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arXiv:0811.3128 · Physical Review Letters

No-Go Theorem for Gaussian Quantum Error Correction

Julien Niset, Jaromı́r Fiurášek, Nicolas J. Cerf

Published Mar 24, 2009• 327 citations
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Abstract

We prove that Gaussian operations are of no use for protecting Gaussian states against Gaussian errors in quantum communication protocols. Specifically, we introduce a new quantity characterizing any single-mode Gaussian channel, called entanglement degradation, and show that it cannot decrease via Gaussian encoding and decoding operations only. The strength of this no-go theorem is illustrated with some examples of Gaussian channels.

DOI
10.1103/physrevlett.102.120501

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We prove that Gaussian operations are of no use for protecting Gaussian states against Gaussian errors in quantum communication protocols. Specifically, we introduce a new quantity characterizing any single-mode Gaussian channel, called entanglement degradation, and show that it cannot decrease via Gaussian encoding and decoding operations only. The strength of this no-go theorem is illustrated with some examples of Gaussian channels.
Photon loss in optical channels fundamentally limits long-range reliable quantum communication. A standard approach to overcoming this limitation is the use of quantum repeater nodes, which typically perform experimentally demanding non-Gaussian operations. However, whether Gaussian repeater protocols can enhance quantum communication rates over bosonic attenuation channels has remained open. In this work, we prove a no-go theorem for Gaussian quantum repeaters in a quantum network. Specifically, we show that any repeater chain composed of Gaussian operations, homodyne measurements, and arbitrary classical communication cannot enhance the quantum capacity of a pure-loss attenuation channel beyond that achievable by direct transmission. Our proof introduces a generalisation of kkk-extendibility to a notion of fractional extendibility for Gaussian states and establishes some of its useful properties, thereby providing a powerful framework for analysing Gaussian quantum networks.
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.
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Joint Gaussian measurements of two quantum systems are important for quantum communication between remote parties and are often used in continuous-variable teleportation or entanglement-swapping protocols. Many of the errors in real-world implementations can be modeled by independent Gaussian error channels acting prior to measurement. In this work we study independent single-mode Gaussian error channels on two modes A\mathsf{A}A and B\mathsf{B}B that take place prior to a joint Gaussian measurement. We determine the set of pairs of such channels that render all Gaussian measurements separable and therefore unsuitable for entanglement swapping or teleportation of arbitrary input states. For example, if the error channels are loss with parameters lA{l}_{\mathsf{A}}lA​ and lB{l}_{\mathsf{B}}lB​ followed by added noise with parameters nA{n}_{\mathsf{A}}nA​ and nB{n}_{\mathsf{B}}nB​, then all Gaussian measurements are separable if and only if ${l}_{\mathsf{A}}+{l}_{\mathsf{B}}+{n}_{\mathsf{A}}+{n}_{\mathsf{B}}\ensuremath{\ge}1$.
Entanglement is a physical property, emerging from the superposition of composite quantum systems, and manifested through non-classical correlations of quantum observables. In the field of Quantum Information, entanglement is considered the resource for various quantum protocols, so the ability to quantify it would allow us to associate it with the success of those protocols. Entanglement of formation is a proper way to quantify entanglement, but an analytical expression for this measure exists only for special cases. In this thesis, we focus on two-mode Gaussian states, and we derive narrow upper and lower bounds for this measure that get tight for several special cases. We further study how quantum teleportation and entanglement distillation can be employed in order to error-correct information encoded on a Gaussian state that suffers from Gaussian noise. In particular, we derive every physical state able to simulate a given phase-insensitive Gaussian channel through teleportation with finite-energy resources, and show how error-correction of a state is related to the simulation of a less decohering channel that the state has to pass through. We also discuss how the premise that the whole environment is under control of the adversary in quantum key distribution can be eliminated if we consider a teleportation-based eavesdropping attack. More specifically, we propose an all-optical teleportation attack that under collective measurements can reach optimality in the limit of infinite amount of entanglement, while for finite entanglement resources it outperforms the corresponding optimal individual attack. Finally, we show how using a class of finite-energy resource states we can increasingly approximate the infinite-energy bounds for decreasing purity, so that they provide tight upper bounds to the secret-key capacity of single-mode phase-insensitive Gaussian channels.
We reexamine a non-Gaussian quantum error-correction code designed to protect optical coherent-state qubits against errors due to an amplitude-damping channel. We improve on a previous result [R. Wickert, N. K. Bernardes, and P. van Loock, Phys. Rev. A 81, 062344 (2010)] by providing a tighter upper bound on the performance attained when considering realistic assumptions, which constrain the operation of the gates employed in the scheme. The quantitative characterization is performed through measures of fidelity and concurrence, the latter obtained by employing the code as an entanglement distillation protocol. We find that, when running the code in fully deterministic error-correction mode, direct transmission can only be beaten for certain combinations of channel and input state parameters. In contrast, in error-detection mode, the usage of higher repetition encodings remains beneficial throughout, however, at the expense of diminishing success probabilities.