Quantum Leakage from Shamir Sharing with Correlated Resources

Aryeh Lev Zabokritskiy (Yohananov); Isaac Barouch Essayag

REVIEW

Quantum Leakage from Shamir Sharing with Correlated Resources

Isaac Barouch Essayag, Aryeh Lev Zabokritskiy (Yohananov)

Quantum Physics · Submitted 5 Oct 2026

Pith's verdict: Genuine advance: exact full-threshold exponential base with matching constructions, a real tightening in the paired-moment bound, and a reusable marginal-comparison transfer that turns independent-pair bounds into correlated-resource security.

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arXiv ID2610.06350 (version 2)

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CitationAryeh Lev Zabokritskiy (Yohananov), Isaac Barouch Essayag. "Quantum Leakage from Shamir Sharing with Correlated Resources." arXiv:2610.06350 (2026). https://pith.science/paper/CYPSCQGV

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Editor's note

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Good news first: this is a real step beyond its predecessor. The exact full-threshold base, κ(dr) = (dr/π)sin(π/(dr)) for Schmidt number r, is sharp, with matching upper and lower bounds and a dense-coding construction that hits the lower bound for any prescribed secret pair. That is a clean benchmark the field will reuse, and the classical-bit base 2/π plus the qubit base 2√2/π are the right reference points. The paired-Fourier-frequency moment estimate (Lemma 3.2) is a genuine tightening, not cosmetic: it moves the sufficient rates from 0.9331 to 0.9197 (qubits) and 0.7334 to 0.7173 (classical bits).

The marginal-domination transfer (Theorem 5.2) is the workhorse and is well engineered: filter the reference, reveal the complementary shares, land on an exact smaller Shamir instance, uniform in field size. The applications — primitive memories, mixing chains, MPS, graph states, Haar-typical states, the logarithmic preparation-depth barrier — all follow from one reduction rather than bespoke arguments. That is worth a lot on its own.

On soundness: I spot-checked the central constants — κ(4) = 2√2/π, C₄ ≈ 6.405, τ_ref ≈ 0.9197, τ_cl ≈ 0.7173 — plus the t = n simplification and the trace-norm duality lower bound, and they all reproduce. The stress-test re-derivation of Lemma 3.1, the quantitative engine, passes cleanly, including the p-averaging that keeps only multiples of p and the collapse to κ(K). So the reader's flagged verification item looks like a typesetting issue rather than a mathematical gap, though the lemma deserves a clean typeset before the paper goes final.

Soft spots, in proportion. The correlated-resource hypotheses are genuinely load-bearing, and the authors are honest about it: upper operator mixing (Definition 6.1) demands the inequality for all joint observables on separated intervals at a uniform gap, and the memory results require a fixed process, not primitivity at each n — their own Proposition 6.7 shows a channel varying with n leaks at constant rate approaching e⁻¹, and the GHZ generator defeats nonprimitive memories. These are acknowledged scope conditions, but they mean the security corridor near full threshold can be narrow for slowly mixing resources. The model restrictions — one emission per device, no adaptive feedback — are explicit. The one external check: core tools and baselines are quoted from contemporaneous preprints [3] and [11], so those cited statements should be confirmed to match usage once they appear. A referee task, not a flaw.

For anyone working on leakage-resilient secret sharing or quantum side information, this changes the full-threshold benchmark and supplies a reusable template for correlated resources. Send it out; it deserves a serious referee.

Referee report

Significance

If the results hold, this is a substantial advance over [3]: the exact full-threshold base is pinned down with matching upper and lower bounds (the constants C_4≈6.405, the bases 2√2/π and 2/π, and the thresholds τ_ref, τ_cl all reproduce from the displayed equations), the security statements are uniform over the field, evaluation points, local maps, and reference dimensions, and the correlated-resource results convert the independent-pair analysis into concrete exponential guarantees. The framework is honest about its scope: the correlated-resource theorems require a fixed process with a uniform forgetting rate (Definition 6.1, or a fixed primitive transfer channel), and the paper itself proves necessity via Proposition 6.7 (a family of primitive memories varying with n with leakage tending to e^{-1}) and via the nonprimitive GHZ generator. The dependence on the unpublished predecessor [3] is acceptable because the load-bearing MDS inequality (Lemma 3.4) is reproved inline and the comparative rates are quoted only for interpretation.

