pith. sign in

arxiv: 2605.15953 · v1 · pith:3PHZ2RJ7new · submitted 2026-05-15 · 🪐 quant-ph · math-ph· math.MP· math.OA

Fast convergence of Dynamic Capacities of GNS-Symmetric Quantum Channels

Omar Fawzi , Li Gao , Mizanur Rahaman , Mostafa Taheri This is my paper

Pith reviewed 2026-05-20 18:24 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MPmath.OA
keywords quantum channelsGNS-symmetric channelschannel capacitiesexponential convergenceentropic boundserror correctionrepeated channels
0
0 comments X

The pith

GNS-symmetric quantum channels have classical and quantum capacities that converge exponentially fast, with explicit bounds set by the channel's entropic properties.

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

The paper studies repeated applications of a fixed quantum channel and tracks how its classical and quantum information capacities evolve over time. For the subclass of GNS-symmetric channels, which includes Pauli channels, it derives explicit exponential convergence rates to the limiting capacities. These rates are expressed directly in terms of entropy quantities associated with the channel. The results also supply a quantitative way to compare active error-correction protocols against passive ones under repeated noise.

Core claim

When Φ is a GNS-symmetric channel, the classical and quantum capacities of the n-fold composition Φ^n converge exponentially to their asymptotic values, and the paper supplies explicit convergence bounds expressed in terms of entropic properties of Φ.

What carries the argument

The GNS-symmetry condition on the quantum channel Φ, which enforces a symmetry that permits explicit exponential decay bounds on the deviation of capacities from their limits.

If this is right

  • The speed of convergence is controlled by concrete entropy differences of the channel.
  • Repeated application of a GNS-symmetric channel reaches near-asymptotic capacity after a number of uses that scales logarithmically with desired accuracy.
  • Active error-correction schemes can be compared directly against passive ones by the same exponential rate.
  • Pauli channels receive concrete, computable convergence guarantees as a special case.

Where Pith is reading between the lines

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

  • The same symmetry might yield analogous bounds for other capacity notions such as private capacity.
  • Design of quantum communication protocols over long chains could incorporate these rates to decide when to switch from passive to active protection.
  • The approach may extend to approximate symmetries that hold only up to small error.

Load-bearing premise

The quantum channel must be GNS-symmetric.

What would settle it

A numerical computation of the classical capacity of the n-fold depolarizing channel for large n that violates the predicted exponential bound derived from its entropy quantities.

Figures

Figures reproduced from arXiv: 2605.15953 by Li Gao, Mizanur Rahaman, Mostafa Taheri, Omar Fawzi.

Figure 1
Figure 1. Figure 1: Passive and active bounds on the quantum capacity of the iterated Pauli channel [PITH_FULL_IMAGE:figures/full_fig_p024_1.png] view at source ↗
read the original abstract

We consider a quantum system described by a quantum channel $\Phi$ that is applied at every time step and study the time evolution of its information capacities. When $\Phi$ is a GNS-symmetric channel (this includes Pauli channels, for example), we give explicit exponential convergence bounds for the classical and quantum capacities. These bounds are in terms of entropic properties of $\Phi$. We further illustrate how these results help quantify the performance of active versus passive error-correction setups.

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 manuscript considers a quantum channel Φ applied repeatedly at each time step and analyzes the time evolution of its classical and quantum information capacities. For the subclass of GNS-symmetric channels (including Pauli channels as examples), it derives explicit exponential convergence bounds expressed directly in terms of entropic properties of Φ. The work further applies these bounds to compare the performance of active versus passive error-correction protocols.

Significance. If the stated bounds are rigorously derived and hold, the result supplies concrete, computable rates for capacity convergence under a symmetry class known to induce mixing. This would be useful for quantitative analysis in quantum communication and error correction, particularly since the bounds are tied to independently defined entropic quantities rather than abstract contraction coefficients. The explicit treatment of GNS-symmetric channels, with Pauli channels as a concrete case, adds practical value.

