We develop two general approaches to characterising the manipulation of quantum states by means of probabilistic protocols constrained by the limitations of some quantum resource theory.

First, we give a general necessary condition for the existence of a physical transformation between quantum states, obtained using a recently introduced resource monotone based on the Hilbert projective metric. In all affine quantum resource theories (e.g. coherence, asymmetry, imaginarity) as well as in entanglement distillation, we show that the monotone provides a necessary and sufficient condition for one-shot resource convertibility under resource-non-generating operations, and hence no better restrictions on all probabilistic protocols are possible. We use the monotone to establish improved bounds on the performance of both one-shot and many-copy probabilistic resource distillation protocols.

Complementing this approach, we introduce a general method for bounding achievable probabilities in resource transformations under resource-non-generating maps through a family of convex optimisation problems. We show it to tightly characterise single-shot probabilistic distillation in broad types of resource theories, allowing an exact analysis of the trade-offs between the probabilities and errors in distilling maximally resourceful states. We demonstrate the usefulness of both of our approaches in the study of quantum entanglement distillation.

中文翻译：

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对量子态概率转换的严格限制

我们开发了两种通用方法来通过概率协议来表征量子态的操纵，这些协议受到一些量子资源理论的限制。

首先，我们给出了量子态之间存在物理变换的一般必要条件，这是使用最近引入的基于希尔伯特射影度量的资源单调获得的。在所有仿射量子资源理论（例如相干性、不对称性、想象性）以及纠缠蒸馏中，我们表明单调性为资源非生成操作下的一次性资源转换提供了充分的必要条件，因此没有更好的对所有概率协议的限制都是可能的。我们使用单调来建立单次和多副本概率资源蒸馏协议的性能改进界限。

作为对这种方法的补充，我们引入了一种通用方法，用于通过一系列凸优化问题来限制资源非生成映射下资源转换中可实现的概率。我们展示了它在广泛类型的资源理论中紧密表征单次概率蒸馏，允许精确分析在最大资源状态中提取概率和错误之间的权衡。我们证明了我们两种方法在量子纠缠蒸馏研究中的有用性。