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arXiv:1512.02468 · Physical Review A

Three-tangle of the nine classes of four-qubit states

Andreas Osterloh

Published Jul 15, 2016• 7 citations
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Abstract

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.

DOI
10.1103/physreva.94.012323

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I calculate the mixed three-tangle \ensuremathτ3[\ensuremathρ]{\ensuremath{\tau}}_{3}[\ensuremath{\rho}]\ensuremathτ3​[\ensuremathρ] 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 \ensuremathτ3{\ensuremath{\tau}}_{3}\ensuremathτ3​ vanishes shrinks to a single point.
In the recent contribution a rather simple example of a rank-two density matrix has posed a problem in calculating upper bounds of the convex roof of the three-tangle from the zero polytope. Here, I show how a three-dimensional view, namely in terms of the Boch-sphere analogy, solves the problem immediately. I therefore calculate the zero-polytope of the three-tangle numerically exactly (up to computer accuracy) and upper bounds to its convex roof. This method can then be applied to the more general case under consideration. Here, I apply this procedure to a superposition of the four qubit GHZ- and W-state.
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.
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 $(2m\ensuremath{-}n)$-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.
We investigate the lower bound obtained from experimental data of a quantum state $\ensuremath{\rho}$, as proposed independently by O. G\"uhne et al. [Phys. Rev. Lett. 98, 110502 (2007)] and J. Eisert et al. [New J. Phys. 9, 46 (2007)], and apply it to mixed states of three qubits. The measure we consider is the convex-roof extended three-tangle. Our findings highlight an intimate relation to lower bounds obtained recently from so-called characteristic curves of a given entanglement measure. We apply the bounds to estimate the three-tangle present in recently performed experiments aimed at producing a three-qubit Greenberger-Horne-Zeilinger (GHZ) state. A nonvanishing lower bound is obtained if the GHZ fidelity of the produced states is larger than 3/43/43/4.
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.