We introduce so-called super/sub-martingale projections as a family of endomorphisms defined on unions of Polish spaces. Such projections allow us to identify martingales as collections of transformations that relate path-valued random variables to each other under conditional expectations. In this sense, super/sub-martingale projections are random functionals that (i) are boundedness preserving and (ii) satisfy a conditional expectation criterion similar to that of the classical martingale theory. As an application to the theory of open quantum systems, we prove (a) that any system-environment interaction that manifests a supermartingale projection on the density matrix gives rise to decoherence, and (b) that any system-environment interaction that manifests a submartingale projection gives rise an increase in Shannon-Wiener information. It follows (c) that martingale projections in an open quantum system give rise both to quantum decoherence and to information gain.

翻译：我们引入所谓的上/下鞅投影作为定义在波兰空间并集上的一族自同态。此类投影使我们能够将鞅识别为在条件期望下将路径值随机变量相互关联的变换集合。在此意义上，上/下鞅投影是满足以下条件的随机泛函：(i) 保持有界性；(ii) 满足类似于经典鞅理论的条件期望准则。作为在开放量子系统理论中的应用，我们证明：(a) 任何在密度矩阵上呈现上鞅投影的系统-环境相互作用都会导致退相干；(b) 任何呈现下鞅投影的系统-环境相互作用都会引起香农-维纳信息的增加。由此可得：(c) 开放量子系统中的鞅投影同时导致量子退相干与信息增益。