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arXiv:1609.08537 · Physical Review A

Exact zeros of entanglement for arbitrary rank-two mixtures derived from a geometric view of the zero polytope

Andreas Osterloh

Published Dec 23, 2016• 5 citations
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Abstract

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 $W$ 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.

DOI
10.1103/physreva.94.062333

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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.
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 show that the quantification of entanglement of any rank-2 state with any polynomial entanglement measure can be recast as a geometric problem on the corresponding Bloch sphere. This approach provides insight into the properties of entanglement and allows us to relate different polynomial measures to each other, simplifying their quantification. In particular, unveiling and exploiting the geometric structure of the concurrence for two qubits, we show that the convex roof of any polynomial measure of entanglement can be quantified exactly for all rank-2 states of an arbitrary number of qubits which have only one or two unentangled states in their range. We give explicit examples by quantifying the three-tangle exactly for several representative classes of three-qubit states. We further show how our methods can be used to obtain analytical results for entanglement of more complex states if one can exploit symmetries in their geometric representation.
We discuss aspects of the convex-roof extension of multipartite entanglement measures, that is, SL(2,C)\mathrm{SL}(2,\mathbb{C})SL(2,C) invariant tangles. We highlight two key concepts that contain valuable information about the tangle of a density matrix: the zero polytope is a convex set of density matrices with vanishing tangle whereas the convex characteristic curve readily provides a nontrivial lower bound for the convex roof and serves as a tool for constructing the convex roof outside the zero polytope. Both concepts are derived from the tangle for superpositions of the eigenstates of the density matrix. We illustrate their application by considering examples of density matrices for two-qubit and three-qubit states of rank 2, thereby pointing out both the power and the limitations of the concepts.
Quantifying entanglement in composite systems is a fundamental challenge, yet exact results are available in only a few special cases. This is because hard optimization problems are routinely involved, such as finding the convex decomposition of a mixed state with the minimal average pure-state entanglement, the so-called convex roof. We show that under certain conditions such a problem becomes trivial. Precisely, we prove by a geometric argument that polynomial entanglement measures of degree 2 are independent of the choice of pure-state decomposition of a mixed state, when the latter has only one pure unentangled state in its range. This allows for the analytical evaluation of convex roof extended entanglement measures in classes of rank-2 states obeying such a condition. We give explicit examples for the square root of the three-tangle in three-qubit states, and we show that several representative classes of four-qubit pure states have marginals that enjoy this property.
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.