## The Modified Trace Distance of Coherence is Constant on Most Pure States

**Abstract:**

Recently, the much-used trace distance of coherence was shown to not be a proper measure of coherence, so a modification of it was proposed. We derive an explicit formula for this modified trace distance of coherence on pure states. Our formula shows that, despite satisfying the axioms of proper coherence measures, it is likely not a good measure to use, since it is maximal (equal to 1) on all except for an exponentially-small (in the dimension of the space) fraction of pure states.

**Authors:**

- Nathaniel Johnston
- Chi-Kwong Li
- Sarah Plosker

**Download:**

- Official publication from Journal of Physics A: Mathematical and Theoretical
- Preprint from arXiv:1708.06967 [quant-ph]
- Local preprint [pdf]

**Cite as:**

- N. Johnston, C.-K. Li, and S. Plosker. The modified trace distance of coherence is constant on most pure states.
*Journal of Physics A: Mathematical and Theoretical,*51:414010, 2018.

**Supplementary material:**

- MATLAB code for computing the modified trace distance of coherence [.zip file of MATLAB .m scripts – requires YALMIP]
- Quantifying the coherence of pure quantum states – an earlier paper about the (non-modified) trace distance of coherence