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arXiv:1803.03516 · Physical Review A

Simulation of Gaussian channels via teleportation and error correction of Gaussian states

Spyros Tserkis, Josephine Dias, Timothy C. Ralph

Published Nov 26, 2018• 32 citations
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Abstract

Gaussian channels are the typical way to model the decoherence introduced by the environment in continuous-variable quantum states. It is known that those channels can be simulated by a teleportation protocol using as a resource state either a maximally entangled state passing through the same channel, i.e., the Choi state, or a state that is entangled at least as much as the Choi state. Since the construction of the Choi state requires infinite mean energy and entanglement, i.e., it is unphysical, we derive instead every physical state able to simulate a given channel through teleportation with finite resources and we further find the optimal ones, i.e., the resource states that require the minimum energy and entanglement. We show that the optimal resource states are pure and equally entangled to the Choi state as measured by the entanglement of formation. We also show that the same amount of entanglement is enough to simulate an equally decohering channel, while even more entanglement can simulate less decohering channels. We finally use that fact to generalize a previously known error-correction protocol by making it able to correct noise coming not only from pure loss but from thermal loss channels as well.

DOI
10.1103/physreva.98.052335

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Gaussian channels are the typical way to model the decoherence introduced by the environment in continuous-variable quantum states. It is known that those channels can be simulated by a teleportation protocol using as a resource state either a maximally entangled state passing through the same channel, i.e., the Choi state, or a state that is entangled at least as much as the Choi state. Since the construction of the Choi state requires infinite mean energy and entanglement, i.e., it is unphysical, we derive instead every physical state able to simulate a given channel through teleportation with finite resources and we further find the optimal ones, i.e., the resource states that require the minimum energy and entanglement. We show that the optimal resource states are pure and equally entangled to the Choi state as measured by the entanglement of formation. We also show that the same amount of entanglement is enough to simulate an equally decohering channel, while even more entanglement can simulate less decohering channels. We finally use that fact to generalize a previously known error-correction protocol by making it able to correct noise coming not only from pure loss but from thermal loss channels as well.
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.
If a quantum channel is commutable with a certain set of operators, a Choi state of the channel becomes a sufficient resource for any entanglement generation achieved by a single use of the channel with the help of local operation and classical communications (LOCC). This property, called the teleportation stretchable, could be useful to determine an upper bound of a fundamental rate-loss trade-off for optical quantum key distribution. Unfortunately, the known formulation of the teleportation-stretching expansion for lossy Gaussian channels is based on an additional assumption of the Choi state with infinite energy. In this paper, we present an entanglement-assisted LOCC protocol with a bounded energy entangled state, and identify a resource state being sufficient for a general entanglement generation protocol through a single use of lossy Gaussian channels. We also extend this protocol for the cases of general single-mode Gaussian channels. Our results provide a regular path to determine an ultimate limit of quantum communication.
Given a certain amount of entanglement available as a resource, what is the most efficient way to accomplish a quantum task? We address this question in the relevant case of continuous variable quantum teleportation protocols implemented using two-mode Gaussian states with a limited degree of entanglement and energy. We first characterize the class of single-mode phase-insensitive Gaussian channels that can be simulated via a Braunstein-Kimble protocol with nonunit gain and minimum shared entanglement, showing that infinite energy is not necessary apart from the special case of the quantum limited attenuator. We also find that apart from the identity, all phase-insensitive Gaussian channels can be simulated through a two-mode squeezed state with finite energy, albeit with a larger entanglement. We then consider the problem of teleporting single-mode coherent states with Gaussian-distributed displacement in phase space. Performing a geometrical optimization over phase-insensitive Gaussian channels, we determine the maximum average teleportation fidelity achievable with any finite entanglement and for any realistically finite variance of the input distribution.
Upper bounds for private communication over quantum channels can be derived by adopting channel simulation, protocol stretching, and relative entropy of entanglement. All these ingredients have led to single-letter upper bounds to the secret-key capacity which can be directly computed over suitable resource states. For bosonic Gaussian channels, the tightest upper bounds have been derived by employing teleportation simulation over asymptotic resource states, namely, the asymptotic Choi matrices of these channels. In this work, we adopt a different approach. We show that teleporting over an analytical class of finite-energy resource states allows us to closely approximate the ultimate bounds for increasing energy, so as to provide increasingly tight upper bounds to the secret-key capacity of one-mode phase-insensitive Gaussian channels. We then show that an optimization over the same class of resource states can be used to bound the maximum secret-key rates that are achievable in a finite number of channel uses.
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.
The ideal realization of quantum teleportation relies on having access to a maximally entangled state; however, in practice, such an ideal state is typically not available and one can instead only realize an approximate teleportation. With this in mind, we present a method to quantify the performance of approximate teleportation when using an arbitrary resource state. More specifically, after framing the task of approximate teleportation as an optimization of a simulation error over one-way local operations and classical communication (LOCC) channels, we establish a semidefinite relaxation of this optimization task by instead optimizing over the larger set of 2-PPT-extendible channels (where PPT denotes positive partial transpose). The main analytical calculations in our paper consist of exploiting the unitary covariance symmetry of the identity channel to establish a significant reduction of the computational cost of this latter optimization. Next, by exploiting known connections between approximate teleportation and quantum error correction, we also apply these concepts to establish bounds on the performance of approximate quantum error correction over a given quantum channel. Finally, we evaluate our bounds for various examples of resource states and channels.