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arXiv:quant-ph/0510237 · Physical Review A

Observables suitable for restricting the fidelity to multipartite maximally entangled states

Koji Nagata, Masato Koashi, Nobuyuki Imoto

Published Apr 1, 2002• 14 citations
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Abstract

We present a class of observables which are suitable for determining the fidelity of a state to the multipartite Greenberger-Horne-Zeilinger (GHZ) state. Given an expectation value of an observable belonging to the class, we give a simple formula that gives a lower bound and an upper bound for the fidelity. Applying the formula to the GHZ-state preparation experiment by Pan et al. [Nature (London) 403, 515 (2000)], we show that the observed state lies outside of the class of biseparable mixed three-qubit states. We also show that for this class of operators, adopting the principle of minimum variance [Phys. Rev. A 60, 4338 (1999)] in the state estimation always results in the state with the minimum fidelity.

DOI
10.1103/physreva.65.042314

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We present a class of observables which are suitable for determining the fidelity of a state to the multipartite Greenberger-Horne-Zeilinger (GHZ) state. Given an expectation value of an observable belonging to the class, we give a simple formula that gives a lower bound and an upper bound for the fidelity. Applying the formula to the GHZ-state preparation experiment by Pan et al. [Nature (London) 403, 515 (2000)], we show that the observed state lies outside of the class of biseparable mixed three-qubit states. We also show that for this class of operators, adopting the principle of minimum variance [Phys. Rev. A 60, 4338 (1999)] in the state estimation always results in the state with the minimum fidelity.
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.
The first characterization of mixed-state entanglement was achieved for two-qubit states in Werner's seminal work [Phys. Rev. A 40, 4277 (1989)]. A physically important extension concerns mixtures of a pure entangled state [such as the Greenberger-Horne-Zeilinger (GHZ) state] and the unpolarized state. These mixed states serve as benchmark for the robustness of multipartite entanglement. They share the symmetries of the GHZ state. We call such states GHZ symmetric. Here we give a complete description of the entanglement in the family of three-qubit GHZ-symmetric states and, in particular, of the three-qubit generalized Werner states. Our method relies on the appropriate parametrization of the states and on the invariance of entanglement properties under general local operations. An application is the definition of a symmetrization witness for the entanglement class of arbitrary three-qubit states.
Mixed states appear naturally in experiment over pure states. So for studying different notions of nonlocality and their relation with entanglement in realistic scenarios, one needs to consider mixed states. In a recent paper [Phys. Rev. Lett. 108, 020502 (2012)], a complete characterization of entanglement of an entire class of mixed three-qubit states with the same symmetry as the Greenberger-Horne-Zeilinger state, known as GHZ-symmetric states, has been achieved. In this paper we investigate different notions of nonlocality of the same class of states. By finding the analytical expressions of maximum violation value of most efficient Bell inequalities we obtain the conditions of standard nonlocality and genuine nonlocality of this class of states. Also the relation between entanglement and nonlocality is discussed for this class of states. Interestingly, genuine entanglement of GHZ-symmetric states is necessary to reveal standard nonlocality. However, it is not sufficient to exploit the same.
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 violation of the Svetlichny's inequality (SI) [Phys. Rev. D 35, 3066 (1987)] is sufficient but not necessary for genuine tripartite nonlocal correlations. Here we quantify the relationship between tripartite entanglement and the maximum expectation value of the Svetlichny operator (which is bounded from above by the inequality) for the two inequivalent subclasses of pure three-qubit states: the Greenberger-Horne-Zeilinger (GHZ) class and the WWW class. We show that the maximum for the GHZ-class states reduces to Mermin's inequality [Phys. Rev. Lett. 65, 1838 (1990)] modulo a constant factor, and although it is a function of the three tangle and the residual concurrence, large numbers of states do not violate the inequality. We further show that by design SI is more suitable as a measure of genuine tripartite nonlocality between the three qubits in the WWW-class states, and the maximum is a certain function of the bipartite entanglement (the concurrence) of the three reduced states, and only when their sum attains a certain threshold value do they violate the inequality.
Device-independent certification of multipartite entangled states plays a central role in a wide range of practical applications, including quantum networks, conference key agreement, and verifiable distributed quantum computation. A particularly important class of multipartite entangled states is the class of Greenberger-Horne-Zeilinger (GHZ) states. Many Bell operators have been proposed to self-test GHZ states. However, in practical scenarios, due to imperfections and the finite collection of statistics, the observed statistics do not satisfy the ideal self-testing relations. Hence, it becomes essential to investigate and compare the robustness of the different self-testing protocols. In this work, we investigate the robustness of self-testing schemes constructed from Bell operators due to Svetlichny and Mermin--Ardehali--Belinskii--Klyshko (MABK), using the analytic operator-inequality framework developed by Kaniewski [\href{https://doi.org/10.1103/PhysRevLett.117.070402}{Phys. Rev. Lett. 117, 070402 (2016)}]. We derive lower bounds on the extractable fidelity as a function of the observed value of these Bell operators. Although these protocols self-test the same underlying state, they exhibit markedly different levels of robustness. By comparing the resulting fidelity bounds, we demonstrate that the self-testing scheme based on the Svetlichny's Bell operator is the more robust among the two. Our results thus identify the Svetlichny operator based self-testing protocol as the most favorable candidate for device-independent certification of GHZ states in realistic, noisy experimental scenarios.