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arXiv:quant-ph/0511116 · Physical Review Letters

Experimental Entanglement Distillation of Two-Qubit Mixed States under Local Operations

Zhiwei Wang, Xiang-Fa Zhou, Yun‐Feng Huang, Yong-Sheng Zhang, Xi‐Feng Ren, Guang‐Can Guo

Published Jun 8, 2006• 51 citations
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Abstract

We experimentally demonstrate optimal entanglement distillation from two forms of two-qubit mixed states under local filtering operations according to the constructive method introduced by [F. Verstraete, Phys. Rev. A 64, 010101(R) (2001)10.1103/PhysRevA.64.010101]. In principle, our setup can be easily applied to distilling entanglement from arbitrary two-qubit partially mixed states. We also test the violation of the Clauser-Horne-Shinmony-Holt inequality for the distilled state from the first form of mixed state to show its "hidden nonlocality."

DOI
10.1103/physrevlett.96.220505

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We experimentally demonstrate optimal entanglement distillation from two forms of two-qubit mixed states under local filtering operations according to the constructive method introduced by [F. Verstraete, Phys. Rev. A 64, 010101(R) (2001)10.1103/PhysRevA.64.010101]. In principle, our setup can be easily applied to distilling entanglement from arbitrary two-qubit partially mixed states. We also test the violation of the Clauser-Horne-Shinmony-Holt inequality for the distilled state from the first form of mixed state to show its "hidden nonlocality."
We experimentally demonstrate optimal entanglement distillation from two forms of two-qubit mixed states under local filtering operations according to the constructive method intruduced by F. Verstraete et al. [Phys. Rev. A 64, 010101(R) (2001)]. In principle, our set-up can be easily applied to distilling entanglement from arbitrary two-qubit partially mixed states. We also test the violation of the Clauser-Horne-Shinmony-Holt(CHSH) inequality for the distilled state from the first form of mixed state to show its "hidden non-locality".
Amplitude damping changes entangled pure states into usually less-entangled mixed states. We show, however, that even local amplitude damping of one or two qubits can result in mixed states more entangled than pure states if one compares the relative entropy of entanglement (REE) for a given degree of the Bell--Clauser-Horne-Shimony-Holt inequality violation (referred to as nonlocality). By applying Monte Carlo simulations, we find the maximally entangled mixed states and show that they are likely to be optimal by checking the Karush-Kuhn-Tucker conditions, which generalize the method of Lagrange multipliers for this nonlinear optimization problem. We show that the REE for mixed states can exceed that of pure states if the nonlocality is in the range (0,0.82) and the maximal difference between these REEs is 0.4. A former comparison [Phys. Rev. A 78, 052308 (2008)] of the REE for a given negativity showed analogous property but the corresponding maximal difference in the REEs is one order smaller (i.e., 0.039) and the negativity range is (0,0.53) only. For appropriate comparison, we normalized the nonlocality measure to be equal to the standard entanglement measures, including the negativity, for arbitrary two-qubit pure states. We also analyze the influence of the phase-damping channel on the entanglement of the initially pure states. We show that the minimum of the REE for a given nonlocality can be achieved by this channel, contrary to the amplitude-damping channel.
Hardy’s theorem on nonlocality has been verified by a series of experiments with two-qubit entangled pure states. However, in this paper we demonstrate the experimental test of the theorem by using the two-photon entangled mixed states. We first investigate the generic logic in Hardy’s proof of nonlocality, which can be applied for arbitrary two-qubit mixed polarization entangled states and can be reduced naturally to the well-known logic tested successfully by the previous pure state experiments. Then, the optimized violations of locality for various experimental parameters are delivered by the numerical method. Finally, the logic argued above for testing Hardy’s theorem on nonlocality is demonstrated experimentally by using the mixed entangled-photon pairs generated via pumping two type-I BBO crystals. Our experimental results shows that Hardy’s proof of nonlocality can also be verified with two-qubit polarization entangled mixed states, with a violation of about 3.4 standard deviations.
In this work, we study how to distill two typical multipartite entangled noisy states, the amplitude-damped W state and the amplitude-damped Greenberger-Horne-Zeilinger state. Unlike the currently existing multipartite entanglement purification schemes, our scheme is a single-copy-based scheme, i.e., only one noisy state is needed in each distillation round, and the complicated multilateral controlled-not (cnot) operations are replaced by measurements of local positive-operator-valued measures (POVMs). The results show that local POVM measurements on one copy of the noisy state can project the initial state onto a new state with higher entanglement and nonlocality. In this scheme, although the final state can violate the multipartite Bell inequality, the corresponding initial noisy state does not violate it, which means that this distillation scheme can reveal the hidden genuine multipartite nonlocality of these initial states. For the case of the amplitude-damped W state, if the POVM parameters are appropriately chosen, the fidelity of the output state can approach 1.01.01.0 for all the initial states with fidelity bigger than zero. So this single-copy-based multipartite entanglement distillation scheme has a wide range of distillable fidelities, and it is much simpler than the two-copy- or multiple-copy-based schemes. Furthermore, if the POVM measurement is carried out only on some member qubits rather than all of them, the distillation scheme can succeed too, which will greatly decrease the implementation complexity of the scheme. So this multipartite entanglement distillation scheme may be implementable in some physical system.
Nonlocality and quantum entanglement constitute two special aspects of the quantum correlations existing in quantum systems, which are of paramount importance in quantum-information theory. Traditionally, they have been regarded as identical (equivalent, in fact, for pure two qubit states, that is, Gisin's Theorem ), yet they constitute different resources. Describing nonlocality by means of the violation of several Bell inequalities, we obtain by direct optimization those states of two qubits that maximally violate a Bell inequality, in terms of their degree of mixture as measured by either their participation ratio R = 1/Tr(ρ 2 ) or their maximum eigenvalue λ max . This optimum value is obtained as well, which coincides with previous results. Comparison with entanglement is performed too. An example of an application is given in the XY model. In this novel approximation, we also concentrate on the nonlocality for linear combinations of pure states of two qubits, providing a closed form for their maximal nonlocality measure. The case of Bell diagonal mixed states of two qubits is also extensively studied. Special attention concerning the connection between nonlocality and entanglement for mixed states of two qubits is paid to the so-called maximally entangled mixed states. Additional aspects for the case of two qubits are also described in detail. Since we deal with qubit systems, we will perform an analogous study for three qubits, employing similar tools. Relation between distillability and nonlocality is explored quantitatively for the whole space of states of three qubits. We finally extend our analysis to four-qubit systems, where nonlocality for generalized Greenberger–Horne–Zeilinger states of arbitrary number of parties is computed.