arXiv subjects

Milad M. Goodarzi

Publications and source records attributed to Milad M. Goodarzi.

5 recordsLinked to original sources

arXiv2026

Submodularity of entropy under quantum convolution

We develop a submodular framework for the von Neumann entropy of discrete quantum convolutions, providing a noncommutative counterpart to the direct side of entropic additive combinatorics. We first introduce globally weighted quantum convolutions, which form compatible families indexed by admissible subsets of a fixed collection of inputs. Our main theorem reveals a polymatroidal geometry underlying their entropy growth: relative to any fixed admissible input block, the entropy gains admit a normalized, monotone, submodular extension to all subsets of the remaining inputs. The theorem yields convolutional strong subadditivity, quantum Ruzsa triangle inequality, and quantum entropic Plünnecke--Ruzsa inequalities for arbitrary input states. For repeated inputs, it gives sharp comparisons of entropy growth across admissible scales; in particular, the quantum doubling constant controls all higher admissible convolution entropies with optimal exponents. Together, these results bring submodular methods from additive combinatorics into the quantum setting and provide a systematic route to broad families of convolutional entropy inequalities.

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arXiv2026

Sharp Data-Processing Region for Quantum Conditional Rényi Entropies

We study the three-parameter family of quantum conditional Rényi entropies $H_{α,z}^λ$ introduced in \textit{Commun. Math. Phys.} 407, 180 (2026). We prove that the previously established data-processing region is sharp. We then characterize the club-sandwiched family through a classical coarse-graining principle: within the full data-processing region, universal monotonicity under deterministic coarse-graining of the classical conditioned register holds if and only if $z = α$. Finally, we establish rigidity of entropic duality by showing that any universal duality relation for finite-dimensional tripartite pure states must satisfy the known parameter relations. Thus, the data-processing region, the club-sandwiched subfamily, and the duality transformation are each uniquely characterized by their respective universal properties.

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arXiv2026

Quantum conditional entropies from convex trace functionals

We study geometric properties of trace functionals that generalize those in [Zhang, Adv. Math. 365:107053 (2020)], arising from a novel family of conditional entropies with applications in quantum information. Building on new convexity results for these functionals, we establish data-processing inequalities and additivity properties for our entropies, demonstrating their operational significance. We further prove completeness under duality, chain rules, and various monotonicity properties for this family. Our proofs draw on tools from complex interpolation theory, multivariate Araki--Lieb and Lieb--Thirring inequalities, variational characterizations of trace functionals, and spectral pinching techniques.

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arXiv2026

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 $ρ^{\boxplus n}$ of an $m$-mode quantum state $ρ$, with zero first moments and finite second moments, converges weakly, as $n$ increases, to a Gaussian state $ρ_G$ with the same first and second moments as those of $ρ$, 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 $ρ^{\boxplus n}$ and $ρ_G$ decays at the optimal rate of $\mathcal{O}(n^{-1/2})$. Furthermore, for states with finite fourth-order moments (order $4+δ$ for an arbitrary small $δ>0$ if $m>1$), we prove that the relative entropy between $ρ^{\boxplus n}$ and $ρ_G$ 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 $ρ$. 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

Operator-valued Schatten spaces and quantum entropies

Operator-valued Schatten spaces were introduced by G. Pisier as a noncommutative counterpart of vector-valued $\ell_p$-spaces. This family of operator spaces forms an interpolation scale which makes it a powerful and convenient tool in a variety of applications. In particular, as the norms coming from this family naturally appear in the definition of certain entropic quantities in Quantum Information Theory (QIT), one may apply Pisier's theory to establish some features of those quantities. Nevertheless, it could be quite challenging to follow the proofs of the main results of this theory from the existing literature. In this article, we attempt to fill this gap by presenting the underlying concepts and ideas of Pisier's theory in a self-contained way which we hope to be more accessible, especially for the QIT community at large. Furthermore, we describe some applications of this theory in QIT. In particular, we prove a new uniform continuity bound for the quantum conditional Rényi entropy.

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