Generalized Pinching Inequality Extends Hayashi’s Inequality with Weighted Projections and Reversed Matrix Ordering

Hayashi’s Pinching Inequality has become a cornerstone in quantum information theory, establishing a fundamental relationship between semidefinite matrices and projective measurements, and finding applications in diverse areas of the field. Now, Andreas Winter from University zu Köln, and colleagues, present a remarkably simple proof of this inequality, opening the door to significant generalisations. The team demonstrates that the inequality extends to scenarios where measurements employ varying weights for different projections, and even allows for reversal of the original matrix inequality. This work culminates in a novel gentle measurement lemma, which replaces traditional approximation methods with a powerful matrix ordering approach, representing a substantial advance in the field.

A semidefinite matrix and a multiple of its modified version appear in many applications within quantum information theory and beyond. This research presents a streamlined proof of a key inequality, extending it to encompass scenarios with weighted measurements and even reversed inequalities. This work expands the original inequality, established by Hayashi, to a broader context, defining a specific set, denoted as An, within an n-dimensional space. The core of the generalisation lies in a lemma demonstrating that if a set of values belongs to An, then for any bounded operators and a positive semidefinite operator, a specific inequality holds relating sums of operator products to a weighted sum. Conversely, the lemma establishes that if this inequality holds for a specific type of measurement, then the values must belong to An.

The proof involves defining a bounded operator and demonstrating its connection to completely positive maps, ultimately linking the inequality to the properties of this operator. Further investigation reveals that membership in An is determined by a simple semidefinite constraint, expressible as a polynomial criterion involving determinants of principal submatrices. The researchers detail a recursive method for constructing these polynomial conditions, starting from the case of n=2, utilising the Schur complement to simplify the calculations. This allows for a systematic determination of the conditions for values to belong to An for increasing values of n. The study also explores a reverse pinching inequality, considering the opposite set to An, and derives a tight bound for the binary case (n=2). Scientists demonstrate that the inequality holds not only for standard projective measurements, but also when different projections are assigned varying weights, significantly broadening its applicability. This generalised inequality leads to a new gentle measurement lemma, replacing traditional approximation methods with matrix ordering, offering a potentially more robust analytical tool. Measurements confirm the validity of the reversed pinching inequality under specific conditions, expanding the scope of the original theorem. This new lemma provides a robust framework for analysing quantum measurements and their associated uncertainties, and represents a significant advancement in the field, providing researchers with powerful new tools for tackling complex problems.