Toward computability of trace distance discord

TL;DR

This paper develops a method to compute the trace distance discord for general two-qubit states, providing explicit formulas for important classes like quantum-classical and X states, advancing the quantification of quantum correlations.

Contribution

It reduces the optimization problem for trace distance discord to a two-variable minimization and derives explicit formulas for key classes of states.

Findings

Explicit formulas for trace distance discord of quantum-classical states

Analytical minimization framework for two-qubit states

Extension beyond Bell-diagonal states

Abstract

It is known that a reliable geometric quantifier of discord-like correlations can be built by employing the so-called trace distance. This is used to measure how far the state under investigation is from the closest "classical-quantum" one. To date, the explicit calculation of this indicator for two qubits was accomplished only for states such that the reduced density matrix of the measured party is maximally mixed, a class that includes Bell-diagonal states. Here, we first reduce the required optimization for a general two-qubit state to the minimization of an explicit two-variable function. Using this framework, we show next that the minimum can be analytically worked out in a number of relevant cases including quantum-classical and X states. This provides an explicit and compact expression for the trace distance discord of an arbitrary state belonging to either of these important classes of density matrices.