Majorization uncertainty relations for mixed quantum states

TL;DR

This paper extends majorization uncertainty relations to mixed quantum states, providing bounds on entropy sums based on the state's spectrum and measurement bases, applicable to bipartite systems.

Contribution

It generalizes majorization uncertainty relations to mixed states and derives entropy bounds involving the state's spectrum and measurement bases.

Findings

Derived a lower bound for the sum of entropies for measurements on mixed states.

Extended the results to bipartite systems and conditional entropies.

Applicable to arbitrary finite-dimensional quantum systems.

Abstract

Majorization uncertainty relations are generalized for an arbitrary mixed quantum state $\rho$ of a finite size $N$. In particular, a lower bound for the sum of two entropies characterizing probability distributions corresponding to measurements with respect to arbitrary two orthogonal bases is derived in terms of the spectrum of $\rho$ and the entries of a unitary matrix $U$ relating both bases. The obtained results can also be formulated for two measurements performed on a single subsystem of a bipartite system described by a pure state, and consequently expressed as uncertainty relation for the sum of conditional entropies.