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arxiv: 2410.21998 · v3 · submitted 2024-10-29 · 🪐 quant-ph

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

Salman Beigi , Milad M. Goodarzi , Hami Mehrabi This is my paper

Pith reviewed 2026-05-23 18:58 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum central limit theoremtrace distancerelative entropyconvergence ratesGaussificationbosonic statescharacteristic functionsEdgeworth expansion
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The pith

Centered m-mode quantum states with finite third moments converge to their Gaussification in trace distance at rate O(n^{-1/2}), and with finite fourth moments the relative entropy converges at O(n^{-1}).

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes optimal convergence rates for the quantum central limit theorem on bosonic systems. It proves that the n-fold convolution of a centered state reaches its Gaussification in trace distance at the rate O(n^{-1/2}) when only third moments are finite. The relative entropy reaches the same limit at rate O(n^{-1}) under fourth-moment assumptions, including a small extra power when there is more than one mode. Both rates are shown to be sharp even if every higher moment exists, and the moment conditions are demonstrated to be essentially minimal by explicit examples. The arguments adapt classical Edgeworth expansions of characteristic functions to the quantum setting and include a general bound on relative entropy to the Gaussification.

Core claim

For a centered m-mode quantum state with finite third-order moments the trace distance between its n-fold convolution and the matching Gaussian state decays at rate O(n^{-1/2}). When fourth-order moments exist (or order 4+δ for small δ>0 when m>1), the relative entropy decays at rate O(n^{-1}). Both rates are optimal, and the moment hypotheses are nearly the weakest possible. The proofs rely on Edgeworth-type expansions of quantum characteristic functions together with an auxiliary upper bound on the relative entropy between an arbitrary state and its Gaussification.

What carries the argument

Edgeworth-type expansions of quantum characteristic functions adapted from the classical setting

If this is right

  • The rates match the best classical central-limit results under the same moment conditions.
  • Stronger moment assumptions used in earlier quantum work can be dropped.
  • Explicit counter-examples confirm that the stated moment conditions are essentially minimal.
  • A general upper bound on relative entropy distance to the Gaussification holds independently of the convolution argument.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same expansion technique may transfer to other quantum distance measures or to limit theorems for non-Gaussian targets.
  • Error scaling in quantum-optical or continuous-variable information protocols that rely on Gaussification approximations can now be predicted from the number of modes and the available moments.
  • Direct numerical simulation of low-mode states with controlled moments offers a concrete test of the predicted exponents.

Load-bearing premise

Classical Edgeworth expansions of characteristic functions can be carried over to the quantum case and used to bound the distances once third or fourth moments are finite.

What would settle it

An explicit m-mode state with finite third moments whose trace distance to the corresponding Gaussification decays slower than order n^{-1/2} for large n.

read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript proves optimal rates in the quantum central limit theorem for m-mode bosonic states. For centered states with finite third moments, the trace distance between the n-fold convolution ρ^{⊞n} and its Gaussification ρ_G is O(n^{-1/2}). For states with finite fourth moments (or 4+δ for m>1), the relative entropy D(ρ^{⊞n} || ρ_G) is O(n^{-1}). Both rates are shown sharp by explicit counterexamples even under all-moment assumptions. Proofs adapt classical Edgeworth expansions of characteristic functions to the quantum Weyl setting; a key independent step is an upper bound on relative entropy to the Gaussification.

Significance. If the derivations hold, the work supplies the first optimal quantitative rates for the quantum CLT in trace distance and relative entropy under minimal moment hypotheses that match the classical setting. The explicit optimality constructions and the standalone relative-entropy bound to Gaussification are notable strengths. The adaptation of Edgeworth techniques to quantum characteristic functions is carried out without introducing extra regularity assumptions beyond those stated.

minor comments (2)
  1. [Introduction] The statement of the key relative-entropy bound (used for the entropic CLT) appears in the abstract and introduction but would benefit from an explicit theorem number and a one-sentence pointer to its proof location for readers interested only in that lemma.
  2. [Section on optimality constructions] In the optimality examples, the precise moment conditions that fail (third for trace distance, fourth for relative entropy) should be stated in a single displayed equation or table to make the sharpness claim immediately verifiable.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, recognition of the optimal rates achieved under minimal moment assumptions, and recommendation to accept. We appreciate the note that the results match the classical setting and that the Edgeworth adaptation and standalone relative-entropy bound are strengths.

Circularity Check

0 steps flagged

Derivation self-contained via classical adaptations

full rationale

The paper derives optimal convergence rates for the quantum CLT by adapting Edgeworth-type expansions of quantum characteristic functions from classical literature, with explicit counterexamples establishing sharpness of the moment conditions. No load-bearing steps reduce by construction to self-citations, fitted parameters renamed as predictions, or self-definitional relations. The upper bound on relative entropy to Gaussification is an independent technical contribution, and the overall chain relies on external classical results plus direct quantum adaptations without internal reduction to the target rates.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on adapting classical Edgeworth expansions to quantum characteristic functions, assuming the quantum version behaves similarly under moment conditions.

axioms (2)
  • domain assumption The n-fold convolution of quantum states is well-defined for bosonic systems with the given moments.
    Fundamental to the quantum CLT setup.
  • standard math Trace distance and relative entropy serve as appropriate metrics for quantifying convergence to Gaussian states.
    Used to state the convergence rates.

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