Figure 1

Bounds on the minimum Rényi entropies ${\mathbb{S}}_z\left({\mathcal{N}}_n\right)$ (in nats) as a function of $z$: ${\mathbb{S}}_z$ is restricted to the gray region. The upper bound 0 is the Rényi output entropy of the vacuum input, i.e., the right-hand side of Eq. (8). Lower bound 1 (dotted staircase) derives from the fact that $S_z$ is a decreasing function of $z$. For $z>1$, it follows from Eq. (25) and the integer-$z$ minimum Rényi entropy calculated in Sec. 2A; for $z⩽1$, it is equal to $\overline{\mathbb{S}}\left({\mathcal{N}}_n\right)$, the best of the the von Neumann output entropy lower bounds from [7] (in this case $\overline{\mathbb{S}}\sim 0.95$). Lower bounds 2 and 4 are given by Eqs. (28, 32), respectively. Lower bound 3 (thick line) is the greater of Eqs. (29, 30). For high values of $z$, the maximum of lower bounds 1–4 asymptotically coincides with the upper bound 0; in the figure lower bound 2 becomes indistinguishable from the upper bound once $z\gtrsim 5$. The dashed line is $S_{\infty }=\ln \left(n+1\right)$.Reuse & Permissions

Figure 2

Same as Fig. 1, but for a different value of the noise parameter $n$. Note that the lower bounds approach the upper bound 0 for high values of $n$, indicating that our conjecture is asymptotically true.Reuse & Permissions

Figure 3

Left: plot of the function $h_z\left(x\right)$ from Eq. (C2) as a function of $x$ for different values of $z$; $z$ increases from $1∕2$ to $5∕2$ in progressing along the direction of the arrow. For $z>1$, $h_z\left(x\right)$ is an increasing function, and for $z<1$ it is decreasing. Note that $h_z\left(1\right)=1$, ${\lim }_{x\to 0}{h}_z\left(x\right)=0$ for $z>1$, and ${\lim }_{x\to 0}{h}_z\left(x\right)=\infty$ for $z<1$. Right: plot of the function $v\left(x\right)$ from Eq. (C6).Reuse & Permissions