Entropy Replacement and Complexity-Sensitive Observer Complementarity in Non-Isometric Holographic Codes

Abstract

Two observers granted independent, cloned access to the same bulk degree of freedom in a non-isometric holographic code can in principle disagree about its entropy. We quantify this disagreement in the AEHPV non-isometric framework with Harlow–Usatyuk–Zhao observer inclusion, and show it is governed by an entropy-replacement principle. Our main result (Theorem 1) is that the von Neumann entropy of an observer’s actual reduced state equals the Shannon entropy of its diagonal in the cloning basis, up to an error $F_{AB}$ with $\mathbb{E}[F_{AB}^2] = O(d_B^{-4} d_M^{-2})$ – a full power of $d_B$ below the two-observer signal. For Haar-random bulk states this principle is established unconditionally, via an exact antisymmetric resolvent representation of $F_{AB}$ together with a fourth-moment bound on the random-projection–induced perturbation. The replacement reduces the disagreement to a moment calculation of the bulk-marginal diagonal, which we carry out for two extreme bulk-state classes. For the Haar class we prove (unconditionally, Theorem 2) $\mathbb{E}|S_A - S_B| \to \sqrt{2/\pi}\,/(d_M\, d_B^{3/2})$; for random product bulk states we obtain, conditionally on the product-class replacement principle (Theorem 3), $\mathbb{E}|S_A - S_B| \to \sqrt{4(\pi^2/3 - 3)/\pi}\, d_B^{-1/2} \approx 0.608\, d_B^{-1/2}$. The two exponents differ by exactly one power of $d_B$, an instance of complexity-sensitive complementarity: the degree to which two cloned observers can disagree is set by the complexity class of the bulk state. All exponents and prefactors are exact asymptotics, supported by an extensive numerical program across the full HUZ+$V$ pipeline.

§1. Introduction

1.1 Observer complementarity in non-isometric codes

In non-isometric holographic codes, the map from the effective (bulk) description to the fundamental (boundary) description is many-to-one: distinct bulk states can map to identical boundary data. This is the defining feature of the AEHPV construction, and it is what allows a black-hole interior to be encoded with far fewer fundamental degrees of freedom than the naive bulk Hilbert space would require. A consequence is that “an observer” is not a passive bystander: including an observer who measures a bulk degree of freedom changes the code, because the observer’s record must itself be encoded. The Harlow–Usatyuk–Zhao (HUZ) prescription makes this precise by cloning the measured degree of freedom into an observer register before the non-isometric map is applied.

When two such observers are included independently – each cloning the same bulk degree of freedom into its own register – the code now carries two records of the same information. Because the non-isometric map is not injective, the two observers’ reduced states need not agree, and in particular their von Neumann entropies need not agree. The size of this disagreement is a sharp, computable diagnostic of how much “room” the non-isometry leaves for observer-dependent descriptions.

1.2 The question

Concretely: let $|\psi\rangle$ be a bulk state on $\mathcal H_A\otimes\mathcal H_B\otimes\mathcal H_C$, included for two observers $A$ and $B$ via HUZ cloning and mapped to the fundamental description by a fixed non-isometry built from a Haar-random isometry $V$. Writing $\rho_{R_A},\rho_{R_B}$ for the two observer-reduced states, we ask for the typical magnitude of

under the joint randomness of $V$ and (in two natural ensembles) the bulk state. How large is the disagreement, and what controls it?

1.3 Main results

The answer comes in two layers. The first is structural and is the technical core of the paper.

Theorem 1 (entropy-replacement principle, informal). The von Neumann entropy of an observer’s actual reduced state equals the Shannon entropy of its diagonal in the cloning basis, up to an error $F_{AB}$ obeying

$$\mathbb{E}\bigl[F_{AB}^2\bigr] = O\!\bigl(d_B^{-4} d_M^{-2}\bigr), \qquad \mathbb{E}\,|F_{AB}| = o\!\bigl(d_B^{-3/2} d_M^{-1}\bigr).$$

For the Haar bulk class this is a theorem, proved unconditionally in Appendix C; for the product class it is a conjecture, numerically supported but not proved. The proof for the Haar class is, we believe, of independent interest: it represents the antisymmetric entropy difference exactly through a resolvent integral, splits it into a linear piece (which reduces to the very bulk-marginal moment that controls the signal) and a nonlinear piece (controlled by a fourth moment of the random-projection–induced perturbation), and closes the fourth moment by a concentration estimate on the unitary group. The replacement error is suppressed relative to the signal by a full power of $d_B$ – which is exactly what is needed to make the entropy difference equal to the Shannon difference at leading order.

