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arXiv:1512.03326 · Physical Review Letters

Entanglement Quantification Made Easy: Polynomial Measures Invariant under Convex Decomposition

Bartosz Regula, Gerardo Adesso

Published Feb 19, 2016• 25 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.116.070504

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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.
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.
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 a powerful method to compute entanglement measures based on convex roof constructions. In particular, our method is applicable to measures that, for pure states, can be written as low order polynomials of operator expectation values. We show how to compute the linear entropy of entanglement, the linear entanglement of assistance, and a bound on the dimension of the entanglement for bipartite systems. We discuss how to obtain the convex roof of the three-tangle for three-qubit states. We also show how to calculate the linear entropy of entanglement and the quantum Fisher information based on partial information or device independent information. We demonstrate the usefulness of our method by concrete examples.
Coherence, the superposition of orthogonal quantum states, is indispensable in various quantum processes. Inspired by the polynomial invariant for classifying and quantifying entanglement, we first define polynomial coherence measure and systematically investigate its properties. Except for the qubit case, we show that there is no polynomial coherence measure satisfying the criterion that its value takes zero if and only if for incoherent states. Then, we release this strict criterion and obtain a necessary condition for polynomial coherence measure. Furthermore, we give a typical example of polynomial coherence measure for pure states and extend it to mixed states via a convex-roof construction. Analytical formula of our convex-roof polynomial coherence measure is obtained for symmetric states which are invariant under arbitrary basis permutation. Consequently, for general mixed states, we give a lower bound of our coherence measure.
We show a powerful method to compute entanglement measures based on convex roof constructions. In particular, our method is applicable to measures that, for pure states, can be written as low order polynomials of operator expectation values. We show how to compute the linear entropy of entanglement, the linear entanglement of assistance, and a bound on the dimension of the entanglement for bipartite systems. We discuss how to obtain the convex roof of the three-tangle for three-qubit states. We also show how to calculate the linear entropy of entanglement and the quantum Fisher information based on partial information or device independent information. We demonstrate the usefulness of our method by concrete examples
The quantification of multipartite entanglement remains a long-standing open challenge in quantum information theory with major implications for quantum foundations and quantum technologies. In the context of three-qubit pure states, the concurrence triangle underlies geometric interpretations for a variety of multipartite entanglement measures. In particular, the concurrence fill, which is proportional to the square root of the area of the concurrence triangle, has been shown to be a valid measure of genuine tripartite entanglement. As a result, there is now significant interest in extending the concurrence triangle and concurrence fill to three-qubit mixed states. In this work, we show that any rank-2 three-qubit mixed state can be visualized in terms of concurrence prisms associated with their pure state decompositions. Thus, the concurrence fill of the mixed state obtained through convex-roof extension is proportional to the total prism fill area, defined as the square root of the base area times the height, associated with the prisms minimized over all pure state decompositions. We then consider three-qubit mixed states that are incoherent mixtures of Greenberger-Horne-Zeilinger (GHZ) and W states and analytically perform their convex-roof extension to obtain a closed-form expression for the concurrence fill of these rank-2 mixtures. Finally, we apply the geometric picture of concurrence prisms to these mixtures and their pure state decomposition corresponding to minimum concurrence fill as a tool for visualization.