Generalised Pinching Inequality

Hayashi's Pinching Inequality, which establishes a matrix inequality between a semidefinite matrix and a multiple of its "pinched" version via a projective measurement, has found many applications in quantum information theory and beyond. Here, we show a very simple proof of it, which lends itself immediately to natural generalisations where the different projections of the measurement have different weights, and where the matrix inequality can be reversed. As an application we show how the generalised pinching inequality in the case of binary measurements gives rise to a novel gentle measurement lemma, where matrix ordering replaces approximation in trace norm.