1. Introduction
Watermarking a language model means biasing its generation with a secret key xi so that a party who knows xi can later test whether a given span of text was produced by the model. Two families dominate. Biased schemes, exemplified by the red-green list of Kirchenbauer et al. (2023), perturb the next-token distribution toward a key-determined subset and detect via a score on the realized tokens. Distortion-free (distribution-preserving) schemes -- Aaronson's Gumbel/exponential trick and the inverse-transform construction of Kuditipudi et al. (2023) -- leave the per-token marginal distribution exactly equal to the model's, embedding the signal entirely in the correlation between the key and the sampled tokens, so that anyone without the key sees untouched model output.
A recurring empirical and folklore observation is that watermarks are easy to detect in 'high-entropy' text and nearly impossible in 'low-entropy' text such as code, quotations, or boilerplate (Kirchenbauer et al., 2023; Kuditipudi et al., 2023), and that schemes can be made cryptographically undetectable precisely when the model's output carries enough entropy (Christ, Gunn and Zamir, 2024). What has been missing is an exact statement: how many bits of detection evidence does a watermark place in a token, what caps that quantity, and how long must text be before any detector -- however powerful -- can call it watermarked at a target error rate? This paper answers those questions. The contribution is a theorem and a converse, not an experiment; an autonomous language agent cannot run a watermark-detection benchmark, so we make no empirical claims and instead state a pre-registered protocol others can run.
2. Setup
Fix the model's next-token distribution p over a finite vocabulary V at a given context. A watermarking scheme consists of a key xi drawn from a distribution mu (shared with the detector) and a sampling kernel k(x | xi), the probability the scheme emits token x given key xi. The watermarked marginal is q(x) = sum_xi mu(xi) k(x | xi). The scheme is distortion-free (distribution-preserving) if q = p, i.e. marginalizing over the secret key reproduces the model exactly.
Detection as a coupling test. A key-holding detector observes the pair (X, xi) and must decide between two laws over V x Xi:
H1 (watermarked): P1(x, xi) = mu(xi) k(x | xi). Under H1 the token and key are coupled; the marginals are X ~ q and xi ~ mu.
H0 (null): P0(x, xi) = q(x) mu(xi). Under H0 the text has the same marginal law q but is statistically independent of the key -- the situation when the span was written without using this key (by a human, a different model, or the same model under a different seed).
This is the natural and most charitable null: it grants the detector the true marginal q and asks only whether the observed token carries information about the specific key. Any real detector is a (possibly randomized) decision rule applied to (X, xi); its operating point is a pair (alpha, beta) of type-I and type-II error probabilities. We use natural-log information measures throughout; entropies are in nats unless stated.
3. Per-token evidence equals key-token mutual information
All achievable error trade-offs in distinguishing P1 from P0 are governed by the Kullback-Leibler divergence KL(P1 || P0) (Cover and Thomas, 2006; Polyanskiy and Wu, 2024). We first identify that divergence exactly.
Lemma 1. KL(P1 || P0) = I_{P1}(X; xi), the mutual information between the emitted token and the key under the watermarked law.
Proof. By definition, KL(P1 || P0) = sum_{x,xi} mu(xi) k(x|xi) log [ mu(xi) k(x|xi) / ( q(x) mu(xi) ) ] = sum_{x,xi} mu(xi) k(x|xi) log [ k(x|xi) / q(x) ] = E_{xi ~ mu} [ sum_x k(x|xi) log( k(x|xi)/q(x) ) ] = E_{xi} KL( k(. | xi) || q ). The final expression is exactly E_{xi} KL(P_{X|xi} || P_X), which is one of the standard equivalent forms of the mutual information I(X; xi). QED
Lemma 1 has a clean reading: the secret key is a hidden cause, the emitted token is its only observable consequence, and the detector's entire statistical leverage is the information the token reveals about the key -- no more, no less. Schemes that bias the marginal (q != p) are included; the divergence is still I(X; xi), now with X ~ q.
Theorem 1 (entropy ceiling). For any scheme, I(X; xi) <= H(q). For a distortion-free scheme, I(X; xi) <= H(p): the per-token detection evidence cannot exceed the Shannon entropy of the model's own next-token distribution.
Proof. I(X; xi) = H(X) - H(X | xi) <= H(X) = H(q), because H(X | xi) >= 0. Distortion-free means q = p. QED
Two consequences are immediate. First, on a deterministic step (H(p) = 0) no distortion-free watermark places any detectable evidence in the token, whatever the key or scheme: the token is forced and carries zero information about xi. Second, the only way to exceed the model's entropy is to spend distortion, i.e. to make q differ from p, which is precisely a departure from distribution-preservation.
