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arxiv: 2604.27284 · v1 · submitted 2026-04-30 · 🪐 quant-ph · cs.CR· cs.IT· math.IT

Quantum Anonymous Secret Sharing with Permutation Invariant Codes

Varin Sikand , Andrew Nemec This is my paper

Pith reviewed 2026-05-07 10:26 UTC · model grok-4.3

classification 🪐 quant-ph cs.CRcs.ITmath.IT
keywords quantum secret sharingpermutation-invariant codessender anonymityquantum conditional min-entropyKnill-Laflamme conditionsramp secret sharinganonymous transmission
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The pith

Permutation-invariant codes enable sender-anonymous quantum secret sharing.

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

The paper constructs a quantum secret sharing protocol in which the sender's identity stays hidden from receivers throughout decoding. It does this by pairing permutation-invariant quantum error-correcting codes with existing anonymous quantum transmission methods. The work also quantifies information leakage in ramp schemes through quantum conditional min-entropy and shows why this quantity is a sound measure by linking it directly to the Knill-Laflamme error-correction conditions. A reader would care because the added sender privacy strengthens quantum cryptographic tools without changing how the secret itself is protected.

Core claim

Using permutation-invariant QEC codes along with a set of anonymous quantum transmission algorithms constructs a quantum anonymous secret sharing scheme that achieves sender-anonymity. Information leakage in ramp quantum secret sharing schemes is quantified via the quantum conditional min-entropy, justified as a valid measure by relating it to the Knill-Laflamme quantum error correction conditions. Several permutation-invariant codes are evaluated using this measure to observe the information leakage of intermediate shares for each scheme.

What carries the argument

Permutation-invariant quantum error-correcting codes, which remain unchanged under any reordering of their qubits, combined with anonymous quantum transmission algorithms that hide the sender's identity while delivering the shares.

Load-bearing premise

That permutation-invariant codes can be combined with anonymous transmission algorithms to achieve sender anonymity without introducing new leakage or violating security, and that the conditional min-entropy measure validly captures leaked information when related to Knill-Laflamme conditions.

What would settle it

An explicit protocol execution or calculation in which the sender identity can be recovered from the received shares despite the anonymous transmission step, or a concrete code where the reported min-entropy value contradicts the leakage predicted by the Knill-Laflamme conditions.

Figures

Figures reproduced from arXiv: 2604.27284 by Andrew Nemec, Varin Sikand.

Figure 1
Figure 1. Figure 1: Overview of ANONQ(|ϕ⟩) (Protocol II.4) with Alice encircled in green and Bob in blue. Each participant holds one qubit of the n-qubit GHZ state, denoted as σi, and all participants besides Alice and Bob apply an H gate to their qubit, leaving only Alice and Bob with an entangled pair. The green arrow represents the teleportation step after the AE protocol. and instead obtain a direct expression for maximum… view at source ↗
Figure 2
Figure 2. Figure 2: Overview of the QASS protocol. We initially assume the secret has been encoded into n shares. In Step 1, each green arrow denotes a share sent from each k participating shareholders via the ANONQ protocol. Once received by Bob, the identities of each share is entirely lost due to ANONQ. Finally, Bob decodes the anonymous set of retrieved shares using the PI code’s decoding circuit to obtain the secret. Let… view at source ↗
Figure 3
Figure 3. Figure 3: Hmin for QASS schemes using the ((4, 2, 2)) Aydin et al. PI code [46] and the ((4, 2, 2)) Hagiwara and Nakayama PI code [13], the ramp QSS scheme using the [[4, 1, 2]] Leung et al. stabilizer code [44], and the HQASS scheme using Aydin et al. PI code [46] for both quantum and classical secrets. All three 4-qubit QSS schemes have identical Hmin values, whereas the HQASS scheme has a threshold of k = 3 with … view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of Hmin across various PI codes of distance 3. (a) The ((7, 2, 3)) codes of Pollatsek and Ruskai [47], [49] and Aydin et al. [46], which have the same Hmin values. (b) The ((9, 2, 3)) code of Ruskai [48] (the gnu (3, 3, 1)-PI code [50]). (c) Three ((11, 2, 3)) codes including the one given by Kubischta and Teixeira [49], the shifted gnu (3, 3, 1)-PI code of Ouyang [45], and the Q4,1,2,−1 code of… view at source ↗
read the original abstract

Quantum secret sharing schemes are a family of quantum cryptographic protocols which provide secure quantum encodings, mapping one secret to multiple shares of information such that the original secret cannot be accessed without an authorized set of shares present for decoding. In this work, we describe a protocol that enables sender-anonymity during the secret decoding process. By using permutation-invariant QEC codes along with a set of anonymous quantum transmission algorithms, we construct a quantum anonymous secret sharing scheme that achieves sender-anonymity. We quantify information leakage in ramp quantum secret sharing schemes via the quantum conditional min-entropy, justifying it as a valid measure of leaked information by relating it to the Knill-Laflamme quantum error correction conditions. Finally, we evaluate several permutation-invariant codes using this measure to make observations on the information leakage of intermediate shares for each quantum anonymous secret sharing scheme.

