Generalised Pinching Inequality

TL;DR

This paper presents a simple proof of Hayashi's Pinching Inequality, extends it to weighted and reversed cases, and introduces a new gentle measurement lemma based on matrix ordering.

Contribution

It provides a straightforward proof and generalizations of the pinching inequality, including weighted and reversed versions, and derives a novel gentle measurement lemma.

Findings

Extended pinching inequality to weighted projections

Reversed matrix inequality established

Introduced a new gentle measurement lemma

Abstract

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.