Renormalising entanglement distillation

It has long been noted in the field of quantum information science that entanglement constitutes the key resource in various information processing and specifically communication tasks [1]. Secure quantum key distribution necessarily relies on entanglement, even in a sense in prepare and measure schemes [2–4]. One of the main efforts in quantum information science has therefore been directed towards getting an understanding of manipulating entanglement and of finding ways to transform less usable forms of entanglement into more suitable ones. The task of entanglement distillation specifically captures the resource character of entanglement, in that it aims at preparing maximally entangled states from noisy or less entangled ones [1]. Distillation steps are part of quantum repeater protocols [5–7], necessary to distribute entanglement over arbitrary distances over noisy quantum channels: In such a scheme, entanglement is being established between different repeater stations and transferred to the final designated nodes via suitable entanglement swapping steps. Distillation schemes thought of in this context are often iterative schemes, such as the recurrence protocol[1, 8] and deterministic protocols based on error-correction codes. While they are usually not achieving the maximum rates set by the distillable entanglement, they are more practically feasible. The silent assumption in almost exclusively all of the proposed distillation schemes, however, is that the initial resources have been identically prepared and they show no correlations whatsoever. Depending on the type of preparation, this may or may not be a reasonable assumption at all. Whenever memory effects or channels [9–11] are involved, one expects some correlations between the involved entangled pairs, going beyond an i.i.d. setting. These correlations are expected to decay rapidly over several pairs sent through a channel – reflecting the natural correlation structure arising from a memory channel (see Fig. 1). Steps have been taken concerning the mathematical definition of distillable entanglement in the presence of correlations [12, 13]. The important practical problem of how to distill entanglement from correlated pairs arising from quantum memory channels is still wide open. In this work, we propose a conceptually novel way forward to solve this problem, bringing together ideas from entanglement theory with those of condensed matter theory, specifically of renormalisation and tensor network states. We start by identifying the natural class of states arising from preparations and memory channels as bi-partite matrix-product operators. Such classes of states are usually considered in the condensed matter context to capture thermal many-body states or those …