Quantum error correction of continuous-variable states against Gaussian noise

Continuous-variable quantum information protocols use quantum operations and measurements acting on states with continuous eigenvalue spectra to perform quantum information tasks such as quantum teleportation, quantum key distribution, and quantum processing [1]. An attraction of continuous-variable protocols is that many require only Gaussian states, operations, and measurements [2]—all of which can be implemented deterministically in optics with current technology. However, many applications also require the ability to error correct the quantum states in order to realize their full potential. Recently it has been proven that error correction of Gaussian noise, imposed on Gaussian states, using Gaussian operations is impossible [3]. This is a significant result as Gaussian noise is the most common source of errors for continuous-variable states. It is thus of considerable interest to determine whether additional, non-Gaussian resources, can be employed to allow error correction of continuous-variable states against Gaussian noise. Here we answer this question in the affirmative by describing an error correction protocol that is effective against the Gaussian noise produced by loss. Although in principle our protocol could be applied to any physical architecture, we will focus particularly on optics given its experimental relevance. The only additional non-Gaussian operation required for our protocol is photon counting. A Gaussian continuous-variable error correction protocols based on a direct generalization of the Shor 9-qubit error correction code [4] has been developed [5,6] and demonstrated experimentally [7]. This code can correct a large range of non-Gaussian errors but not Gaussian errors. Other protocols for correcting more specific types of non-Gaussian noise imposed on Gaussian states, using Gaussian operations have also been proposed and demonstrated [8,9]. Methods for correcting Gaussian noise imposed on specific non-Gaussian code states have also been described [10,11]. We consider a situation in which we wish to transmit an ensemble of quantum states through a channel of loss η [see Fig. 1(a)]. The loss inevitably couples the system to the environment and reduces the distinguishability of the states, producing errors in any quantum information encoded in the ensemble. Successful error correction should reduce the effective loss on the channel thus leading to a lower error rate. Instead of considering error correction based on error correction codes such as those discussed above, we will consider error correction based on the distillation of entanglement [12], and the subsequent use of the distilled entanglement for teleportation [see Fig. 1(b)]. Distillation of continuous-variable entanglement is known to be possible using photon counting [13,14]. Here we will use a convenient distillation approach based on heralded noiseless linear amplification that has been demonstrated recently [15]. Although a well known equivalence between error correction and distillation exists for discrete variables the situation is not so straightforward for continuous variables. The problem is that continuous-variable entanglement is only strictly maximal in the limit of infinite energy. The effect of nonmaximal entanglement is to add noise in the teleportation protocol, and thus potentially compromise any error correction achieved via the distillation. However, here we show that by modifying the teleportation protocol, error correction can be achieved in the absence of maximal entanglement.