Quantum error correction of continuous-variable states with realistic resources

Josephine Dias and T. C. Ralph

Phys. Rev. A 97, 032335 – Published 23 March 2018

Abstract

Gaussian noise induced by loss on Gaussian states may be corrected by distributing Einstein-Podolsky-Rosen entanglement through the loss channel, purifying the entanglement using a noiseless linear amplifier (NLA), and then using it for continuous-variable teleportation of the input state. Linear optical implementations of the NLA unavoidably introduce small amounts of excess noise and detection, and source efficiency will be limited in current implementations. In this paper, we analyze the error-correction protocol with nonunit efficiency sources and detectors and show the excess noise may be partially compensated by adjusting the classical gain of the teleportation protocol. We present a strong case for the potential of demonstrable error-correction with current technology.

Received 6 December 2017

DOI:https://doi.org/10.1103/PhysRevA.97.032335

Physics Subject Headings (PhySH)

Quantum communication

Quantum Information, Science & Technology

Authors & Affiliations

Josephine Dias^* and T. C. Ralph

Centre for Quantum Computation and Communication Technology, School of Mathematics and Physics, University of Queensland, Brisbane, Queensland 4072, Australia

^*josephine.dias@uqconnect.edu.au

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Issue

Vol. 97, Iss. 3 — March 2018

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Images

Figure 1
(a) A lossy channel of transmission $η$ takes an input coherent state $| α 〉$ to $| \sqrt{η} α 〉$ . (b) Protocol for quantum error-correction of CV states. EPR entanglement is distributed through the lossy channel. The NLA distills the entanglement, which is then used for CV teleportation of the input state. Ideally, this protocol takes an input coherent state $| α 〉$ to output state $| g \sqrt{η} χ α 〉$ [19].
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Figure 2
NLA with a single quantum scissor. Successful operation is heralded when a single photon is detected at $D 1$ and none at $D 2$ or vice versa. The gain $g$ is controlled by the tunable beam splitter ratio $ξ$ , related to the gain by $g = \sqrt{(1 - ξ) / ξ}$ .
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Figure 3
Improving the protocol by adjusting the classical teleportation gain. (a) Entanglement of formation of a Gaussian two-mode squeezed state (EPR strength $ζ = 0.5$ ), where one arm is distributed through a loss channel of transmission $η = 0.01$ (black, solid line). EPR strength in the CV teleporter has been set to $χ = 0.5$ (corresponding to $r \approx 0.55$ where $χ = tanh r$ ). The light-blue dashed line represents the error-correction protocol operating with the classical gain scaling factor $λ = g \sqrt{η} χ$ . The dark-blue dashed line uses a numerically optimized teleportation gain $λ_{opt}$ to produce higher entanglement of formation and error correction over a larger range of effective transmission. (b) The success probability for the same case as in (a). By optimizing $λ$ , the NLA gain may be reduced resulting in an improvement in the success probability.
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Figure 4
The error-correction protocol for an initial loss channel of transmission $η = 0.01$ . (a) The entanglement of formation of a Gaussian two-mode squeezed state (EPR strength $ζ = 0.5$ ) where one arm is distributed through loss $η$ (black solid line). The gray, dot-dashed line represents error-correction with the NLA operating in the unphysical limit of $N \to \infty$ and passes the black line at $g_{\min} > 1 / χ$ (4), where EPR strength in the CV teleporter has been set to $χ = 0.5$ . The dashed dark-blue line is the EOF of the same state distributed through the error-correction protocol with a single quantum scissor NLA (with perfect sources and detectors). The solid dark-blue line includes realistic nonunit efficiency sources and detectors, using homodyne detection efficiencies of $τ = 0.98$ , single photon source efficiency of $ε = 0.7$ and single photon detection efficiency $δ = 0.9$ . (b) The probability of successful operation of the NLA with the same parameters as Fig. 4.
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Figure 5
The effect of reduced source and detection efficiencies. (a) The entanglement of formation of a Gaussian two-mode squeezed state (EPR strength $ζ = 0.5$ ) where one arm is distributed through a loss channel of transmission $η = 0.01$ (black solid line). The solid dark-blue line corresponds to that of the same state distributed through the error-correction protocol with the same parameters as in Fig. 4. The dashed green line has reduced homodyne efficiency ( $τ = 0.95$ ) but maintains the same NLA efficiency of the dark blue line. The dot-dashed red line has reduced single photon source efficiency ( $ε = 0.5$ ), and the dashed yellow line has reduced single photon detector efficiency ( $δ = 0.8$ ). The EOF of the dashed yellow and the solid blue lines overlap. (b) Log plot of the probability of successful operation of the NLA with the same parameters as Fig. 5. The success probability of dot-dashed red and the solid blue lines overlap.
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Figure 6
An example of the parameters required to beat the deterministic bound of entanglement. These results use an initial loss channel of transmission $η = 0.005$ . EPR strength in the CV teleporter has been set to $χ = 0.5$ . (a) The entanglement of formation of a Gaussian two-mode squeezed state (EPR strength $ζ = 0.5$ ) where one arm is distributed through loss $η$ (black solid line). The deterministic bound representing the maximum entanglement achieved by an infinitely squeezed state passing through the same loss channel $η$ (red solid line). The solid purple line corresponds to the error-correction protocol with homodyne detection efficiencies of $τ = 0.98$ , single photon source and detection efficiency has been set to $ε = δ = 0.9$ . (b) Log plot of the probability of successful operation of the NLA with the same parameters as Fig. 6.
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