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arXiv:1612.06063 · Physical Review A

Monogamy relation in multipartite continuous-variable quantum teleportation

Jaehak Lee, Se-Wan Ji, Jiyong Park, Hyunchul Nha

Published Dec 15, 2016• 6 citations
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Abstract

Quantum teleportation (QT) is a fundamentally remarkable communication protocol that also finds many important applications for quantum informatics. Given a quantum entangled resource, it is crucial to know to what extent one can accomplish the QT. This is usually assessed in terms of output fidelity, which can also be regarded as an operational measure of entanglement. In the case of multipartite communication when each communicator possesses a part of an $N$-partite entangled state, not all pairs of communicators can achieve a high fidelity due to the monogamy property of quantum entanglement. We here investigate how such a monogamy relation arises in multipartite continuous-variable (CV) teleportation, particularly when using a Gaussian entangled state. We show a strict monogamy relation, i.e., a sender cannot achieve a fidelity higher than optimal cloning limit with more than one receiver. While this seems rather natural owing to the no-cloning theorem, a strict monogamy relation still holds even if the sender is allowed to individually manipulate the reduced state in collaboration with each receiver to improve fidelity. The local operations are further extended to non-Gaussian operations such as photon subtraction and addition, and we demonstrate that the Gaussian cloning bound cannot be beaten by more than one pair of communicators. Furthermore, we investigate a quantitative form of monogamy relation in terms of teleportation capability, for which we show that a faithful monogamy inequality does not exist.

DOI
10.1103/physreva.94.062318

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Quantum teleportation (QT) is a fundamentally remarkable communication protocol that also finds many important applications for quantum informatics. Given a quantum entangled resource, it is crucial to know to what extent one can accomplish the QT. This is usually assessed in terms of output fidelity, which can also be regarded as an operational measure of entanglement. In the case of multipartite communication when each communicator possesses a part of an NNN-partite entangled state, not all pairs of communicators can achieve a high fidelity due to the monogamy property of quantum entanglement. We here investigate how such a monogamy relation arises in multipartite continuous-variable (CV) teleportation, particularly when using a Gaussian entangled state. We show a strict monogamy relation, i.e., a sender cannot achieve a fidelity higher than optimal cloning limit with more than one receiver. While this seems rather natural owing to the no-cloning theorem, a strict monogamy relation still holds even if the sender is allowed to individually manipulate the reduced state in collaboration with each receiver to improve fidelity. The local operations are further extended to non-Gaussian operations such as photon subtraction and addition, and we demonstrate that the Gaussian cloning bound cannot be beaten by more than one pair of communicators. Furthermore, we investigate a quantitative form of monogamy relation in terms of teleportation capability, for which we show that a faithful monogamy inequality does not exist.
We demonstrate the existence of general constraints on distributed quantum correlations, which impose a trade-off on bipartite and multipartite entanglement at once. For all N-mode Gaussian states under permutation invariance, we establish exactly a monogamy inequality, stronger than the traditional one, that by recursion defines a proper measure of genuine N-partite entanglement. Strong monogamy holds as well for subsystems of arbitrary size, and the emerging multipartite entanglement measure is found to be scale invariant. We unveil its operational connection with the optimal fidelity of continuous variable teleportation networks.
In this work we point out how the continuous variable teleportation protocol takes advantage of the quadrature entanglement in different ways, depending on the preparation of the entangled state. For a given amount of the entanglement resource, we describe the best production scheme for a two-mode Gaussian state, which enables quantum teleportation with optimal fidelity. We extend this study to multiparty entangled Gaussian states and define an operative measure of multipartite entanglement related to the optimal fidelity in a quantum teleportation network experiment. This optimal fidelity is shown to be equivalent to the entanglement of formation for the standard two-user protocol, and to the multipartite localizable entanglement for the multiuser protocol.
Quantum teleportation is one of the crucial protocols in quantum information processing. It is important to accomplish an efficient teleportation under practical conditions, aiming at a higher fidelity desirably using fewer resources. The continuous-variable (CV) version of quantum teleportation was first proposed using a Gaussian state as a quantum resource, while other attempts were also made to improve performance by applying non-Gaussian operations. We investigate the CV teleportation to find its ultimate fidelity under energy constraint identifying an optimal quantum state. For this purpose, we present a formalism to evaluate teleportation fidelity as an expectation value of an operator. Using this formalism, we prove that the optimal state must be a form of photon-number entangled states. We further show that Gaussian states are near optimal, while non-Gaussian states make a slight improvement and therefore are rigorously optimal, particularly in the low-energy regime.
We study a continuous-variable dense coding protocol, originally proposed to employ a two-mode squeezed state, using a general two-mode Gaussian state as a quantum channel. We particularly obtain conditions to manifest quantum advantage by beating two well-known single-mode schemes, namely, the squeezed-state scheme (best Gaussian scheme) and the number-state scheme (optimal scheme achieving the Holevo bound). We then extend our study to a multipartite Gaussian state and investigate the monogamy of operational entanglement measured by the communication capacity under the dense coding protocol. We show that this operational entanglement represents a strict monogamy relation, by means of Heisenberg's uncertainty principle among different parties; i.e., the quantum advantage for communication can be possible for only one pair of two-mode systems among many parties.
We devise the optimal form of Gaussian resource states enabling continuous-variable teleportation with maximal fidelity. We show that a nonclassical optimal fidelity of N-user teleportation networks is necessary and sufficient for N-party entangled Gaussian resources, yielding an estimator of multipartite entanglement. The entanglement of teleportation is equivalent to the entanglement of formation in a two-user protocol, and to the localizable entanglement in a multiuser one. Finally, we show that the continuous-variable tangle, quantifying entanglement sharing in three-mode Gaussian states, is defined operationally in terms of the optimal fidelity of a tripartite teleportation network.
In this chapter we review the characterization of entanglement in Gaussian states of continuous variable systems. For two-mode Gaussian states, we discuss how their bipartite entanglement can be accurately quantified in terms of the global and local amounts of mixedness, and efficiently estimated by direct measurements of the associated purities. For multimode Gaussian states endowed with local symmetry with respect to a given bipartition, we show how the multimode block entanglement can be completely and reversibly localized onto a single pair of modes by local, unitary operations. We then analyze the distribution of entanglement among multiple parties in multimode Gaussian states. We introduce the continuous-variable tangle to quantify entanglement sharing in Gaussian states and we prove that it satisfies the Coffman-Kundu-Wootters monogamy inequality. Nevertheless, we show that pure, symmetric three-mode Gaussian states, at variance with their discrete-variable counterparts, allow a promiscuous sharing of quantum correlations, exhibiting both maximum tripartite residual entanglement and maximum couplewise entanglement between any pair of modes. Finally, we investigate the connection between multipartite entanglement and the optimal fidelity in a continuous-variable quantum teleportation network. We show how the fidelity can be maximized in terms of the best preparation of the shared entangled resources and, viceversa, that this optimal fidelity provides a clearcut operational interpretation of several measures of bipartite and multipartite entanglement, including the entanglement of formation, the localizable entanglement, and the continuous-variable tangle.