Towards Resource Theory of Coherence in Distributed Scenarios

The search for a simple description of fundamental physical processes is an important part of quantum theory. One example for such an abstraction can be found in the distance lab paradigm: if two separated parties are connected via a classical channel, it is notoriously difficult to characterize all possible operations these parties can perform. This class of operations is widely known as local operations and classical communication. Surprisingly, the situation becomes comparably simple if the more general class of separable operations is considered, a finding that has been extensively used in quantum information theory for many years. Here, we propose a related approach for the resource theory of quantum coherence, where two distant parties can perform only measurements that do not create coherence and can communicate their outcomes via a classical channel. We call this class local incoherent operations and classical communication. While the characterization of this class is also difficult in general, we show that the larger class of separable incoherent operations has a simple mathematical form, yet still preserves the main features of local incoherent operations and classical communication. We demonstrate the relevance of our approach by applying it to three different tasks: assisted coherence distillation, quantum teleportation, and single-shot quantum state merging. We expect that the results we obtain in this work also transfer to other concepts of coherence that are discussed in recent literature. The approach we present here opens new ways to study the resource theory of coherence in distributed scenarios.

Popular Summary

Quantum coherence and entanglement are among the most puzzling features in nature. The former encompasses the notion of superposition, where a particle can exist in many states at once, and the latter accounts for states that are shared between two widely separated particles. Researchers have only recently started to understand how entanglement and coherence can be applied in quantum technology. In the last two decades, this research has focused on entanglement. However, recent results show that other phenomena, such as coherence and discord, are required to describe all features of a quantum system. In this work, we study a framework for manipulating coherence as a resource.

In our scenarios, two (or more) spatially separated parties are allowed to share classical information and are restricted to local operations in their labs, which cannot create coherence. The most fundamental question in this context concerns possible transformations that a quantum state can undergo. While those transformations are difficult to characterize in general, our work shows that a larger class of transformations has a simple mathematical form while preserving the main features of the process. This superclass is shown to be useful in several scenarios, such as when one party can enhance coherence distillation with the help of a second party, as well as when one party aims to send her part of a quantum state to another party while preserving correlations with a third party. We also show that standard quantum teleportation does not require local coherence.

The tools presented here pave the way to a rigorous resource theory of coherence in distributed scenarios and can also be applied to quantum thermodynamics, processes that respect symmetries, and related research fields.

Figure 1

Hierarchy of LICC, SI, LOCC, and separable operations $\mathrm{S}$. The set of LICC operations is the weakest set, and $\mathrm{S}$ is the most powerful set. The crossed region displays operations that are in LOCC and SI, but not in LICC. It remains open if such operations exist. For simplicity, we do not display the sets LQICC and SQI.