polynomial SL-invariant multipartite entanglement measure EEE. This study exemplifies how this algorithm works based on a method for calculating convex roofs of rank-2 density matrices. It iteratively considers the decompositions of the density matrix into two states each, exploiting the knowledge for the rank-2 case. The algorithm is therefore quasiexact as far as the rank-2 case is concerned, and it also hints where it should include more states in the decomposition of the density matrix. Focusing on the measure of three-way entanglement of qubits (called three-tangle), I show the results the algorithm gives for two states, one of which is the Greenberger-Horne-Zeilinger--Werner (GHZ-WWW) state, for which the exact convex roof is known. It overestimates the three-tangle in the state, thereby giving insight into the optimal decomposition the GHZ-WWW state has. As a proof of principle, I have run the algorithm for the three-tangle on the transverse quantum Ising model. I give qualitative and quantitative arguments why the convex roof should be close to the upper bound found here.
Some mixed states composed of only Greenberger-Horne-Zeilinger (GHZ) states can be expressed in terms of only WWW states. This fact implies that such states have vanishing three-tangle. One of such rank-3 states, ${\ensuremath{\Pi}}_{GHZ}$, is explicitly presented in this Rapid Communication. These results are used to compute analytically the three-tangle of a rank-4 mixed state $\ensuremath{\sigma}$ composed of four GHZ states. This analysis with considering Bloch sphere S16{S}^{16}S16 of d=4d=4d=4 qudit system allows us to derive the hyperpolyhedron. It is shown that the states in this hyperpolyhedron have vanishing three-tangle. Computing the one-tangles for ${\ensuremath{\Pi}}_{GHZ}$ and $\ensuremath{\sigma}$, we prove the monogamy inequality explicitly. Making use of the fact that the three-tangle of ${\ensuremath{\Pi}}_{GHZ}$ is zero, we try to explain why the WWW class in the whole mixed states is not of measure zero contrary to the case of pure states.
We have studied mixed quantum states in the system of three qubits with the property that all their partial transposes are positive; these are called PPT states. We classify a PPT state by the ranks of the state itself and its three single partial transposes. We have studied especially the rank 4444 and rank 5555 entangled PPT states.We find two distinct classes of rank four states, identified by a real valued quadratic expression invariant under local sln\\slnsln transformations, mathematically equivalent to continuous Lorentz transformations. This quadratic Lorentz invariant is non-zero for one class of states (type 1) and zero for the other class (type 2). We present analytical constructions of states of both types, general enough to reproduce all the rank four PPT states we have found numerically.There are six product vectors in a generic five dimensional subspace. The product vectors are used to define SL invariants of the state and its partial transposes. We find four distinct classes of states based which invariants are conserved or not under partial transposition.We find that a state is in the equivalence class of a symmetric state when its invariants are conserved under a partial transposition, and we have shown how to transform the state into its symmetric form through Lorentz transformations.The dimensions and geometry of extremal rank 5555 PPT states has also been examined. We find that states of different types lie on surfaces of rank 5555 different dimensions. And the surfaces touch the simplex of separable states in different ways