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arxiv: 2605.16782 · v1 · pith:6LNS7VXOnew · submitted 2026-05-16 · 🪐 quant-ph

Central Limit Theorem for Bosonic Quantum Channels

Hami Mehrabi , Ludovico Lami , Mark M. Wilde This is my paper

Pith reviewed 2026-05-19 21:09 UTC · model grok-4.3

classification 🪐 quant-ph
keywords central limit theorembosonic quantum channelsGaussian extremalityuncertainty relationsquantum capacitycontinuous-variable systemsquantum information
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The pith

The central limit theorem extends to bosonic quantum channels and establishes Gaussian channels as extremal objects.

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

This paper develops an extension of the central limit theorem to bosonic quantum channels. The extension shows that repeated composition or use of such channels leads to Gaussian bosonic channels under suitable conditions. A sympathetic reader would care because the result recovers both the classical central limit theorem and the central limit theorem for bosonic quantum states, creating a single framework that links classical probability to continuous-variable quantum systems. It further supplies necessary uncertainty relations that every physical bosonic channel must obey. The same limit theorems yield tight lower bounds on the energy-constrained quantum capacity of linear bosonic channels by comparison with their Gaussian counterparts.

Core claim

The authors prove a central limit theorem for bosonic quantum channels. This theorem shows that Gaussian bosonic channels arise as the limiting objects when sequences of physical bosonic channels are composed or applied repeatedly. The result recovers the classical central limit theorem and the central limit theorem for bosonic states as special cases, thereby connecting classical probability theory with continuous-variable quantum systems. It also implies that Gaussian channels are extremal among all physical bosonic channels and supplies concrete uncertainty relations that non-Gaussian channels must satisfy.

What carries the argument

The central limit theorem for bosonic quantum channels, which maps sequences of channels obeying finite-moment and complete-positivity conditions to Gaussian channel limits while preserving the bosonic structure.

If this is right

  • The classical central limit theorem is recovered as a special case.
  • The central limit theorem for bosonic quantum states is recovered as a special case.
  • Necessary uncertainty relations must hold for every physical bosonic quantum channel, Gaussian or otherwise.
  • Tight lower bounds follow for the energy-constrained quantum capacity of any linear bosonic channel relative to its Gaussian associate.

Where Pith is reading between the lines

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

  • The same limiting argument might be adapted to derive central limit theorems for other families of quantum channels beyond the bosonic case.
  • The extremality property could simplify variational problems that appear in quantum communication and metrology.
  • Laboratory checks of the uncertainty relations on engineered non-Gaussian channels would provide direct experimental tests of the theorem.

Load-bearing premise

Bosonic quantum channels must satisfy technical conditions such as finite moments and complete positivity so that the limiting process preserves the bosonic structure and physical validity.

What would settle it

A physical bosonic quantum channel that violates one of the derived uncertainty relations or whose energy-constrained quantum capacity falls below the lower bound set by its associated Gaussian channel would disprove the central claim.

Figures

Figures reproduced from arXiv: 2605.16782 by Hami Mehrabi, Ludovico Lami, Mark M. Wilde.

Figure 1
Figure 1. Figure 1: FIG. 1: Symmetric convolution of channel [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: for a visual depiction. Using the expression in the penultimate line of (9) for linear bosonic channels, we obtain the no-signalling FIG. 2: Extension of the 2-fold symmetric convolution. property that the channel N˜ 𝐴1→𝐵1 is independent of the input on 𝐴2. By definition, the reduced channel N˜ 𝐴1→𝐵1 = N ⊞2 . It is also straightforward to see that the same property holds for the channel N˜ 𝐴2→𝐵2 that there… view at source ↗
read the original abstract

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.

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

1 major / 2 minor

Summary. The paper develops an extension of the Central Limit Theorem to bosonic quantum channels. It establishes Gaussian bosonic channels as extremal objects, recovers both the classical CLT and the CLT for bosonic quantum states from this extension, derives necessary uncertainty relations that every physical (possibly non-Gaussian) bosonic channel must satisfy, and obtains tight lower bounds on the energy-constrained quantum capacity of linear bosonic channels by relating them to associated Gaussian channels.

Significance. If the central limit theorem for channels holds under the stated technical conditions, the work supplies a unified perspective linking classical probability theory to continuous-variable quantum systems and reinforces the extremal role of Gaussian channels. The applications to uncertainty relations and capacity bounds constitute a substantive contribution to quantum information theory.

major comments (1)
  1. [§3, Theorem 1] §3, Theorem 1: The precise technical conditions (e.g., moment bounds or complete positivity requirements) under which the channel CLT converges while preserving the bosonic structure are not stated explicitly enough; this is load-bearing for the claim that the result applies to every physical bosonic channel and for the subsequent derivation of uncertainty relations.
minor comments (2)
  1. [§2] The notation for the limiting Gaussian channel in the main theorem could be introduced with a short table of symbols to improve readability.
  2. [Figure 2] Figure 2 caption: the diagram of channel composition would benefit from an explicit label indicating the scaling parameter n.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment of our work and for the constructive comment, which will help improve the clarity of the technical conditions in the manuscript. We address the point below.

read point-by-point responses

  1. Referee: [§3, Theorem 1] §3, Theorem 1: The precise technical conditions (e.g., moment bounds or complete positivity requirements) under which the channel CLT converges while preserving the bosonic structure are not stated explicitly enough; this is load-bearing for the claim that the result applies to every physical bosonic channel and for the subsequent derivation of uncertainty relations.

    Authors: We agree that the technical conditions should be stated more explicitly to support the claims. In the revised version, we will add a dedicated paragraph in §3 immediately preceding Theorem 1 that specifies the required assumptions: (i) uniform bounds on the second moments of the input states with respect to the energy observable, (ii) complete positivity and trace preservation of the sequence of channels, and (iii) a uniform continuity condition on the channel action in the weak operator topology that preserves the bosonic commutation relations in the limit. These conditions ensure convergence to a Gaussian bosonic channel while maintaining physicality. With these clarifications, the applicability to every physical bosonic channel (under the stated moment bounds) and the subsequent derivations of uncertainty relations and capacity bounds will be fully justified. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The abstract and description present an extension of the CLT to bosonic quantum channels that recovers prior classical and state versions while deriving uncertainty relations and capacity bounds for physical channels. No equations, self-definitional constructions, fitted inputs renamed as predictions, or load-bearing self-citation chains are exhibited in the provided text. The derivation is self-contained as an independent extension with external applications, consistent with the default expectation that most papers are not circular.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities can be identified from the provided text.

pith-pipeline@v0.9.0 · 5665 in / 1163 out tokens · 24368 ms · 2026-05-19T21:09:16.365596+00:00 · methodology

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Reference graph

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