Triangle Inequality for the Square Root of Quantum Jensen–Shannon Divergence

Determine whether the square root of the quantum Jensen–Shannon divergence satisfies the triangle inequality for mixed quantum states.

Background

The paper uses the square root of the quantum Jensen–Shannon divergence as a distance for defining basis-independent, collective, and localized coherence. The text states that the triangle inequality is established for pure states, whereas its validity for mixed states has only been supported by numerical verification and remains conjectural. Establishing this property would clarify whether the quantity is a genuine metric on the full space of mixed density matrices.

References

Its square root, \begin{equation} \mathcal{D}(\rho,\sigma) = \sqrt{J(\rho,\sigma)}, \end{equation} satisfies the triangle inequality for pure states and is conjectured, and numerically verified, to do so for mixed states as well .

— Quantum correlations and Basis-Independent Coherence Distribution in Two Gravitational Cat States  (2608.13493 - Mansour et al., 13 Aug 2026) in Section 2, Subsection 2.3, “The quantum Jensen--Shannon divergence”