Progress on Quantum Discord of Two Qubit states: Optimization and Upper Bound

TL;DR

This paper introduces a new relation simplifying quantum discord calculation for two-qubit states, enabling direct computation and optimization without extensive minimization, and establishes a tight upper bound on quantum discord.

Contribution

It provides a simple relation for quantum conditional entropy, facilitating easier calculation and optimization of quantum discord in two-qubit states, and derives a tight upper bound.

Findings

Derived a relation expressing quantum conditional entropy as the difference of Shannon entropies.

Provided an analytical optimization procedure for quantum discord.

Established a tight upper bound on quantum discord.

Abstract

Calculation of the quantum discord requires to find the minimum of the quantum conditional entropy $S(\rho^{AB}|\{\Pi^B_{k}\})$ over all measurements on the subsystem $B$. In this paper, we provide a simple relation for the conditional entropy as the difference of two Shannon entropies. The relation is suitable for calculation of the quantum discord in the sense that it can by used to obtain the quantum discord for some classes of two-qubit states, without the need for minimization. We also present an analytical procedure of optimization and obtain conditions under which the quantum conditional entropy of a general two-qubit state is stationary. The presented relation is also used to find a tight upper bound on the quantum discord.