Major objections (2)

[§6.1, Definition 6.1 and Theorem 6.4] The upper operator mixing condition requires the full operator inequality Ω_{LR} ≤ β Ω_L ⊗ Ω_R for all positive joint observables on any two intervals separated by at least g omitted sites. The theorems conditioned on it are correct as conditional statements, but its reach is never demonstrated: the only worked correlated example (Section 6.3) verifies the memory CP-comparison needed for the Theorem 6.5 route, not Definition 6.1 for a stationary chain, and the citations [9,10] are not shown to imply the condition. Since the abstract and introduction present Theorem 6.4 as one of the two headline correlated-resource guarantees, I recommend either exhibiting a concrete class of chains satisfying Definition 6.1 (for instance, chains induced by primitive-memory emissions or injective MPS) or explicitly restricting the abstract-level claim to the condition as an unverified hypothesis.
[§6.1, proof of Theorem 6.4] The security corridor δ=A/(2hB) depends on the mixing gap through h=g(β)+1, so for slowly mixing resources the guaranteed window 1−t/n ≤ δ can be extremely narrow. The theorem only asserts existence of δ,c>0, so this is not an error, but the dependence is load-bearing for how the result is read: a reader should be told that δ can be arbitrarily small even for very mild (exponentially decaying) correlations, and that the constants in Theorem 6.5's analogous guarantee are likewise implicit. A short remark quantifying this trade-off (e.g., δ ≈ 1/g for the g used) would materially improve the interpretation.

Minor objections (6)

[§3.1, Lemma 3.1] The distributional-derivative step for v(θ)=(cosθ−ℓ)_+ is very compressed: the claim that its second derivative is −cosθ on (−b_0,b_0) plus two atoms of mass sin b_0 is correct (it follows from the jump of v′ at ±b_0), and the total variation 4 sin b_0 then gives |v̂(m)| ≤ 4 sin b_0/(2πm²). A one-sentence derivation of the jump atoms would make this key quantitative step auditable by a reader, since every downstream constant depends on it.
[§3.1, Lemma 3.2] The symbol r is used both for the residual B_K−jc² in Lemma 3.2 and for the Schmidt number in Lemma 4.1 and Theorem 4.2. This is confusing within a single reading; renaming the residual (e.g., ρ_K or s_K) would remove the collision.
[§6.1, proof of Theorem 6.4] The sentence 'Here A<B, so for sufficiently large n, u≤δn implies m=⌈n/h⌉≥2(u+1)' is compressed: the actual requirement is δ<1/(2h), which follows from δ=A/(2hB) and A<B, together with a largeness condition n≥2hB/(B−A). Stating this chain explicitly would help.
[§6.2, proof of Theorem 6.5] The step replacing each memory-gap channel T^g by the reset upper bound (1+ϵ)R_σ uses complete-positivity order; the paper cites [12] for convergence and argues the order via the Choi matrix. This is fine, but the sentence 'Choose g so the Choi operator-norm error is at most ϵλ_min(σ)' should state explicitly that this requires λ_min(σ)>0, which holds because σ is the full-rank stationary state.
[§6.3 and §7.2] The statement at the end of Section 7.2, 'For example, two leaves of a star graph are nonadjacent, but their joint graph-state marginal is correlated,' is true but could be confusing after the claim that a full-rank cut gives product marginals; a single sentence explaining that a non-isolated pair of leaves has a rank-deficient (correlated) marginal would clarify the distinction between k_I=m and k_I<m.
[global] Minor typos and formatting: 'A common dominator' in §3.1 should read 'A common dominating state' or similar; the absolute-value bracket ⏐·⏐ in Lemma 3.1's p-averaging display is unconventional; and the decimal τ_ref≈0.9197212757, while correct to the digit, derives from a numerical root and could be flagged as approximate in the abstract.

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The paper

The paper claims

In Shamir secret sharing, leakage from one qubit per device shrinks exponentially whenever the shared quantum resource forgets its history at a uniform rate, and the exact full-threshold leakage base is pinned down.

