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Optimal processing of quantum coherence at zero energy cost

Giulio Chiribella, Yuxiang Yang

Published Feb 1, 2015• 2 citations
OpenAlex

Abstract

Quantum technologies are rapidly enhancing our ability to generate and manipulate coherent superpositions of different energy levels. Envisaging a new generation of energy-efficient quantum devices, here we ask how well coherence can be manipulated without drawing energy from the surrounding environment. We start from the task of converting a coherent superposition of energy eigenstates into another. For this task we identify the optimal energy-preserving operations, both in the deterministic and in the probabilistic scenario. We then design a recursive protocol, wherein a branching sequence of probabilistic filters increase the probability of success while reaching maximum fidelity at each iteration. Building on the recursive protocol, we construct efficient approximations of the optimal fidelity-probability tradeoff, by taking coherent superpositions of the different branches generated by probabilistic filtering. The benefits of this construction are illustrated in a number of applications to phase estimation, quantum cloning, coherent state amplification, and ancilla-driven computation. Finally, we extend our results to transitions where the input state is generally mixed and we apply our findings to the task of purifying quantum coherence.

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Quantum technologies are rapidly enhancing our ability to generate and manipulate coherent superpositions of different energy levels. Envisaging a new generation of energy-efficient quantum devices, here we ask how well coherence can be manipulated without drawing energy from the surrounding environment. We start from the task of converting a coherent superposition of energy eigenstates into another. For this task we identify the optimal energy-preserving operations, both in the deterministic and in the probabilistic scenario. We then design a recursive protocol, wherein a branching sequence of probabilistic filters increase the probability of success while reaching maximum fidelity at each iteration. Building on the recursive protocol, we construct efficient approximations of the optimal fidelity-probability tradeoff, by taking coherent superpositions of the different branches generated by probabilistic filtering. The benefits of this construction are illustrated in a number of applications to phase estimation, quantum cloning, coherent state amplification, and ancilla-driven computation. Finally, we extend our results to transitions where the input state is generally mixed and we apply our findings to the task of purifying quantum coherence.
Quantum technologies are developing powerful tools to generate and manipulate coherent superpositions of different energy levels. Envisaging a new generation of energy-efficient quantum devices, here we explore how coherence can be manipulated without exchanging energy with the surrounding environment. We start from the task of converting a coherent superposition of energy eigenstates into another. We identify the optimal energy-preserving operations, both in the deterministic and in the probabilistic scenario. We then design a recursive protocol, wherein a branching sequence of energy-preserving filters increases the probability of success while reaching maximum fidelity at each iteration. Building on the recursive protocol, we construct efficient approximations of the optimal fidelity-probability trade-off, by taking coherent superpositions of the different branches generated by probabilistic filtering. The benefits of this construction are illustrated in applications to quantum metrology, quantum cloning, coherent state amplification, and ancilla-driven computation. Finally, we extend our results to transitions where the input state is generally mixed and we apply our findings to the task of purifying quantum coherence.
Quantum technologies are developing powerful tools to generate and manipulate coherent superpositions of different energy levels. Envisaging a new generation of energy-efficient quantum devices, here we explore how coherence can be manipulated without exchanging energy with the surrounding environment. We start from the task of converting a coherent superposition of energy eigenstates into another. We identify the optimal energy-preserving operations, both in the deterministic and in the probabilistic scenario. We then design a recursive protocol, wherein a branching sequence of energy-preserving filters increases the probability of success while reaching maximum fidelity at each iteration. Building on the recursive protocol, we construct efficient approximations of the optimal fidelity-probability trade-off, by taking coherent superpositions of the different branches generated by probabilistic filtering. The benefits of this construction are illustrated in applications to quantum metrology, quantum cloning, coherent state amplification, and ancilla-driven computation. Finally, we extend our results to transitions where the input state is generally mixed and we apply our findings to the task of purifying quantum coherence.
Quantum-limited amplifiers increase the amplitude of quantum signals at the price of introducing additional noise. Quantum purification protocols operate in the reverse way, by reducing the noise while attenuating the signal. Here we investigate a scenario that interpolates between these two extremes. We search for the optimal physical process that generates MMM approximate copies of a pure and amplified coherent state, starting from NNN copies of a noisy coherent state with Gaussian modulation. We prove that the optimal deterministic processes are always Gaussian, whereas non-Gaussianity powers up probabilistic advantages in suitable parameter regimes. The optimal processes are experimentally realizable with current technology, both in the deterministic and in the probabilistic scenario. In view of this fact, we provide benchmarks that can be used to certify the experimental demonstration of the quantum-enhanced amplification and purification of coherent states.
Pure quantum states are a fundamental resource for quantum information processing, but unavoidable noise degrades their purity and limits the performance of quantum tasks. Quantum state purification attempts to restore purity by jointly processing several noisy copies of an unknown state, typically in a probabilistic way. Whether purification is possible, and how much it can achieve, depends strongly on the set of operations that are physically allowed. In this work, we study universal purification under \textit{energy-preserving operations}, which cannot exchange energy with an external environment. We identify exactly when the energy-preserving constraint forbids any universal purification advantage, and we provide concrete examples in which an advantage available to unrestricted operations disappears under energy conservation. When energy-preserving purification remains possible, we determine its optimal fidelity and success probability and construct the corresponding protocols. The framework includes unrestricted purification as a special case and extends naturally to purification assisted by an external energy resource.
The ability to distill quantum coherence is pivotal for optimizing the performance of quantum technologies; however, such a task cannot always be accomplished with certainty. Here we develop a general framework of probabilistic distillation of quantum coherence in a one-shot setting, establishing fundamental limitations for different classes of free operations. We first provide a geometric interpretation for the maximal success probability, showing that under maximally incoherent operations (MIO) and dephasing-covariant incoherent operations (DIO) the problem can be simplified into efficiently computable semidefinite programs. Exploiting these results, we find that DIO and its subset of strictly incoherent operations have equal power in the probabilistic distillation of coherence from pure input states, while MIO are strictly stronger. We then prove a fundamental no-go result: Distilling coherence from any full-rank state is impossible even probabilistically. We further find that in some conditions the maximal success probability can vanish suddenly beyond a certain threshold in the distillation fidelity. Finally, we consider probabilistic coherence distillation assisted by a catalyst and demonstrate, with specific examples, its superiority to the unassisted and deterministic cases.
Abstract The resource theory of coherence studies the operational value of superpositions in quantum technologies. A key question in this theory concerns the efficiency of manipulation and interconversion of the resource. Here, we solve this question completely for qubit states by determining the optimal probabilities for mixed-state conversions via stochastic incoherent operations. Extending the discussion to distributed scenarios, we introduce and address the task of assisted incoherent state conversion, where the process is enhanced by making use of correlations with a second party. Building on these results, we demonstrate experimentally that the optimal state-conversion probabilities can be achieved in a linear optics setup. This paves the way towards real world applications of coherence transformations in current quantum technologies.