Figure 1.

Bounds for the sum of two Shannon entropies rescaled to the logarithm with base 2. (a,b) Plots of the bounds versus the angle $\theta$ at fixed unsharpness parameters (a) $\mu =1$ and (b) $\mu =0.8$; (c,d) Plots of the bounds versus the unsharpness parameter $\mu$ at fixed angles (c) $\theta =\pi /2$ and (d) $\theta =\pi /3$. (Blue solid curves: our direct-sum majorization bound ${\mathcal{B}}_{d2}$ in Equation (22); red dashed curves: Maassen–Uffink bound ${\mathcal{B}}_{MU}$ in Equation (16); orange dotted curves: previous direct-sum majorization bound ${\mathcal{B}}_{d1}$ in Equation (13); and purple dot-dashed curves: tensor-product majorization bound ${\mathcal{B}}_t$ in Equation (8)).