Monotonicity of the von Neumann Entropy under Quantum Convolution

The quantum entropy power inequality, proven by K\"onig and Smith (2012), states that $\exp(S(\rho \boxplus \sigma)/m)\geq \frac 12 (\exp(S(\rho)/m) + \exp(S(\sigma)/m))$ for two $m$-mode bosonic quantum states $\rho$ and $\sigma$. One direct consequence of this inequality is that the sequence $\big\{ S(\rho^{\boxplus n}): n\geq 1 \big\}$ of von Neumann entropies of symmetric convolutions of $\rho$ has a monotonically increasing subsequence, namely, $S(\rho^{\boxplus 2^{k+1}})\geq S(\rho^{\boxplus 2^{k}})$. In the classical case, it has been shown that the whole sequence of entropies of the normalized sums of i.i.d.~random variables is monotonically increasing. Also, it is conjectured by Guha (2008) that the same holds in the quantum setting, and we have $S(\rho^{\boxplus n}) \geq S(\rho^{\boxplus (n-1)})$ for any $n$. In this paper, we resolve this conjecture by establishing this monotonicity. We in fact prove generalizations of the quantum entropy power inequality, enabling us to compare the von Neumann entropy of the $n$-fold symmetric convolution of $n$ arbitrary states $\rho_1, \cdots, \rho_n$ with the von Neumann entropy of the symmetric convolution of subsets of these quantum states. Additionally, we propose a quantum-classical version of this entropy power inequality, which helps us better understand the behavior of the von Neumann entropy under the convolution action between a quantum state and a classical random variable.