  In this paper, we show that the minimal quantum discord over
"cross-symmetric" state extensions is an entanglement monotone. In particular,
we show that the minimal Bures distance of discord over cross-symmetric
extensions is equivalent to the Bures distance of entanglement. At last, we
refute a long-held but unstated convention that only contractive distances can
be used to construct entanglement monotones by showing that the entanglement
quantifier induced by the Hilbert-Schmidt distance, which is not contractive
under quantum operations, is also an entanglement monotone.
