Institute of Fundamental Technological Research

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.

Quantum coherence is an essential ingredient in quantum information processing and plays a central role in emergent fields such as nanoscale thermodynamics and quantum biology. However, our understanding and quantitative characterization of coherence as an operational resource are still very limited. Here we show that any degree of coherence with respect to some reference basis can be converted to entanglement via incoherent operations. This finding allows us to define a novel general class of measures of coherence
for a quantum system of arbitrary dimension, in terms of the maximum bipartite entanglement that can be generated via incoherent operations applied to the system and an incoherent ancilla. The resulting measures are proven to be valid coherence monotones satisfying all the requirements dictated by the resource theory
of quantum coherence. We demonstrate the usefulness of our approach by proving that the fidelity-based geometric measure of coherence is a full convex coherence monotone, and deriving a closed formula for it on arbitrary single-qubit states. Our work provides a clear quantitative and operational connection between coherence and entanglement, two landmark manifestations of quantum theory and both key enablers for
quantum technologies.