**5. Conclusions**

In conclusion, we present an explicit expression of the non-commutativity measure of quantum discord in the two-qubit case. We also compare the non-commutativity measure of quantum discord with the geometric quantum discord in the models of two qubits passing through the dephasing channel and the multimode vacuum field coupling the qubits, respectively. Our study shows that the non-commutativity measures of quantum discord lose some important features of the geometric quantum discords, such as the freezing phenomenon in the dephasing channel and the revival in the multimode vacuum field coupling the qubits.

**Author Contributions:** All the authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

**Funding:** The work was supported partly by the NSF of China (11571229).

**Acknowledgments:** The authors would like to thank the reviewers very much for valuable comments and suggestions.

**Conflicts of Interest:** The authors declare that they have no competing interests.