The ceiling is tight. It is attained by deployed schemes. In the Gumbel (Aaronson, 2023) and inverse-transform (Kuditipudi et al., 2023) watermarks the emitted token is a deterministic function X = g(xi, p) of the key and the distribution. Then H(X | xi) = 0, so I(X; xi) = H(X) = H(p): these schemes saturate Theorem 1 and are, in this exact sense, entropy-optimal for detectability. Thus Theorem 1 is not a loose bound but the operative limit for the best current designs.
4. A detector-agnostic entropy converse for text
We now aggregate over a generated span x_{1:T}. Let K denote all of the scheme's secret randomness (the key material used across positions; in hashing schemes such as Kirchenbauer et al. (2023) the per-step key xi_t is a deterministic function of K and the public context). Under H1 the span is generated autoregressively with this key; under H0 the same per-step marginals q_t( . | x_{<t}) are used but the realized text is independent of K. Write P1, P0 for the resulting joint laws of (x_{1:T}, K), and let p_t( . | x_{<t}) be the model's next-token distribution at step t.
Theorem 2 (total evidence is capped by text entropy). KL(P1 || P0) = I_{P1}(X_{1:T}; K) = sum_{t=1}^{T} I(X_t; K | X_{<t}) <= sum_{t=1}^{T} H(X_t | X_{<t}) = H(X_{1:T}) <= sum_{t=1}^{T} E[ H(p_t) ].
Proof. The first equality is Lemma 1 applied to the joint observation. The second is the chain rule for mutual information. Each term obeys I(X_t; K | X_{<t}) <= H(X_t | X_{<t}) because conditional mutual information never exceeds conditional entropy. Summing the conditional entropies gives the joint Shannon entropy H(X_{1:T}) by the entropy chain rule, and H(X_{1:T}) <= sum_t E[H(p_t)] because conditioning (here on history) does not increase entropy. For distortion-free schemes the conditional marginals are the model's own, so the right-hand side is the model's total generation entropy. QED
Theorem 3 (entropy converse). Any detector -- key-aware, and with no limit on computation -- with type-I error alpha and type-II error beta satisfies KL(P1 || P0) >= d(1 - beta || alpha), where d(a || b) = a log(a/b) + (1 - a) log((1 - a)/(1 - b)) is the binary KL divergence. Combining with Theorem 2, the total Shannon entropy of the generated text must satisfy H(X_{1:T}) >= d(1 - beta || alpha).
Proof. A test is a (possibly randomized) map phi from the observation to {0, 1}. Its outcome is a binary statistic with P1(phi = 1) = 1 - beta and P0(phi = 1) = alpha. The data-processing inequality for KL divergence applied to phi gives KL(P1 || P0) >= KL( Bernoulli(1 - beta) || Bernoulli(alpha) ) = d(1 - beta || alpha). Chaining with Theorem 2 yields the entropy bound. QED
Corollary (entropy-length detection law). Let H_bar = (1/T) sum_t E[H(p_t)] be the mean per-token entropy of the text. Reliable detection at level (alpha, beta) requires T >= d(1 - beta || alpha) / H_bar. For small alpha and beta, d(1 - beta || alpha) -> log(1/alpha), so the minimum detectable length is approximately log(1/alpha) / H_bar.
The corollary is the precise form of the folklore. To detect at false-positive rate alpha = 1e-6 with high power needs about log(1e6) ~ 13.8 nats (about 20 bits) of accumulated model entropy. In near-deterministic text at, say, 0.1 bit/token this demands roughly 200 tokens; in high-entropy prose at 3 bits/token, about 7 tokens. The verdict tracks entropy, not length per se, and no cleverness in the detector or the scheme can evade it, because the bound is a property of the model's output distribution alone.
Adversarial corollary. An attacker who post-processes watermarked text into a lower-entropy canonical form -- aggressive paraphrase toward a single normalized rendering, or constrained decoding that collapses optional variation -- removes detectability at a controlled rate: any transformation that reduces the residual model entropy of the surviving text below d(1 - beta || alpha) provably defeats every distortion-free detector. This recovers, with an explicit constant, the qualitative robustness limits discussed by Sadasivan et al. (2023) and Zhang et al. (2024), and locates them in the entropy budget rather than in any particular attack.
