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arXiv:1203.1099 · Physical Review A

Quantifying mixed-state quantum entanglement by optimal entanglement witnesses

Seung‐Sup B. Lee, H.-S. Sim

Published Feb 21, 2012• 12 citations
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Abstract

We develop an approach of quantifying entanglement in mixed quantum states by the optimal entanglement witness operator. We identify the convex set of mixed states for which a single witness provides the exact value of an entanglement measure and show that the convexity, properties, and symmetries of entanglement or of a target state considerably fix the form of the optimal witness. This greatly reduces the difficulty in computing and experimentally determining entanglement measures. As an example, we show how to experimentally quantify bound entanglement in four-qubit noisy Smolin states and three-qubit Greenberger-Horne-Zeilinger entanglement under white noise. For general measures and states, we provide a numerical method to efficiently optimize the witness.

DOI
10.1103/physreva.85.022325

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We develop an approach of quantifying entanglement in mixed quantum states by the optimal entanglement witness operator. We identify the convex set of mixed states for which a single witness provides the exact value of an entanglement measure and show that the convexity, properties, and symmetries of entanglement or of a target state considerably fix the form of the optimal witness. This greatly reduces the difficulty in computing and experimentally determining entanglement measures. As an example, we show how to experimentally quantify bound entanglement in four-qubit noisy Smolin states and three-qubit Greenberger-Horne-Zeilinger entanglement under white noise. For general measures and states, we provide a numerical method to efficiently optimize the witness.
We develop a numerical approach for quantifying entanglement in mixed quantum states by convex-roof entanglement measures, based on the optimal entanglement witness operator and the minimax optimization method. Our approach is applicable to general entanglement measures and states and is an efficient alternative to the conventional approach based on the optimal pure-state decomposition. Compared with the conventional one, it has two important merits: (i) that the global optimality of the solution is quantitatively verifiable, and (ii) that the optimization is considerably simplified by exploiting the common symmetry of the target state and measure. To demonstrate the merits, we quantify Greenberger-Horne-Zeilinger (GHZ) entanglement in a class of three-qubit full-rank mixed states composed of the GHZ state, the WWW state, and the white noise, the simplest mixtures of states with different genuine multipartite entanglement, which have not been quantified before this work. We discuss some general properties of the form of the optimal witness operator and of the convex structure of mixed states, which are related to the symmetry and the rank of states.
Whether entanglement in a state can be detected, distilled, and quantified without full state reconstruction is a fundamental open problem. We demonstrate a new scheme encompassing these three tasks for arbitrary two-qubit entanglement, by constructing the optimal entanglement witness for polarization-entangled mixed-state photon pairs without full state reconstruction. With better efficiency than quantum state tomography, the entanglement is maximally distilled by newly developed tunable polarization filters and quantified by the expectation value of the witness, which equals the concurrence. This scheme is extendible to multiqubit Greenberger-Horne-Zeilinger entanglement.
Recently, a new type of symmetry for three-qubit quantum states was introduced, the so-called Greenberger-Horne-Zeilinger (GHZ) symmetry. It includes the operations which leave the three-qubit standard GHZ state unchanged. This symmetry is powerful as it yields families of mixed states that are, on the one hand, complex enough from the physics point of view and, on the other hand, simple enough mathematically so that their properties can be characterized analytically. We show that by using the properties of GHZ-symmetric states it is straightforward to derive optimal witnesses for three-qubit entanglement.
We introduce an approach which allows a full structural and quantitative analysis of multipartite entanglement. The sets of states with different structures are convex and nested. Hence, they can be distinguished from each other using appropriate measurable witnesses. We derive equations for the construction of optimal witnesses and discuss general properties arising from our approach. As an example, we formulate witnesses for a 4-cluster state and perform a full quantitative analysis of the entanglement structure in the presence of noise and losses. The strength of the method in multimode continuous variable systems is also demonstrated for a dephased Greenberger-Horne-Zeilinger-type state.
Recently, a new type of symmetry for three-qubit quantum states was introduced, the so-called Greenberger-Horne-Zeilinger (GHZ) symmetry. It includes the operations which leave the three-qubit standard GHZ state unchanged. This symmetry is powerful as it yields families of mixed states that are, on the one hand, complex enough from the physics point of view and, on the other hand, simple enough mathematically so that their properties can be characterized analytically. We show that by using the properties of GHZ-symmetric states it is straightforward to derive optimal witnesses for detecting class-specific entanglement in arbitrary three-qubit states.
The study of Entanglement of Formation of a mixed state of a bipartite system in high-dimensional Hilbert space is not easy in general. So, we focus on determining the amount of entanglement for a bipartite mixed state based on the concept of decomposable optimal entanglement witness (DOEW), that can be calculated as a minimum distance of an entangled state from the edge of positive partial transpose (PPT) states which has the most negative (positive) expectation value for non-PPT (bound) entangled states. We have constructed DOEWs based on the convex optimization method, then by using of it we quantify the degree of entanglement for two spin half particles under the Lorentz transformations. For convenience, we restrict ourselves to 2D momentum subspace and under this constraint when the momentum and the Lorentz boost are parallel, we have shown that the entanglement is not relativistic invariant. Keywords : Relativistic entanglement, Measure of entanglement, Optimal entanglement witnesses, Convex optimization PACS numbers: 03.67.Hk, 03.65.Ta