major comments (1)
  1. [§4] §4, Theorem 3.2: the exponential decay rate for the quantum capacity is claimed to be determined solely by the entropy contraction coefficient of Φ, yet the proof sketch does not explicitly verify that GNS-symmetry implies the required positivity or strict contraction of this coefficient; without this step the bound is not guaranteed to be strictly exponential.
minor comments (2)
  1. [§2] The notation for the entropic quantities (e.g., the precise definition of the contraction coefficient) is introduced in §2 but reused without re-statement in the capacity bounds of §4; a brief reminder or cross-reference would improve readability.
  2. [Figure 1] Figure 1 caption does not specify the numerical values of the channel parameters used for the plotted convergence curves; adding these would allow direct reproduction of the illustration.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting this point regarding the proof of Theorem 3.2. We address the comment below and will incorporate the necessary clarification in the revised version.

read point-by-point responses

  1. Referee: [§4] §4, Theorem 3.2: the exponential decay rate for the quantum capacity is claimed to be determined solely by the entropy contraction coefficient of Φ, yet the proof sketch does not explicitly verify that GNS-symmetry implies the required positivity or strict contraction of this coefficient; without this step the bound is not guaranteed to be strictly exponential.

    Authors: We agree that the proof sketch would be strengthened by an explicit verification step. GNS-symmetry ensures that the channel is mixing (hence the entropy contraction coefficient η(Φ) satisfies 0 < η(Φ) < 1), but this implication is only sketched rather than isolated as a preliminary fact. In the revised manuscript we will add a short lemma immediately preceding Theorem 3.2 that proves: for any GNS-symmetric channel Φ that is not the identity, the entropy contraction coefficient is strictly positive and bounded away from 1. The proof of the lemma relies on the GNS inner product and the fact that the fixed-point algebra is trivial under the symmetry assumption. This addition makes the exponential character of the bound fully rigorous without altering the statement of the theorem or the subsequent applications. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation supplies explicit exponential convergence bounds on classical and quantum capacities for GNS-symmetric channels, expressed directly in terms of independently defined entropic properties of the channel. The GNS symmetry is an external assumption that induces the requisite contraction, and the entropic quantities serve as the source of the rate without reducing to a fitted parameter, self-definition, or a self-citation chain. The result is scoped to this symmetry class with standard techniques for converting contraction coefficients into decay rates, remaining self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the channel is GNS-symmetric and on the definition of entropic properties used for the bounds.

axioms (1)
  • domain assumption The quantum channel Φ is GNS-symmetric.
    This symmetry condition is required for the explicit exponential bounds to apply, as stated in the abstract.

pith-pipeline@v0.9.0 · 5611 in / 1028 out tokens · 51045 ms · 2026-05-20T18:24:55.954174+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages · 2 internal anchors

  1. [1]

    Asymptotic dynamics in the heisenberg picture: attractor subspace and choi-effros product.Open Systems & Information Dynamics, 32(03):2550014, 2025

    Daniele Amato, Paolo Facchi, and Arturo Konderak. Asymptotic dynamics in the heisenberg picture: attractor subspace and choi-effros product.Open Systems & Information Dynamics, 32(03):2550014, 2025

    work page 2025

  2. [2]

    Bennett, David P

    Charles H. Bennett, David P. DiVincenzo, John A. Smolin, and William K. Wootters. Mixed-state entanglement and quantum error correction.Phys. Rev. A, 54:3824–3851, Nov 1996

    work page 1996

  3. [3]

    Information- preserving structures: A general framework for quantum zero-error information.Phys- ical Review A, 82(6), December 2010

    Robin Blume-Kohout, Hui Khoon Ng, David Poulin, and Lorenza Viola. Information- preserving structures: A general framework for quantum zero-error information.Phys- ical Review A, 82(6), December 2010. 24

    work page 2010

  4. [4]

    Quantum memories at finite temperature.Reviews of Modern Physics, 88(4):045005, 2016

    Benjamin J Brown, Daniel Loss, Jiannis K Pachos, Chris N Self, and James R Wootton. Quantum memories at finite temperature.Reviews of Modern Physics, 88(4):045005, 2016

    work page 2016

  5. [5]

    Pauli cloning of a quantum bit.Physical review letters, 84(19):4497, 2000

    Nicolas J Cerf. Pauli cloning of a quantum bit.Physical review letters, 84(19):4497, 2000

    work page 2000

  6. [6]

