Although quantum channels underlie the dynamics of quantum states, maps which are not physical channels — that is, not completely positive — can often be encountered in settings such as entanglement detection, non-Markovian quantum dynamics, or error mitigation. We introduce an operational approach to the quantitative study of the non-physicality of linear maps based on different ways to approximate a given linear map with quantum channels. Our first measure directly quantifies the cost of simulating a given map using physically implementable quantum channels, shifting the difficulty in simulating unphysical dynamics onto the task of simulating linear combinations of quantum states. Our second measure benchmarks the quantitative advantages that a non-completely-positive map can provide in discrimination-based quantum games. Notably, we show that for any trace-preserving map, the quantities both reduce to a fundamental distance measure: the diamond norm, thus endowing this norm with new operational meanings in the characterisation of linear maps. We discuss applications of our results to structural physical approximations of positive maps, quantification of non-Markovianity, and bounding the cost of error mitigation.

中文翻译：

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钻石范数和相关措施在量化量子映射的非物理性方面的操作应用

尽管量子通道是量子态动力学的基础，但在诸如纠缠检测、非马尔可夫量子动力学或误差缓解等环境中，经常会遇到不是物理通道的映射——也就是说，不完全是正的。我们引入了一种操作方法来定量研究线性映射的非物理性，该方法基于使用量子通道近似给定线性映射的不同方法。我们的第一个措施直接量化了使用物理上可实现的量子通道模拟给定映射的成本，将模拟非物理动力学的困难转移到模拟量子态线性组合的任务上。我们的第二个措施对非完全正映射在基于判别的量子游戏中可以提供的定量优势进行了基准测试。尤其，我们表明，对于任何保留痕迹的地图，数量都减少到一个基本的距离度量：菱形范数，从而赋予这个范数在线性地图的表征中具有新的操作意义。我们讨论了我们的结果在正映射的结构物理近似中的应用、非马尔可夫性的量化以及限制错误缓解的成本。