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arXiv:1307.7415 · Physical Review A

Optimal architecture for a nondeterministic noiseless linear amplifier

Nathan A. McMahon, Austin P. Lund, Timothy C. Ralph

Published Feb 27, 2014• 42 citations
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Abstract

Nondeterministic quantum noiseless linear amplifiers are a new technology with interest in both fundamental understanding and new applications. With a noiseless linear amplifier it is possible to perform tasks such as improving the performance of quantum key distribution, purifying lossy channels, and distilling entanglement. Previous designs for noiseless linear amplifiers involving linear optics and photon counting are nonoptimal because they have a probability of success lower than the bound given by the theory of generalized quantum measurement. This paper develops a theoretical model using unitary interactions and projective measurements which reaches this limit. We calculate the fidelity and probability of success of this model for coherent states and Einstein-Podolsky-Rosen entangled states. Finally, we explore some examples of the complex interplay between the fidelity, probability, and the distilling and purifying power of the model.

DOI
10.1103/physreva.89.023846

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Nondeterministic quantum noiseless linear amplifiers are a new technology with interest in both fundamental understanding and new applications. With a noiseless linear amplifier it is possible to perform tasks such as improving the performance of quantum key distribution, purifying lossy channels, and distilling entanglement. Previous designs for noiseless linear amplifiers involving linear optics and photon counting are nonoptimal because they have a probability of success lower than the bound given by the theory of generalized quantum measurement. This paper develops a theoretical model using unitary interactions and projective measurements which reaches this limit. We calculate the fidelity and probability of success of this model for coherent states and Einstein-Podolsky-Rosen entangled states. Finally, we explore some examples of the complex interplay between the fidelity, probability, and the distilling and purifying power of the model.
Linear amplifiers are necessarily affected by a minimal amount of noise, which is needed in order to preserve the linearity and the unitarity prescribed by quantum mechanics. Such a limitation might be partially overcome if the process is realized by conditioning its operation on a trigger event, for instance, the result of a measurement. Here we present a detailed analysis of a noiseless amplifier, implemented using linear optics, a down-conversion-based single-photon source, and single-photon detection. Our results demonstrate an amplification adding a level of noise lower than the minimum allowed by quantum mechanics for deterministic amplifiers. This is made possible by the nondeterministic character of our device, whose success rate is sufficiently low not to violate any fundamental limit. We compare our experimental data to a model taking into account the main imperfections of the setup and find a good agreement.
We study the operational regime of a noiseless linear amplifier (NLA) based on quantum scissors that can nondeterministically amplify the one photon component of a quantum state with weak excitation. It has been shown that an arbitrarily large quantum state can be amplified by first splitting it into weak excitation states using a network of beamsplitters. The output states of the network can then be coherently recombined. In this paper, we analyse the performance of such a device for distilling entanglement after transmission through a lossy quantum channel, and look at two measures to determine the efficacy of the NLA. The measures used are the amount of entanglement achievable and the final purity of the output amplified entangled state. We study the performances of both a single and a two-element NLA for amplifying weakly excited states. Practically, we show that it may be advantageous to work with a limited number of stages.
Most of the schemes for ``noiseless'' amplification of coherent states, which have recently been attracting theoretical and experimental interest, share a common trait: The amplification is not truly noiseless, or perfect, for nonzero success probability. While this must hold true for all phase-independent amplification schemes, in this work we point out that truly noiseless amplification is indeed possible, provided that the states which we wish to amplify come from a finite set. Perfect amplification with unlimited average gain is then possible with finite success probability, for example, using techniques for unambiguously distinguishing between quantum states. Such realizations require only linear optics, no single-photon sources, and no photon counting. We also investigate the optimal success probability of perfect amplification of a symmetric set of coherent states. There are two regimes: low-amplitude amplification, where the target amplitude is below one, and general amplification. For the low-amplitude regime, analytic results for the optimal amplification success probabilities can be obtained. In this case a natural bound imposed by the ratio of success probabilities of optimal unambiguous discrimination of the source and amplified states can always be reached. We also show that for general amplification this bound cannot always be satisfied.
The unavoidable addition of noise during amplification is a well known signature of quantum mechanics. It is at the heart of fundamental results such as the no-cloning theorem, quantum limited metrology, quantum key distribution and the impossibility of increasing entanglement by local operations. Nonetheless one can still avoid the unavoidable by moving to a non-deterministic protocol. This novel concept and a linear optics implementation have been proposed [1] and experimentally realised for the case of amplifying coherent states [2-4], qubits [5,6] and the concentration of phase information [7]. All these were extremely challenging experiments, with only [2] demonstrating entanglement distillation and none directly showing the EPR distillation necessary for application to CV QKD. Furthermore the success probability of these experiments was substantially worse than the theoretical considerations would imply. However as has been noted in [8,9] it is possible to virtually implement noiseless amplification (NLA) and hence entanglement distillation via post-selective measurements, achieving significant distillation with a much improved probability of success.
Fundamental laws of quantum mechanics impose that arbitrary quantum states cannot be perfectly cloned or amplified without introducing some unavoidable noise in the process. The quantum noise intrinsic to the functioning of a linear phase-insensitive amplifier can however be avoided by relaxing the requirement of a deterministic operation. Non-deterministic noiseless linear amplifiers that do not violate any fundamental quantum law are therefore possible and here we present the first experimental realization of a scheme that allows noiseless amplification of coherent states at the best level of effective gain and final state fidelity ever reached. This scheme, based on a sequence of photon addition and subtraction, and characterized by a significant amplification and low distortions, may become a useful tool for quantum communications and metrology, by enhancing the discrimination between partially overlapping quantum states or by recovering the information transmitted over lossy channels.