  An entanglement witness is an observable with the property that a negative
expectation value signals the presence of entanglement. The question arises how
a witness can be improved if the expectation value of a second observable is
known, and methods for doing this have recently been discussed as so-called
ultrafine entanglement witnesses. We present several results on the
characterization of entanglement given the expectation values of two
observables. First, we explain that this problem can naturally be tackled with
the method of the Legendre transformation, leading even to a quantification of
entanglement. Second, we present necessary and sufficient conditions that two
product observables are able to detect entanglement. Finally, we explain some
fallacies in the original construction of ultrafine entanglement witnesses [F.
Shahandeh et al., Phys. Rev. Lett. 118, 110502 (2017)].
