Linking entanglement to discord with state extensions

One of the fundamental problem to explore the potential and advantage of quantum technology is the char acterization and quantification of the quantum correlations, especially entanglement. In [Phys. Rev. A 94, 032129 (2016)], the author proposed the minimal discord over state extensions as the measure of entanglement. In this work, we show that the minimal Bures distance of discord over state extensions is equivalent to the Bures distance of entanglement and its convex roof. This equivalence puts discord in a more primitive place than entanglement conceptually, that is, entanglement can be interpreted as the irreducible part of discord over all state extensions. Moreover, for bipartite state, we also show that the minimal quantum discord over a kind of symmetric state extensions is an entanglement monotone, i.e., non-increasing under local operation and classical communications. The results presented here show that a large class of discord measures can be used to construct entanglement measures. In particular, although Hilbert-Schmidt distance is not contractive, our result show that the corresponding quantification is an entanglement monotone in our framework.