Conjecture: Subadditivity of specific reduced functions

Show that the reduced functions $h_{\mathcal{F}'}(\rho)=1-(\mathrm{Tr}\,\rho^2)^2$, $h_{A\mathcal{F}}(\rho)=1-\sqrt{\mathrm{Tr}\,\rho^3}$, and $\hat{h}(\rho)$ are subadditive, and that $\hat{h}(\rho)$ is concave.

Background

Subadditivity of the reduced function is crucial to proving tight complete monogamy for certain global measures.

The authors explicitly conjecture subadditivity for these reduced functions.

References

We conjecture that $h_{\mathcal{F}'}$, $h_{A\mathcal{F}$, and $\hat{h}$ are subadditive, and that $\hat{h}$ is subadditive. In what follows, we always assume that $h_{\mathcal{F}'}$, $h_{A\mathcal{F}$, and $\hat{h}$ are subadditive, and that $\hat{h}$ is concave.

Measure of entanglement and the monogamy relation: a topical review  (2512.21992 - Guo et al., 26 Dec 2025) in Section 9.3 Subadditivity of the reduced function