Measuring Quantum Coherence with Entanglement

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.

Figure 1

(a) Incoherent operations cannot generate entanglement from incoherent input states. (b) Conversely, we show that any nonzero coherence in the input state of a system $S$ can be converted to entanglement via incoherent operations on $S$ and an incoherent ancilla $A$. Input coherence and output entanglement are quantitatively equivalent: For any entanglement monotone $E$, the maximum entanglement generated between $S$ and $A$ by incoherent operations defines a faithful coherence monotone $C_E$ on the initial state of $S$.