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arXiv:1009.4888 · Physical Review A

Distillation of mixed-state continuous-variable entanglement by photon subtraction

Sheng Li Zhang, Peter van Loock

Published Dec 14, 2010• 86 citations
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Abstract

We present a detailed theoretical analysis for the distillation of one copy of a mixed two-mode continuous-variable entangled state using beam splitters and coherent photon-detection techniques, including conventional on-off detectors and photon-number-resolving detectors. The initial Gaussian mixed-entangled states are generated by transmitting a two-mode squeezed state through a lossy bosonic channel, corresponding to the primary source of errors in current approaches to optical quantum communication. We provide explicit formulas to calculate the entanglement in terms of logarithmic negativity before and after distillation, including losses in the channel and the photon detection, and show that one-copy distillation is still possible even for losses near the typical fiber channel attenuation length. A lower bound for the transmission coefficient of the photon-subtraction beam splitter is derived, representing the minimal value that still allows to enhance the entanglement.

DOI
10.1103/physreva.82.062316

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We present a detailed theoretical analysis for the distillation of one copy of a mixed two-mode continuous-variable entangled state using beam splitters and coherent photon-detection techniques, including conventional on-off detectors and photon-number-resolving detectors. The initial Gaussian mixed-entangled states are generated by transmitting a two-mode squeezed state through a lossy bosonic channel, corresponding to the primary source of errors in current approaches to optical quantum communication. We provide explicit formulas to calculate the entanglement in terms of logarithmic negativity before and after distillation, including losses in the channel and the photon detection, and show that one-copy distillation is still possible even for losses near the typical fiber channel attenuation length. A lower bound for the transmission coefficient of the photon-subtraction beam splitter is derived, representing the minimal value that still allows to enhance the entanglement.
O(η)O(\sqrt\eta)O(η​) efficiency entanglement distillation with one-time photon subtraction over near-zero transmittance bosonic channel
Entanglement distillation is an efficient method to retrieve high-quality entanglement from a large number of noise-damped entanglement states. Many investigations on continuous variable entanglement distillation have focused on low-loss or medium-loss channels. Herein, we primarily study entanglement distillation in extremely lossy (total channel transmittance η ≪ 1 ) or even near-zero transmittance ( η ≈ 0 ) channels. In particular, we analyze the manner in which entanglement and success probability decrease as η → 0 . The performances of conventional biside photon subtraction and one-time photon subtraction on two optical modes separated by L km are investigated. The motivation for our study includes the recent advancements in twin-field quantum key distribution, wherein single-photon detection is sufficient to provide a useful private key. In the same vein, we propose an entanglement distillation scheme with an efficiency of O ( η ) using one-time photon subtraction and weak coherent state injection. Thus, our proposed scheme provides a method for distilling extremely damped entanglement states for continuous variable quantum information processing.
Recently, the strong single-mode squeezed vacuum state of light with −15-15−15−15 dB of classic noise has been created [Phys. Rev. Lett.117, 110801 (2016)10.1103/PhysRevLett.117.110801]. This is an important resource in continuous variable (CV) quantum communication and fault-tolerant quantum computing. Nevertheless, strong squeezing means a non-negligible population in high-photon-number subspace and may pose a strong challenge to photon-based technology, for example, entanglement distillation. Entanglement distillation is an efficient method for retrieving a higher quality of entanglement from a weakly entangled state. Up till now, almost all the schemes for entanglement distillation are restricted to the low-squeezing regime. The distillation of strong-squeezing induced entanglement is an interesting but open topic in the near future. Here, we take the single-mode squeezed entangled state (SMSE) as an example and show that conventional photon subtraction based distillation fails to generate a higher quality of entanglement with experimentally feasible optical beam splitters. To this point, we show that the superposition of photon annihilation and creation could be an effective method. A practical scheme for implementing such a superposition with on-line squeezing is suggested. Our method is verified numerically by the calculation of a 25 dB SMSE state with an ideal photon detector and for distillation of 20.5 dB SMSE state with practical dichotic on-off detectors. The analysis of the distillation of the strong squeezing effect could be extended straightforwardly to other squeezing-based quantum information processing, such as quantum metrology and quantum illumination.
Subtracting photons from a two-mode squeezed state is a well-known method to increase entanglement. We analyze different strategies of local photon subtraction from a two-mode squeezed state in terms of entanglement gain and success probability. We develop a general framework that incorporates imperfections and losses in all stages of the process: before, during, and after subtraction. By combining all three effects into a single efficiency parameter, we provide analytical and numerical results for subtraction strategies using photon-number-resolving and threshold detectors. We compare the entanglement gain afforded by symmetric and asymmetric subtraction scenarios across the two modes. For a given amount of loss, we identify an optimized set of parameters, such as initial squeezing and subtraction beam splitter transmissivity, that maximize the entanglement gain rate. We identify regimes for which asymmetric subtraction of different Fock states on the two modes outperforms symmetric strategies. In the lossless limit, subtracting a single photon from one mode always produces the highest entanglement gain rate. In the lossy case, the optimal strategy depends strongly on the losses on each mode individually, such that there is no general optimal strategy. Rather, taking losses on each mode as the only input parameters, we can identify the optimal subtraction strategy and required beam splitter transmissivities and initial squeezing parameter. Finally, we discuss the implications of our results for the distillation of continuous-variable quantum entanglement.
Entanglement distillation is a probabilistic quantum operation which tries to repair entanglement after environment-induced decoherence. All known distillations of continuous variable entanglement always assume that the entanglement is distributed via a standard attenuating channel with fixed photon loss rate, which is fairly idealized without many realistic effects taken into account. Here we investigate the problem of distillation of entanglement transmitted via atmospheric channel, with transmittance being highly probabilistic and unfixed. Two typical distillation schemes, i.e., single-sided photon subtraction and bi-sided photon subtraction, are investigated.
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.
In this paper, we study entanglement distillation of multipartite continuous-variable Gaussian entangled states. Following Opatrný \emph{et al.}'s photon subtraction (PS) scheme, the probability of successful distillation decreases exponentially with the number of parties NNN. However, here, we shall propose an entanglement distillation scheme whose success probability scales as a constant with NNN. Our protocol employs several local squeezers, but it requires only a single PS operation. Using the logarithmic negativity as a measure of entanglement, we find that both the success probability and the distilled entanglement can be improved at the same time. Moreover, an NNN-mode transfer theorem (transferring states from phase space to Hilbert space) is presented.