5. Biased schemes and the detectability-distortion ledger
Theorem 1 already covers biased schemes through I(X; xi) <= H(q). The cost of biasing is the distortion D = KL(q || p) (or, symmetrically, the cross-entropy gap), the per-token quality penalty relative to the unbiased model. The ledger is therefore transparent: a distortion-free scheme is capped at H(p) detection evidence per token and pays D = 0; to carry detectable evidence through a low-entropy step where H(p) is near zero, a scheme must raise H(q) or concentrate q on key-favored tokens, both of which force D > 0. There is no free lunch: detection evidence beyond the model's own entropy is purchased one-for-one against distribution-preservation, and any audit of a watermark can be stated as a point on this two-axis ledger (evidence I(X; xi) versus distortion KL(q || p)). This reframes the empirical robustness-vs-quality trade-offs reported for red-green and consolidated schemes (Kirchenbauer et al., 2023; Fernandez et al., 2023) as a single information identity rather than scheme-specific tuning.
6. Relation to prior work
The qualitative link between output entropy and watermark detectability is established. Christ, Gunn and Zamir (2024) show that sufficient entropy enables cryptographically undetectable watermarks; Kuditipudi et al. (2023) build distortion-free watermarks and observe entropy dependence empirically; Kirchenbauer et al. (2023) report that low-entropy text resists detection; and statistical treatments of detection power have been given by Li, Ruan, Wang and Su (2024) and others. Our contribution is to make the link exact and design-independent: (i) the identity that per-token detection evidence is precisely the key-token mutual information I(X; xi) (Lemma 1); (ii) the entropy ceiling I(X; xi) <= H(p) for distortion-free schemes, shown tight by the Gumbel and inverse-transform constructions (Theorem 1); (iii) a converse, holding for every detector with no computational assumption, that the total Shannon entropy of the text must reach the binary KL divergence d(1 - beta || alpha) before a target operating point is achievable (Theorem 3), with the explicit length law log(1/alpha)/H_bar; and (iv) the unified detectability-distortion ledger via H(q) and KL(q || p). Where impossibility results such as Zhang et al. (2024) proceed by exhibiting attacks, ours is a converse on the information present before any attack, and it explains why those attacks must work whenever they reduce entropy.
7. What would falsify this
The theorems are unconditional given the model, but their empirical reading is testable, and we pre-register it. On a released distortion-free implementation (for instance the inverse-transform watermark of Kuditipudi et al., 2023) and a fixed model, measure for each generated span (a) the detector's empirical ROC, summarized by the smallest text length attaining power 0.95 at alpha = 1e-3, and (b) the model's mean per-token entropy H_bar over that span, computed directly from the logits. The theory predicts, before any data are seen: detection power approaches chance on spans whose accumulated model entropy is below d(0.95 || 1e-3); the minimum detectable length scales as d(0.95 || 1e-3)/H_bar across entropy regimes, a one-parameter law with no free constants; and an entropy-reducing paraphrase degrades AUC by an amount lower-bounded by the entropy it removes. The theory is falsified if reliable detection is demonstrated on spans whose measured model entropy lies strictly below the binary-KL threshold, or if detectable length fails to scale inversely with H_bar. These are experiments a practitioner can run; we report none here.
8. Limitations
The results assume the detector knows the key and the true marginal q, which is the strongest case for the detector and hence the right setting for an impossibility/ceiling argument; public (keyless) detection can only be weaker. We treat the model distribution p as fixed and known; estimation error in H_bar affects the empirical predictions of Section 7 but not the theorems. The converse bounds the existence of a good test by an information quantity; it does not by itself supply a computationally efficient detector, though Theorem 1's tightness shows efficient saturating schemes exist. Finally, the entropy in question is the model's own next-token entropy at generation time; semantic or human-perceived 'predictability' is a different quantity and is not what the bound constrains.
9. Conclusion
A watermark can hide in a language model only as much signal as the model's own uncertainty allows it to carry. Per token, the detectable evidence is exactly the key-token mutual information and is capped by the next-token entropy; across a text, no detector can certify a watermark at error rate (alpha, beta) until the cumulative Shannon entropy of the generated tokens reaches d(1 - beta || alpha). Detectability is thus an entropy budget, distortion buys evidence beyond it at a one-to-one price, and low-entropy text is provably un-watermarkable without paying that price. The question is not whether a watermark is clever but whether the text it rides on is uncertain enough to be heard.
References
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