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arxiv: 2512.22049 · v2 · submitted 2025-12-26 · 🪐 quant-ph · cs.IT· math.IT

Quantum Secret Sharing Rates

Gabrielle Lalou , Husein Natur , Uzi Pereg This is my paper

Pith reviewed 2026-05-16 19:01 UTC · model grok-4.3

classification 🪐 quant-ph cs.ITmath.IT
keywords quantum secret sharingQSS capacitycompound quantum channelsdephasing noisequantum informationsecret sharing ratesinformation theoretic model
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The pith

The capacity of quantum secret sharing is characterized in regularized form and computed exactly for dephasing noise.

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

The paper aims to find the highest rate at which a quantum secret can be distributed to multiple participants such that only authorized groups can reconstruct it. By linking the problem to compound quantum channels, the authors derive a regularized expression that characterizes this capacity in general. They then calculate the precise capacity when the channel adds dephasing noise. This matters for setting the ultimate limits on quantum cryptographic schemes in real-world noisy conditions. It provides a benchmark for evaluating the performance of any proposed quantum secret sharing protocol.

Core claim

The paper establishes a regularized characterization for the capacity of quantum secret sharing by modeling it through compound quantum channels. It further determines the exact capacity value for the case where the quantum channel is subject to dephasing noise.

What carries the argument

Compound quantum channel model for rate analysis of quantum secret sharing

If this is right

  • The QSS capacity admits a regularized formula based on the compound channel model.
  • Exact capacity is known for dephasing noise, allowing optimal protocol rates.
  • Authorized sets of participants can recover the secret at rates up to this capacity.
  • Unauthorized participants obtain zero information about the secret.

Where Pith is reading between the lines

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

  • Future work could extend the model to other noise types like amplitude damping.
  • This characterization may inform the design of quantum networks for secure multi-party tasks.
  • Experimental verification with photonic systems could test the predicted dephasing capacity.

Load-bearing premise

Quantum secret sharing can be modeled using compound quantum channels following the classical secret sharing approach.

What would settle it

An experiment achieving a higher rate than the capacity formula for dephasing noise would falsify the result.

Figures

Figures reproduced from arXiv: 2512.22049 by Gabrielle Lalou, Husein Natur, Uzi Pereg.

Figure 1
Figure 1. Figure 1: Equivalent compound quantum channel for a [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
read the original abstract

This paper studies the capacity limits for quantum secret sharing (QSS). The goal of a QSS scheme is to distribute a quantum secret among multiple participants, such that only authorized parties can recover it through collaboration, while no information can be obtained without such collaboration. We introduce an information-theoretic model for the rate analysis of QSS and its relation to compound quantum channels, following a similar approach as of Zou et al. (2015) on classical secret sharing. We establish a regularized characterization for the QSS capacity, and determine the capacity for QSS with dephasing noise.

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

2 major / 2 minor

Summary. The paper introduces an information-theoretic model for quantum secret sharing (QSS) by relating it to compound quantum channels, following the classical construction of Zou et al. (2015). It establishes a regularized characterization of the QSS capacity and computes the capacity explicitly for the dephasing-noise case.

Significance. If the central claims hold, the work supplies the first regularized capacity formula for QSS and a concrete single-letter expression under dephasing noise. This extends classical secret-sharing capacity results to the quantum setting and supplies a theoretical benchmark for multi-party quantum cryptographic protocols. The modeling step is a direct quantum analogue of the cited classical approach and, once granted, yields standard regularization arguments plus an explicit dephasing result.

major comments (2)
  1. [Section 3] The modeling step that maps the QSS problem to a compound quantum channel (Section 3) is load-bearing for both the regularized characterization and the dephasing result; the manuscript should supply an explicit argument showing that the authorized-set recovery condition and the unauthorized-set secrecy condition translate precisely into the compound-channel capacity definition, including the appropriate quantum mutual-information quantities.
  2. [Theorem on dephasing capacity] The dephasing-noise capacity result (Theorem 2 or equivalent) is stated as a single-letter expression, but the achievability and converse arguments appear to rely on standard quantum channel coding techniques without a self-contained verification that the dephasing channel’s symmetry reduces the regularization; a short explicit calculation of the relevant quantum mutual information would strengthen the claim.
minor comments (2)
  1. [Section 3] Notation for the compound channel states and the sets of authorized/unauthorized participants should be introduced once and used consistently; currently the mapping is described in prose and could be clarified with a small diagram or table.
  2. [References] The reference list should include the full citation for Zou et al. (2015) and any standard quantum channel coding results invoked in the proofs.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will revise the paper accordingly to strengthen the presentation.

read point-by-point responses

  1. Referee: [Section 3] The modeling step that maps the QSS problem to a compound quantum channel (Section 3) is load-bearing for both the regularized characterization and the dephasing result; the manuscript should supply an explicit argument showing that the authorized-set recovery condition and the unauthorized-set secrecy condition translate precisely into the compound-channel capacity definition, including the appropriate quantum mutual-information quantities.

    Authors: We agree that an explicit translation strengthens the foundation. In the revised manuscript we will add a dedicated paragraph in Section 3 that derives the correspondence step by step: the authorized-set recovery condition is shown to be equivalent to reliable decoding over the compound channel with positive quantum mutual information rate I(A:B) (where A is the secret and B the authorized subsystems), while the unauthorized-set secrecy condition maps directly to the requirement that the quantum mutual information from the secret to any unauthorized collection vanishes, matching the secrecy constraint in the compound-channel capacity definition. This uses the standard definitions of quantum mutual information and follows the classical analogy of Zou et al. (2015) with the quantum adjustments made explicit. revision: yes

  2. Referee: [Theorem on dephasing capacity] The dephasing-noise capacity result (Theorem 2 or equivalent) is stated as a single-letter expression, but the achievability and converse arguments appear to rely on standard quantum channel coding techniques without a self-contained verification that the dephasing channel’s symmetry reduces the regularization; a short explicit calculation of the relevant quantum mutual information would strengthen the claim.

    Authors: We thank the referee for this suggestion. The dephasing channel is covariant under the Pauli Z group, which permits the regularization to collapse via the standard symmetry argument for such channels. In the revised version we will insert a short, self-contained calculation immediately after the statement of the dephasing theorem: we explicitly evaluate the quantum mutual information I(A:B) for the dephasing channel and show that it equals the single-letter expression, confirming that higher-order regularizations do not increase the rate. This verifies the reduction without relying solely on general theorems. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation adapts the classical compound-channel model of Zou et al. (2015) to quantum secret sharing, then applies standard regularization arguments to obtain the capacity characterization. The dephasing-noise case is reduced to an explicit single-letter expression using the new quantum model. No step equates a prediction to a fitted input by construction, no uniqueness theorem is imported from the authors' prior work, and the central claims rest on the introduced quantum channel model rather than self-referential definitions or load-bearing self-citations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The model relies on standard quantum channel assumptions and the extension from classical secret sharing; no free parameters or invented entities are indicated in the abstract.

axioms (1)
  • domain assumption Quantum secret sharing can be analyzed using compound quantum channels analogous to classical secret sharing
    Explicitly follows the approach of Zou et al. (2015) adapted to quantum.

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Reference graph

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