The gap compares one channel with its two-copy Bell input
The quantity at stake is Δp(Φ; Ωn) = 2Hmin,p(Φ) − Hp(Φ⊗2(Ωn)), where Hmin,p is the minimum-output Rényi entropy of a single channel and Ωn is the standard Bell input. Product inputs always satisfy Hmin,p(Φ⊗Ψ) ≤ Hmin,p(Φ) + Hmin,p(Ψ), so additivity asks whether equality is forced. A positive Δp means it is not: the two-copy Bell input produces less output entropy than twice the single-channel minimum.
This distinction matters because the gap is not a statement about typical inputs. It is a witness built from one specific entangled input, and the paper's channels are constructed so that this witness works. Entropy gaps here are measured in nats, since all logarithms in the paper are natural unless a base is shown.
One channel covers every order above the cutoff
Theorem 2.1 states that for every finite p0 > 0 and every integer r ≥ 1, explicit formulas prescribe a channel Φr,p0 from M2N(r,p0)(C) to M215r(C) with real algebraic entries such that inf over p in [p0, ∞] of Δp(Φr,p0; Ω2N(r,p0)) exceeds r/2. The channel and its Bell input may depend on the prescribed p0 and on the target gap, but not on the running order p.
That uniformity is the paper's central claim. Earlier finite-dimensional counterexamples typically establish nonadditivity at a fixed order or over a restricted range; here a single channel and a single Bell input certify a gap at every order from p0 upward, including the limiting order p = ∞. The integer N(r, p0) is given exactly in equation (113) and bounded by 9216r²30ʳ + 36864r(45r + 20 + (r + 16)/min{p0, 1/2}).
Choosing r converts the theorem into a target gap
The theorem's r/2 margin is what makes the result a statement about arbitrary gaps. For any g > 0, setting r = ⌈2g⌉ gives one channel and the same Bell input with a gap greater than g at every order in [p0, ∞]. So the growth of the violation is controlled by a single integer parameter, and the cost of a larger gap is paid in dimension.
The dimension cost is asymmetric. The construction uses 15r output qubits, which grows linearly in r, while the input-qubit count grows as O(r²30ʳ + r²(1 + p0⁻¹)). The exponential factor 30ʳ sits in the input dimension, not the output dimension, so the output side stays comparatively small even when the certified gap is large.
Moment matching replaces strong convergence
The channels have the form ΦF(X)st = (1/D)Tr(WsFXFWt*) + δst(1/D)Tr((In − F²)X), where the Ws are prescribed real permutation matrices and F is a prescribed real symmetric positive contraction. The first summand is a complementary mixed-unitary channel applied to FXF; the second is a completely positive replacement map. Their traces sum to Tr X, so ΦF is a channel.
The technical route, as the authors describe it, is to replace the strong-convergence input normally used to transfer free-group estimates to finite dimensions with finite-order moment matching, together with a deterministic filter that suppresses exceptional directions while preserving the two-copy correlations needed for the Bell-state witness. Fixed-field high-girth graphs supply real permutation matrices that reproduce the required short free-group moments exactly. Filters built from Chebyshev polynomials and spherical averages then convert those moment identities into the uniform one-copy bounds needed for the entropy comparison. The paper states that the numerical parameters are not intended to be optimal; they are chosen to give transparent, fully analytic margins, so the resulting dimensions should be read as explicit certified bounds.
Fewer output qubits than the Shou–Gorshkov construction
Section 8 compares this finite-dimensional realization and its dimension bounds with those of Shou and Gorshkov. The abstract states that this construction uses fewer output qubits for every target gap and has a smaller asymptotic input-qubit bound as the gap grows with the positive cutoff fixed. For fixed p0 > 0, the exact prescriptions give, as g → ∞, B_in^ours = Θ(g²60ᵍ) against B_in^SG = Θ(g²285ᵍ/(2 log 2)). Since 85/(2 log 2) > 60, the large-gap growth rate of the explicit input-qubit bound improves as well.
The paper is explicit about the scope of that comparison. It concerns fixed p0 > 0 and should not be interpreted as uniform input-dimension superiority under arbitrary joint scalings of g and p0, especially since neither construction is dimensionally optimized. The comparison is therefore a statement about how the certified bounds scale in one regime, not a claim that this construction dominates in every parameter setting.
A separate specialization targets the von Neumann order
At p = 1, a separate specialization gives an explicit real algebraic channel Φ* from M2N*(C) to M256(C) with N* = 37608898, whose standard Bell input satisfies 2Hmin,1(Φ*) − H1(Φ*⊗2(Ω2N*)) > 1/1024. That is eight output qubits and 37,608,898 input qubits for a gap just above 1/1024.
The corollary is not simply Theorem 2.1 applied at that gap. Applying the theorem directly to g = 1/1024 would require r = ⌈1/512⌉ = 1 and would therefore guarantee the much larger gap 1/2. The corollary instead uses estimates tailored to the single order p = 1 to obtain a smaller explicit instance. This is a useful reminder that the theorem's uniform guarantee and the specialized construction answer different questions: one certifies a gap across an order range, the other minimizes the instance at one order.
What the certified bounds do and do not say
The construction is closed-form and finite-dimensional, with real algebraic entries and fully prescribed parameters. The channels are built from prescribed real permutation matrices and a prescribed real symmetric positive contraction, and the entropy comparison rests on the uniform one-copy compression bounds in equation (125) together with a small normalized trace bound. Those are the conditions under which the gap statement holds.
The dimensions are certified bounds rather than optimized ones, and the paper says so directly. The asymptotic input-qubit comparison with Shou–Gorshkov holds for fixed p0 > 0 as the gap grows, not under arbitrary joint scalings of g and p0. The most decision-relevant reading is therefore a trade-off between the order range and the instance size: Theorem 2.1 buys a gap that holds simultaneously on all of [p0, ∞] at the cost of an input dimension with a 30ʳ factor, while the von Neumann corollary buys a small eight-output-qubit instance at the cost of restricting to a single order and a gap above 1/1024.
Sources used in this answer
Closed-Form Arbitrary-Gap Rényi Nonadditivity Above Any Positive Order via Direct Products of Free Groups and High-Girth Graphs
For every p0 > 0 and g > 0, the paper constructs one channel and one Bell input whose minimum-output Rényi entropy additivity gap exceeds g simultaneously for all p in [p0, ∞], using 15r output qubits and O(r²30ʳ + r²(1 + p0⁻¹)) input qubits with r = ⌈2g⌉; a separate specialization at p = 1 gives a gap above 1/1024 with 37608898 input qubits and 8 output qubits.