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arXiv:quant-ph/0702247 · Physical Review A

Teleportation capability, distillability, and nonlocality on three-qubit states

Soojoon Lee, Jaewoo Joo, Jaewan Kim

Published Jul 13, 2007• 27 citations
arXivHTMLPDF

Abstract

In this paper, we consider teleportation capability, distillability, and nonlocality on three-qubit states. In order to investigate some relations among them, we first find the explicit formulas of the quantities about the maximal teleportation fidelity on three-qubit states. We show that if any three-qubit state is useful for three-qubit teleportation then the three-qubit state is distillable into a Greenberger-Horne-Zeilinger state, and that if any three-qubit state violates a specific form of Mermin inequality then the three-qubit state is useful for three-qubit teleportation.

DOI
10.1103/physreva.76.012311

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In this paper, we consider teleportation capability, distillability, and nonlocality on three-qubit states. In order to investigate some relations among them, we first find the explicit formulas of the quantities about the maximal teleportation fidelity on three-qubit states. We show that if any three-qubit state is useful for three-qubit teleportation then the three-qubit state is distillable into a Greenberger-Horne-Zeilinger state, and that if any three-qubit state violates a specific form of Mermin inequality then the three-qubit state is useful for three-qubit teleportation.
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.
We derive a Bell inequality for testing violation of local realism. Quantum nonlocality of several four-qubit states is investigated. These include the Greenberger-Zeilinger-Horne (GHZ) state, WWW state, linear cluster state, and the state $\ensuremath{\mid}\ensuremath{\chi}⟩$ that has recently been proposed in [Phys. Rev. Lett. 96, 060502 (2006)]. The Bell inequality is optimally violated by $\ensuremath{\mid}\ensuremath{\chi}⟩$ but not violated by the GHZ state. The linear cluster state also violates the Bell inequality though not optimally. The state $\ensuremath{\mid}\ensuremath{\chi}⟩$ can thus be discriminated from the linear cluster state by using the inequality. Different aspects of four-partite entanglement are also studied by considering the usefulness of a family of four-qubit mixed states as resources for two-qubit teleportation. Our results generalize those in [Phys. Rev. Lett. 72, 797 (1994)].
The effects of distributing entanglement through the amplitude damping channel or the phase damping channel on the teleportation of a single-qubit state via the Greenberger–Horne–Zeilinger state and the W state are discussed. It is found that the average fidelity of teleportation depends on the type and rate of the damping in the channel. For the one-qubit affected case, the Greenberger–Horne–Zeilinger state is as robust as the W state, i.e., the same quantum information is preserved through teleportation. For the two-qubit affected case, the W state is more robust when the entanglement is distributed via the amplitude damping channel; if the entanglement is distributed via the phase damping channel, the W state is more robust when the noisy parameter is small while the Greenberger–Horne–Zeilinger state becomes more robust when it is large. For the three-qubit affected case, the Greenberger–Horne–Zeilinger state is more robust than the W state.
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.