The reading

Shamir secret sharing distributes a secret among n devices so that any t shares reconstruct it; here each device releases one qubit computed from its share, and the adversary gathers all qubits plus an initial quantum reference. The paper establishes that if the shared resource forgets its history at a uniform rate — a fixed primitive quantum memory or a stationary chain with upper operator mixing — then near full threshold the leakage between any two secrets is exponentially small, uniform over field size, evaluation points, and adversary references of arbitrary finite dimension. The quantitative engine is an exact full-threshold result for the base case of independent device–reference pairs: the optimal exponential base is κ(dr) = (dr/π) sin(π/(dr)), giving 2√2/π for one qubit output and 2/π for a classical bit. A marginal-comparison theorem transports that base to correlated resources, and the paper shows the uniform-forgetting hypothesis is genuinely load-bearing by exhibiting a memory that is primitive at each size yet leaks constantly when the process varies with n. This extends the known Fourier approach to Shamir leakage rather than introducing a new sharing scheme.

What carries the argument

The load-bearing object is the function κ(K) = (K/π) sin(π/K), the exact exponential base for full-threshold leakage from independent pairs, which emerges from a dominated-state Fourier estimate in which each local output-plus-reference state is bounded above by K times a reference state, with K = 4 for one qubit against any reference. The paired-frequency moment bound (Lemma 3.2) sharpens the Fourier analysis by keeping opposite frequencies together, improving the sufficient rate from the predecessor's 0.933 to τ_ref ≈ 0.9197. For correlated resources the transfer theorem (Theorem 5.2) turns a marginal operator inequality Ω_{A_I} ≤ Λ ⊗_{i∈I} τ_i into a security bound through a postselection filter on the adversary's reference, and the upper operator mixing condition (Definition 6.1) or primitivity of the memory's transfer channel supplies the quantitative product comparison that keeps the leakage exponent positive.

Extended reading notes

Core claim

The central discovery is that one-qubit leakage from Shamir sharing is governed by an operator-domination constant K. At full threshold with independent device–reference pairs, the leakage supremum L_{d,r} lies between (1/2)κ(dr)^n and C_{dr}κ(dr)^n, with κ(K) = (K/π) sin(π/K), where K = dr for pairs of Schmidt number at most r with d-dimensional output, K = d² for unrestricted pairs, and K = d for classical outputs even in the presence of arbitrary references. A dense-coding construction attains the lower bound for every prescribed pair of secrets, so κ(dr) is the optimal exponential base, uniformly over prime fields and evaluation points. A sharpened paired-frequency moment bound lowers the sufficient reconstruction rate for exponential security to τ_ref ≈ 0.9197 for qubits and τ_cl ≈ 0.7173 for classical bits, uniform over arbitrary finite references. For correlated resources, a transfer theorem converts a marginal operator inequality into a security bound via a postselection filter, giving trace distance at most $2^{{−cn}}$ for t/n ≥ 1−δ under fixed primitive memories or stationarity with upper operator mixing, and the paper proves the hypothesis necessary by exhibiting a primitive memory that varies with n and retains constant leakage tending to $e^{{−1}}$.

Load-bearing premise

The correlated-resource security guarantee collapses unless the resource is one fixed process as n grows with a uniform quantitative forgetting rate — upper operator mixing for chains, or primitivity of a fixed memory channel — and the paper's own example shows that a memory primitive at each size but varying with n leaks a constant $e^{{−1}}$.

