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arXiv:1304.0186 · Physical Review A

Entanglement distillation for continuous variables in a thermal environment: Effectiveness of a non-Gaussian operation

Jaehak Lee, Hyunchul Nha

Published Mar 7, 2013• 29 citations
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Abstract

We study the task of distilling entanglement by a coherent superposition operation $t\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}+r{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}^{\ifmmode\dagger\else\textdagger\fi{}}$ applied to a continuous-variable state under a thermal noise. In particular, we compare the performances of two different strategies; i.e., the non-Gaussian operation $t\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}+r{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}^{\ifmmode\dagger\else\textdagger\fi{}}$ is applied before or after the noisy Gaussian channel. This is closely related to a fundamental problem of whether Gaussian or non-Gaussian entanglement can be more robust under a noisy channel and also provides a useful insight into the practical implementation of entanglement distribution for a long-distance quantum communication. We specifically look into two entanglement characteristics, the logarithmic negativity as a measure of entanglement and the teleportation fidelity as a usefulness of entanglement, for each distilled state. We find that the non-Gaussian operation after (before) the thermal noise becomes more effective in the low- (high-) temperature regime.

DOI
10.1103/physreva.87.032307

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We study the task of distilling entanglement by a coherent superposition operation ta\ifmmode^\elseˆ\fi+ra\ifmmode^\elseˆ\fi\ifmmode†\else\textdagger\fit\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}+r{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}^{\ifmmode\dagger\else\textdagger\fi{}}ta\ifmmode^\elseˆ\fi+ra\ifmmode^\elseˆ\fi\ifmmode†\else\textdagger\fi applied to a continuous-variable state under a thermal noise. In particular, we compare the performances of two different strategies; i.e., the non-Gaussian operation ta\ifmmode^\elseˆ\fi+ra\ifmmode^\elseˆ\fi\ifmmode†\else\textdagger\fit\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}+r{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}^{\ifmmode\dagger\else\textdagger\fi{}}ta\ifmmode^\elseˆ\fi+ra\ifmmode^\elseˆ\fi\ifmmode†\else\textdagger\fi is applied before or after the noisy Gaussian channel. This is closely related to a fundamental problem of whether Gaussian or non-Gaussian entanglement can be more robust under a noisy channel and also provides a useful insight into the practical implementation of entanglement distribution for a long-distance quantum communication. We specifically look into two entanglement characteristics, the logarithmic negativity as a measure of entanglement and the teleportation fidelity as a usefulness of entanglement, for each distilled state. We find that the non-Gaussian operation after (before) the thermal noise becomes more effective in the low- (high-) temperature regime.
Many different quantum-information communication protocols such as teleportation, dense coding, and entanglement-based quantum key distribution are based on the faithful transmission of entanglement between distant location in an optical network. The distribution of entanglement in such a network is, however, hampered by loss and noise that is inherent in all practical quantum channels. Thus, to enable faithful transmission one must resort to the protocol of entanglement distillation. In this paper we present a detailed theoretical analysis and an experimental realization of continuous variable entanglement distillation in a channel that is inflicted by different kinds of non-Gaussian noise. The continuous variable entangled states are generated by exploiting the third order nonlinearity in optical fibers, and the states are sent through a free-space laboratory channel in which the losses are altered to simulate a free-space atmospheric channel with varying losses. We use linear optical components, homodyne measurements, and classical communication to distill the entanglement, and we find that by using this method the entanglement can be probabilistically increased for some specific non-Gaussian noise channels.
We address the degradation of continuous variable (CV) entanglement distillation in the presence of thermal environment and nonunity detecting efficiency. We explore the logarithmic negativity as a criteria and compare the output entanglement between the schemes of single-photon subtraction (PS) and two-PS. The probability of success is also investigated. We show that compared with detector efficiency, thermal noise plays a more dominant role in Gaussian entanglement distillation. Our results shed more light on the noise mechanism that influences realistic CV entanglement distillation and is of experimental relevance in developing a more efficient entanglement distillation scheme.
Entanglement distillation is a fundamental building block in long-distance quantum communication. Though known to be useless on their own for distilling Gaussian entangled states, local Gaussian operations may still help to improve non-Gaussian entanglement distillation schemes. Here we show that by applying local squeezing operations both the performance and the efficiency of existing distillation protocols can be enhanced. We find that such an enhancement through local Gaussian unitaries can be obtained even when the initially shared Gaussian entangled states are mixed, as, for instance, after their distribution through a lossy-fiber communication channel.
In spite of their outstanding experimental relevance, Gaussian operations in continuous-variable quantum systems are subjected to fundamental limitations, as it is known that general resources cannot be distilled within the Gaussian paradigm. We show that these limitations can be overcome by considering a collaborative setting where one party distills resources with the aid of another party, whose operations are assumed to be Gaussian but are otherwise unrestricted; the two parties can only communicate classically. We show that in single-shot scenarios, unlike in the well-known case of entanglement theory, two-way classical communication does not lead to any improvement over one-way classical communication from the aiding party to the aided party. We then provide a concise general expression for the Gaussian resource of assistance, i.e., the distillable resource when the aiding party holds a purification of the aided party's state. To demonstrate the usefulness of this general expression, we apply it to two important kinds of resources, squeezing and entanglement, and find some simple analytic solutions. In the case of entanglement theory, we are able to find general upper bounds on the regularized Gaussian entanglement of assistance, and to establish additivity for tensor powers of thermal states. This allows us to draw a quantitative and enlightening comparison with the performance of assisted entanglement distillation in the non-Gaussian setting. On the technical side, we develop some variational expressions to handle functions of symplectic eigenvalues that may be of independent interest. Our results suggest further potential for Gaussian operations to play a major role in practical quantum information processing protocols.
Single-photon-level non-Gaussian operations—photon addition, photon subtraction, and their coherent superposition—are powerful tools with which to increase entanglement in continuous-variable optical states. Although such operations are typically not deterministic, there may be other advantages such as noiseless manipulation. Therefore, to fully account for the efficacy of a particular non-Gaussian operation in a practical scenario, we develop figures of merit which trade off the advantages of such protocols against their success probability. Specifically, we define 'entanglement enhancement rate' as the increase in entanglement per trial of a generic non-Gaussian operation on a two-mode squeezed vacuum (TMSV) state. We consider states generated by photon-subtraction, photon-addition and a coherent superposition of subtraction and addition. We compare each strategy when applied to one or both modes of a TMSV state, and also in the presence of channel losses prior to the operation. In many cases, additional properties are analytically calculable, including excess noise arising from the operation and the fidelity of the resulting states to particular Gaussian and non-Gaussian states. Finally, by incorporating loss, we derive optimal interaction parameters for each non-Gaussian operation which maximize the effectiveness of the particular protocol under investigation.
Gaussian quantum information processing with continuous-variable (CV) quantum information carriers holds significant promise for applications in quantum communication and quantum internet. However, applying Gaussian state distillation and quantum error correction (QEC) faces limitations imposed by no-go results concerning local Gaussian unitary operations and classical communications. This paper introduces a Gaussian QEC protocol that relies solely on local Gaussian resources. A pivotal component of our approach is CV gate teleportation using entangled Gaussian states, which facilitates the implementation of the partial transpose operation on a quantum channel. Consequently, we can efficiently construct a two-mode noise-polarized channel from two noisy Gaussian channels. Furthermore, this QEC protocol naturally extends to a nonlocal Gaussian state distillation protocol.