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 proposes a protocol for quantum anonymous secret sharing that achieves sender anonymity by combining permutation-invariant quantum error-correcting codes with anonymous quantum transmission algorithms. It quantifies information leakage in ramp quantum secret sharing schemes via the quantum conditional min-entropy, justifies this measure by relating it to the Knill-Laflamme conditions, and evaluates several permutation-invariant codes to observe leakage properties of intermediate shares.

Significance. If the central construction is sound and the min-entropy quantification is rigorously justified, the work would contribute a concrete framework for sender-anonymous quantum secret sharing with measurable leakage bounds in ramp schemes. The use of established permutation-invariant codes and the provision of explicit code evaluations are strengths that allow falsifiable observations on leakage; these elements support reproducibility and could inform future quantum cryptographic protocols requiring anonymity.

major comments (1)
  1. The justification that quantum conditional min-entropy validly measures leaked information in ramp schemes by relating it to Knill-Laflamme conditions is load-bearing for the leakage claims but lacks an explicit derivation. While KL conditions characterize perfect error correction with no environment information gain, ramp schemes permit partial leakage to intermediate (neither authorized nor fully unauthorized) sets; the manuscript must show how KL satisfaction in the relevant subspace implies a concrete min-entropy lower bound on that partial leakage, rather than assuming the relation carries over directly.
minor comments (2)
  1. The description of the anonymous quantum transmission algorithms should include explicit references or a brief self-contained definition to clarify whether they are drawn from prior literature or newly adapted for this setting.
  2. Notation for the conditional min-entropy and the specific permutation-invariant codes evaluated should be introduced with a table or equation reference early in the manuscript for reader convenience.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need for a more explicit justification of the leakage measure. We address the major comment below and have prepared revisions to strengthen the presentation.

read point-by-point responses

  1. Referee: The justification that quantum conditional min-entropy validly measures leaked information in ramp schemes by relating it to Knill-Laflamme conditions is load-bearing for the leakage claims but lacks an explicit derivation. While KL conditions characterize perfect error correction with no environment information gain, ramp schemes permit partial leakage to intermediate (neither authorized nor fully unauthorized) sets; the manuscript must show how KL satisfaction in the relevant subspace implies a concrete min-entropy lower bound on that partial leakage, rather than assuming the relation carries over directly.

    Authors: We agree that an explicit derivation is required to rigorously connect the Knill-Laflamme (KL) conditions to a concrete lower bound on the quantum conditional min-entropy for partial leakage in ramp schemes. In the revised manuscript we will insert a dedicated derivation subsection. The argument proceeds by considering the action of the intermediate-share projectors on the code subspace: when the KL conditions hold for the relevant error operators (i.e., the inner products satisfy the required orthogonality and proportionality on that subspace), the post-measurement state of the environment is independent of the secret up to a bounded distinguishability. We then invoke the definition of conditional min-entropy in terms of the maximum guessing probability and obtain an explicit lower bound H_min(S|E) ≥ -log(1 - δ), where δ quantifies the residual overlap permitted by the partial KL violation for intermediate sets. This bound is derived directly from the code’s permutation-invariance properties and does not rely on perfect correction. The new subsection will also include a short example with the [[4,2,2]] permutation-invariant code to illustrate the numerical evaluation of the bound. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper constructs a quantum anonymous secret sharing scheme by combining permutation-invariant QEC codes with anonymous quantum transmission algorithms, both treated as established external components. Leakage in ramp schemes is quantified using quantum conditional min-entropy and justified via relation to the standard Knill-Laflamme conditions (an independent, externally defined criterion for perfect error correction and zero environment information). No load-bearing step reduces by definition, by fitting a parameter then relabeling it a prediction, or by a self-citation chain that supplies the result. The derivation chain is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, no explicit free parameters, ad-hoc axioms, or invented entities are identified; the work relies on standard quantum information concepts.

axioms (1)
  • standard math Standard principles of quantum mechanics, quantum error correction, and information theory including Knill-Laflamme conditions
    The protocol and leakage measure are built on these established foundations.

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Reference graph

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