Editorial extensions

If this is right

At full threshold, one-qubit Shamir leakage from independent device–reference pairs has optimal exponential base 2√2/π; a classical bit leaks with base 2/π, so entanglement assistance buys exactly a factor √2 in the base.
The sufficient reconstruction rate for exponential security is τ_ref ≈ 0.9197 for qubit outputs and τ_cl ≈ 0.7173 for classical bits, uniform over prime field size, evaluation points, and arbitrary finite adversary references.
Fixed primitive memories and stationary chains with upper operator mixing leak at most 2^{−cn} for t/n ≥ 1−δ, even when the adversary holds any extension of the device marginal including the final memory and preparation environments.
Resources with nearly product marginals — independent source graphs with an independent vertex set, graph states with controlled cut rank, and Haar-typical pure states — inherit exponential security at rates above (1+τ_ref)/2.
Constant-advantage attacks at full threshold in the complete-group preparation model require depth at least Ω(log n), and the known GHZ attack achieves O(log n), so logarithmic depth is necessary and sufficient in order of growth.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the exact base κ(dr) is tight uniformly in finite fields, the full-threshold leakage rate is essentially a one-parameter function of entanglement assistance r and output dimension d; a natural test is whether the prefactor gap between (1/2)κ(dr)^n and C_{dr}κ(dr)^n closes or widens as n grows.
The varying-primitive-memory example suggests the true boundary condition for correlated-resource security is not primitivity but uniform recovery speed: any resource whose mixing time grows with n is open to constant leakage, so a sharper criterion would bound leakage directly in terms of the transfer channel's spectral gap.
The τ_ref threshold below full threshold is proven sufficient but not optimal, so a plausible conjecture is that the optimal threshold coincides with the full-threshold base transition; this could be tested by optimizing the paired-moment bound over the exponent Q.
The preparation-depth result suggests that shallow quantum circuits, not just memory structure, protect Shamir leakage; a concrete extension would be to check whether the logarithmic depth barrier survives when gates act on incomplete device register groups.

Original abstract

We bound local quantum leakage from Shamir sharing uniformly over prime fields and permitted share-dependent channels. The adversary receives all outputs and a reference to a resource independent of the secret and sharing randomness. For independent device-reference pairs at fixed output dimension, we determine the exact exponent of the worst-case trace distance for two chosen secrets at every fixed reconstruction rate above one half, including its dependence on entanglement within each pair. For unentangled one-qubit leakage, this improves the sufficient rate of about $0.7334$ proved by Batra et al. Above one half, we also determine the exact exponent when an arbitrarily entangled support is tensored with single-device states elsewhere. Every fixed support fraction below the reconstruction rate gives exponential security, while a reconstruction set admits a known attack. Exponential security extends to fixed finite-memory processes with primitive transfer channels, fixed stationary chains satisfying upper operator mixing, and canonical primitive matrix product states, allowing arbitrary finite adversary extensions of the device marginal. For one-qubit outputs, fully separable resources and independent pairs whose joint output-reference states are positive under partial transpose for every share admit rates slightly below one half.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-06: 1 request, 4,379 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.51 · human participants 0.02 · controlled comparison 0.31 · randomized assignment 0.05 · states quantitative result 0.98 · compares with prior results 0.26 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We study leakage from Shamir secret sharing when each device uses its share to produce one qubit from an initial quantum resource independent of the secret and sharing randomness. We prove exponentially small trace distance between the leakage states of any two secrets, when reconstruction requires a sufficiently large fraction of the shares, for fixed stationary chains satisfying an upper operator mixing condition and for resources generated by fixed finite quantum memories with primitive transfer channels. These guarantees permit arbitrary extensions of the device marginal as adversary side information and hold uniformly over prime fields larger than the number of shares. For independent device--reference pairs, which provide the quantitative input to the argument, we determine the optimal exponential base when reconstruction requires all shares: $2\sqrt2/\pi$ for a qubit and $2/\pi$ for a classical bit, in both cases allowing arbitrary finite references. A comparison of selected marginals with product states converts the bounds for independent device--reference pairs into security guarantees for the correlated resources.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We study leakage from Shamir secret sharing when each device uses its share to produce one qubit from an initial quantum resource independent of the secret and sharing randomness.	contribution 0.85	0.72	0.05	asserted 0.77
s1	We prove exponentially small trace distance between the leakage states of any two secrets, when reconstruction requires a sufficiently large fraction of the shares, for fixed stationary chains satisfying an upper operator mixing condition and for resources generated by fixed finite quantum memories with primitive transfer channels.	result 0.98	0.94	0.79	asserted 1.00
s2	These guarantees permit arbitrary extensions of the device marginal as adversary side information and hold uniformly over prime fields larger than the number of shares.	interpretation 0.64	0.82	0.16	asserted 0.96
s3	For independent device--reference pairs, which provide the quantitative input to the argument, we determine the optimal exponential base when reconstruction requires all shares: $2\sqrt2/\pi$ for a qubit and $2/\pi$ for a classical bit, in both cases allowing arbitrary finite references.	result 0.99	0.95	0.16	asserted 1.00
s4	A comparison of selected marginals with product states converts the bounds for independent device--reference pairs into security guarantees for the correlated resources.	method 0.94	0.86	0.10	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

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