  We establish relations between geometric quantum discord and entanglement
quantifiers obtained by means of optimal witness operators. In particular, we
prove a relation between negativity and geometric discord in the
Hilbert-Schmidt norm, which is slightly different from a previous conjectured
one [Phys. Rev. A 84, 052110 (2011)].We also show that, redefining the
geometric discord with the trace norm, better bounds can be obtained. We
illustrate our results numerically.
