Quantum divergences, channel discrimination and strong data-processing

信息论的分歧（也称为相对熵）是经典和量子信息论中大量与操作相关的量的母量。它们可以用来表示熵（数据压缩的相关量）、互信息（经典数据传输和信道模拟中的相关量）和量化鉴别任务中状态的可鉴别性。理解这些分歧的数学性质通常会立即转化为在相关操作问题上取得进展。虽然经典信息论中的分歧已经被研究了很长时间，并且基本上已经被完全分类和理解，但在量子环境中，由于不同的算子排序，存在无限多的可能性，并且不断引入新的有用变体，这为研究开辟了新的途径。通过建立在量子信息理论的最新进展所提供的数学工具和框架上，我们希望进一步研究量子发散的性质，并特别关注这两个领域的问题：1。量子通道的熵量：发散度在辨别任务中作为相关的可分辨性度量，虽然状态辨别理论已经很好地理解，但关于通道辨别（即状态之间的物理变换）的许多问题仍然是开放的。这些通常又与潜在通道发散的数学性质有关，这些数学性质涉及对输入状态的优化，因此处理起来更棘手。这里的一个重要问题是找出何时（以及如何具体）的歧视策略，其中通道被顺序使用，并在其中输入到连续的通道使用，选择基于先前的通道输出可以给出一个优势，比简单的策略，其中通道被并行使用。后者与通道发散的“链式规则”性质有关。我们希望对此进行详细的研究，目的是充分理解并行策略和顺序策略之间的关系，以及有时一种策略比另一种策略更有优势。2.强大的数据处理能力：量子发散的一个定义（也可以说是最重要的）性质是数据处理不等式：一对状态的可重复性只有在它们受到处理操作（即量子通道的作用）时才会降低。强大的数据处理涉及到寻找条件下，可扩展性严格减少，并在理想情况下也限制这种减少。最近已经获得了几个下界的差异的分歧，这取决于是否可以近似逆转的通道的作用。这在许多后续应用中非常有用。这仍然是一个悬而未决的问题，如果和在何种程度上这些界限是紧的，与一些迹象表明，他们往往不是，我们想调查。我们还想看看在应用通道之前和之后的发散商的边界，而不是限制差异，在输入状态上进行优化。这些所谓的收缩系数与快速混合（即开放量子系统向其平衡态的指数快速收敛）的非常实际的问题密切相关，特别是在量子动力学半群（即与无记忆环境相互作用的系统的量子演化）的背景下。虽然经典上这些收缩系数是很好理解的，但量子力学仍然有许多悬而未决的问题，并且已经找到了收缩系数小于1的简单充分条件，这对许多应用来说将是一个重大的进步。

Information-theoretic divergences (also called relative entropies) serve as parent quantities for a plethora of operationally relevant quantities both in classical and quantum information theory. They can be used to express entropy (relevant quantity for data-compression), mutual information (relevant quantity in classical data transmission and channel simulation) and quantify distinguishability of states in discrimination tasks. Understanding mathematical properties of these divergences often immediately translates to making progress on related operational questions. While divergences in classical information theory have been long studied and are essentially fully classified and understood, in the quantum setting, due to different operator orderings, there are infinitely many more possibilities, and new useful variants keep getting introduced, which open up new pathways in research. By building on mathematical tools and frameworks provided by recent advances in quantum information theory, we want to further investigate properties of quantum divergences, and specifically focus on these two areas of questions:1. Entropic quantities for quantum channels: Divergences serve as relevant distinguishability measures in discrimination tasks, and while the theory of state discrimination is very well understood, many questions regarding discrimination of channels (i.e. physical transformations between states) are still open. These are again often tied to mathematical properties of the underlying channel divergences which involve an optimization over input states and are hence trickier to deal with. One important question here is to find out when (and how specifically) discrimination strategies in which channels are used sequentially, and in which inputs to successive channel uses, are chosen based on previous channel outputs can give an advantage over simpler strategies, in which channels are used in parallel. The latter is related to a "chain-rule" property of the channel divergence. We want to study this in detail with the aim of fully understanding how parallel and sequential strategies are related and how there can sometimes be an advantage of one over the other. 2. Strong data processing: One of the defining (and arguably most important) properties of a quantum divergence is the data processing inequality: distinguishability of a pair of states can only decrease if they are subjected to processing operation (i.e. the action of a quantum channel). Strong data processing involves finding conditions under which distinguishability strictly decreases, and ideally also bound this decrease. Recently several lower bounds have been obtained on the difference of the divergences, which depend on whether the action of the channel can be approximately reversed. This has turned out to be very useful in many subsequent applications. It is still an open question if and to what degree these bounds are tight, with some indication that they are often not, which we would like to investigate. Instead of bounding the difference, we also want to look at bounding the quotient of the divergences before and after application of a channel, optimized over input states. These so-called contraction coefficients are closely related to very practical questions of rapid mixing (i.e. exponentially fast convergence of an open quantum system towards its equilibrium state), especially relevant in the context of quantum dynamical semigroups (i.e. quantum evolution of a system interacting with a memoryless environment). While classically these contraction coefficients are well understood, quantum-mechanically there are still many open questions, and already finding simple sufficient conditions for a contraction coefficient to be less than one would be a major step forward for many applications.