Majorization entropic uncertainty relations for quantum operations

TL;DR

This paper develops majorization-based entropic uncertainty relations for quantum operations, extending previous basis-based results to more general quantum maps using Kraus operators, and provides bounds on measurement entropy.

Contribution

It introduces a novel approach to derive majorization uncertainty relations for arbitrary quantum operations using submatrices of Kraus operators, extending prior basis-focused formulations.

Findings

Derived majorization relations for quantum operations.

Reduced to known results for orthogonal measurements.

Provided bounds on entropy for generalized measurements.

Abstract

Majorization uncertainty relations are derived for arbitrary quantum operations acting on a finite-dimensional space. The basic idea is to consider submatrices of block matrices comprised of the corresponding Kraus operators. This is an extension of the previous formulation, which deals with submatrices of a unitary matrix relating orthogonal bases in which measurements are performed. Two classes of majorization relations are considered: one related to tensor product of probability vectors and another one related to their direct sum. We explicitly discuss an example of a pair of one-qubit operations, each of them represented by two Kraus operators. In the particular case of quantum maps describing orthogonal measurements the presented formulation reduces to earlier results derived for measurements in orthogonal bases. The presented approach allows us also to bound the entropy characterizing results of a single generalized measurement.