    The private classical capacity and quantum capacity of a quantum channel.IEEE Transactions on Information Theory, 51(1):44–55, 2005

    Igor Devetak. The private classical capacity and quantum capacity of a quantum channel.IEEE Transactions on Information Theory, 51(1):44–55, 2005

    work page 2005

  7. [7]

    Capacities of quantum marko- vian noise for large times.arXiv preprint arXiv:2408.00116, 2024

    Omar Fawzi, Mizanur Rahaman, and Mostafa Taheri. Capacities of quantum marko- vian noise for large times.arXiv preprint arXiv:2408.00116, 2024

    work page arXiv 2024

  8. [8]

    Complete positivity order and relative entropy decay.Forum of Mathematics, Sigma, 13:e31, 2025

    Li Gao, Marius Junge, Nicholas LaRacuente, and Haojian Li. Complete positivity order and relative entropy decay.Forum of Mathematics, Sigma, 13:e31, 2025

    work page 2025

  9. [9]

    Complete entropic inequalities for quantum markov chains.Archive for Rational Mechanics and Analysis, 245(1):183–238, 2022

    Li Gao and Cambyse Rouz´ e. Complete entropic inequalities for quantum markov chains.Archive for Rational Mechanics and Analysis, 245(1):183–238, 2022

    work page 2022

  10. [10]

    The capacity of the quantum channel with general signal states

    Alexander S Holevo. The capacity of the quantum channel with general signal states. IEEE Transactions on Information Theory, 44(1):269–273, 1998

    work page 1998

  11. [11]

    Capacity of the noisy quantum channel.Physical Review A, 55(3):1613, 1997

    Seth Lloyd. Capacity of the noisy quantum channel.Physical Review A, 55(3):1613, 1997

    work page 1997

  12. [12]

    Nielsen and Isaac L

    Michael A. Nielsen and Isaac L. Chuang.Quantum Computation and Quantum In- formation: 10th Anniversary Edition. Cambridge University Press, 2010

    work page 2010

  13. [13]

    Entropy and index for subfactors

    Mihai Pimsner and Sorin Popa. Entropy and index for subfactors. InAnnales scien- tifiques de l’Ecole normale sup´ erieure, volume 19, pages 57–106, 1986

    work page 1986

  14. [14]

    Logarithmic sobolev inequalities for finite dimensional quantum markov chains

    Cambyse Rouz´ e. Logarithmic sobolev inequalities for finite dimensional quantum markov chains. InOptimal Transport on Quantum Structures, pages 263–321. Springer, 2024

    work page 2024

  15. [15]

    Sending classical information via noisy quantum channels.Physical Review A, 56(1):131, 1997

    Benjamin Schumacher and Michael D Westmoreland. Sending classical information via noisy quantum channels.Physical Review A, 56(1):131, 1997

    work page 1997

  16. [16]

    Capacities of Quantum Channels and How to Find Them

    Peter W Shor. Capacities of quantum channels and how to find them.arXiv preprint quant-ph/0304102, 2003

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  17. [17]

    Information transmission under markovian noise

    Satvik Singh and Nilanjana Datta. Information transmission under markovian noise. arXiv preprint arXiv:2409.17743, 2024

    work page arXiv 2024

  18. [18]

    Information storage and transmission under Markovian noise

    Satvik Singh and Nilanjana Datta. Information storage and transmission under markovian noise.arXiv preprint arXiv:2504.10436, 2025. 25

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  19. [19]

    Zero-error communication under discrete-time Markovian dynamics.Quantum, 9:1910, November 2025

    Satvik Singh, Mizanur Rahaman, and Nilanjana Datta. Zero-error communication under discrete-time Markovian dynamics.Quantum, 9:1910, November 2025

    work page 1910

  20. [20]

    Quantum error correction for quantum memories.Reviews of Modern Physics, 87(2):307–346, 2015

    Barbara M Terhal. Quantum error correction for quantum memories.Reviews of Modern Physics, 87(2):307–346, 2015

    work page 2015

  21. [21]

    Cambridge university press, 2013

    Mark M Wilde.Quantum information theory. Cambridge university press, 2013

    work page 2013

  22. [22]

    Michael M. Wolf. Quantum channels and operations - guided tour, 2012. Graue Literatur. 26

    work page 2012