Quantum discord for two-qubit X states: Analytical formula with very small worst-case error

TL;DR

This paper evaluates the accuracy of an analytical formula for quantum discord in two-qubit X states, showing it is highly accurate with minimal worst-case errors, thus supporting its widespread use despite minor inaccuracies.

Contribution

The paper provides a numerical validation of an existing analytical formula for quantum discord in two-qubit X states, quantifying its small worst-case errors and clarifying its applicability.

Findings

The formula is valid with a worst-case absolute error of 0.0021 for general two-qubit X states.

For symmetric two-qubit X states, the formula has an even smaller worst-case error of 0.0006.

The formula has been widely used in research, and the results are approximately correct despite minor inaccuracies.

Abstract

Quantum discord is a measure of quantum correlation beyond entanglement. Computing quantum discord for simple quantum states is a basic problem. An analytical formula of quantum discord for two-qubit X states is first claimed in [Ali, Rau, and Alber, Phys. Rev. A 81, 042105 (2010)], but later found to be not always correct. I observe numerically that the formula is valid with worst-case absolute error 0.0021. For symmetric two-qubit X states, I give a counterexample to the analytical formula derived in [F. F. Fanchini et al., Phys. Rev. A 81, 052107 (2010)], but observe that the formula is valid with worst-case absolute error 0.0006. The formula has been used in many research papers. The results in all these works are approximately correct, even if they may not be exactly correct.