Relation between quantum coherence and quantum entanglement in quantum measurements

Quantum measurement is a class of quantum channels that sends quantum states to classical states. We set up resource theories of quantum coherence and quantum entanglement for quantum measurements and find relations between them. For this, we conceive a relative entropy-type quantity to account for the quantum resources of quantum measurements. The quantum coherence of a quantum measurement can be converted into the entanglement in a bipartite quantum measurement through coherence nongenerating transformations. Conversely, a quantum entanglement monotone of quantum measurements induces a quantum coherence monotone of quantum measurements. Our results confirm that the understanding on the link between quantum coherence and quantum entanglement is valid even for quantum measurements which do not generate any quantum resource.

Figure 1

Transformation of a quantum measurement ${\mathcal{M}}_A$ to a quantum measurement ${\mathcal{N}}_A$ through a preprocessing channel ${\mathcal{E}}_A$ and a classical postprocessing channel ${\mathcal{S}}_R$. The double line means classical data.

Figure 2

Building a bipartite quantum measurement ${\mathcal{M}}_{AB}^{\prime }$ from two quantum measurements ${\mathcal{M}}_A$ and ${\mathcal{E}}_B$ with a preprocessing channel ${\mathcal{N}}_{AB}$ and a classical postprocessing channel ${\mathcal{S}}_R$. The double line means classical data.