arXiv subjects

Hami Mehrabi

Publications and source records attributed to Hami Mehrabi.

4 recordsLinked to original sources

arXiv2026

Central Limit Theorem for Bosonic Quantum Channels

In this paper, we develop an extension of the Central Limit Theorem (CLT) to the setting of bosonic quantum channels. This extension provides a deeper understanding of Gaussian bosonic channels as extremal objects. Using our CLT for bosonic quantum channels, we recover both the classical CLT and the CLT for bosonic quantum states, thereby offering a unified perspective that connects classical probability theory with continuous-variable quantum systems. Moreover, using our result, we can provide necessary uncertainty relations that every physical (possibly non-Gaussian) bosonic quantum channel must satisfy. As another application of our limit theorems, we derive tight lower bounds on the energy-constrained quantum capacity of linear bosonic channels by relating it to the capacity of their associated Gaussian bosonic channels, further reinforcing the role of Gaussian channels as extremal.

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arXiv2025

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 $\rho_1, \dots, \rho_{n}$ 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.

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arXiv2024

Optimal convergence rates in trace distance and relative entropy for the quantum central limit theorem

A quantum analogue of the Central Limit Theorem (CLT) for bosonic system, first introduced by Cushen and Hudson (1971), states that the $n$-fold convolution $\rho^{\boxplus n}$ of an $m$-mode quantum state $\rho$, with zero first moments and finite second moments, converges weakly, as $n$ increases, to a Gaussian state $\rho_G$ with the same first and second moments as those of $\rho$, called its Gaussification. Recently, this result has been extended with estimates of the convergence rate in various distance measures. In this paper, we establish optimal rates of convergence in both the trace distance and quantum relative entropy. Specifically, we show that for a centered $m$-mode quantum state with finite third-order moments, the trace distance between $\rho^{\boxplus n}$ and $\rho_G$ decays at the optimal rate of $\mathcal{O}(n^{-1/2})$. Furthermore, for states with finite fourth-order moments (order $4+\delta$ for an arbitrary small $\delta>0$ if $m>1$), we prove that the relative entropy between $\rho^{\boxplus n}$ and $\rho_G$ decays at the optimal rate of $\mathcal{O}(n^{-1})$. Both of these rates are proven to be optimal, even when assuming the finiteness of all moments of $\rho$. These results relax previous assumptions on higher-order moments, yielding convergence rates that match the best known results in the classical setting. By giving explicit examples we also show that our moment assumptions are essentially minimal. Our proofs draw on techniques from the classical literature, including Edgeworth-type expansions of quantum characteristic functions, adapted to the quantum context. A key technical step in the proof of our entropic CLT is establishing an upper bound on the relative entropy distance between a general quantum state and its Gaussification, which is of independent interest.

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arXiv2023

Towards Optimal Convergence Rates for the Quantum Central Limit Theorem

The quantum central limit theorem for bosonic quantum systems states that the sequence of states $\rho^{\boxplus n}$ obtained from the $n$-fold convolution of a centered quantum state $\rho$ converges to a quantum Gaussian state $\rho_G$ that has the same first and second moments as $\rho$. In this paper, we contribute to the problem of finding the optimal rate of convergence for this quantum central limit theorem. We first show that if an $m$-mode quantum state has a finite moment of order $\max\{3, 2m\}$, then we have $\|\rho^{\boxplus n} - \rho_G\|_1=\mathcal O(n^{-1/2})$. We also introduce a notion of Poincar\'e inequality for quantum states and show that if $\rho$ satisfies this Poincar\'e inequality, then $D(\rho^{\boxplus n}\| \rho_G)= \mathcal O(n^{-1})$. By giving an explicit example, we verify that both these convergence rates are optimal.

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