Given Theorem 1, the disagreement reduces to the variance of the Shannon entropy of the bulk-marginal diagonal, a classical random-matrix calculation. Carrying it out for two extreme bulk-state classes gives the second layer:

• Haar bulk states (maximal complexity), Theorem 2, unconditional: $\mathbb{E}|S_A - S_B| \to \sqrt{2/\pi}\,/(d_M\, d_B^{3/2}) \approx (0.798/d_M)\,d_B^{-3/2}$.
• Random product bulk states (minimal complexity), Theorem 3, conditional on the product-class form of Theorem 1: $\mathbb{E}|S_A - S_B| \to \sqrt{4(\pi^2/3-3)/\pi}\,d_B^{-1/2} \approx 0.608\,d_B^{-1/2}$.

1.4 Physical interpretation: complexity-sensitive complementarity

The two exponents differ by exactly one power of $d_B$. We read this as complexity-sensitive complementarity: the degree to which two cloned observers can disagree about a bulk degree of freedom is controlled by the complexity class of the bulk state. Highly scrambled (Haar) bulk states leave little room for disagreement – the disagreement falls off as $d_B^{-3/2}$ – while simple (product) bulk states leave an order-$\sqrt{d_B}$ more room, falling off only as $d_B^{-1/2}$. The non-isometry’s tolerance for observer-dependent descriptions is thus not a fixed property of the code but a function of what is encoded in it. Section 6 develops this reading and connects the exponent gap to the fluctuation structure of the bulk marginal.

1.5 Organization

Section 2 fixes the AEHPV/HUZ setup, the two-observer scenario, and the two bulk classes. Section 3 is the technical heart: it states and proves (modulo Appendix C) the entropy-replacement theorem, building it from the structural identity (Lemma 1) and the off-diagonal collapse, and verifies it numerically (Figure 2). Sections 4 and 5 derive the two scaling laws as consequences – the Haar law (Theorem 2, unconditional) first, then the product law (Theorem 3, conditional). Section 6 develops the complexity-sensitive reading of the exponent gap. Section 7 presents the numerical landscape (Table 1, Figures 1 and 5). Section 8 situates the results against EGH, HUZ, the Colorado rule, and the quantum-reference-frame literature. Appendix A records generalized EGH formulas; Appendix B documents reproducibility; Appendix C proves the entropy-replacement theorem for the Haar class – the resolvent representation, the linear and nonlinear bounds (Lemmas C.1–C.2), and the fourth-moment projector estimate (Lemmas C.3–C.5).

§2. Setup

This section fixes notation and conventions. We follow the AEHPV non-isometric-code framework [AEHPV 2207.06536], adapted to the two-observer scenario introduced by EGH 2507.06046 and HUZ 2501.02359. Readers familiar with these constructions may skip to §3.

2.1 Non-isometric maps and the AEHPV framework

The bulk effective theory and the fundamental (boundary) theory are two finite-dimensional Hilbert spaces connected by a linear map:

When $d_{\rm eff} > d_{\rm fund}$, $V$ is non-isometric: there are bulk “null states” in the kernel of $V$. Following AEHPV, we take $V$ to be the first $d_{\rm fund}$ rows of a Haar-random unitary on $U(d_{\rm eff})$. This ensures $V V^\dagger = I_{\rm fund}$ exactly, while $V^\dagger V$ is a Haar-random rank-$d_{\rm fund}$ projector on $\mathcal{H}_{\rm eff}$. The non-isometry parameter is

Throughout this paper we fix $\rho = 1/2$.

2.2 Observer-included states via HUZ cloning

The HUZ 2501.02359 rule specifies an observer’s perspective on a bulk state by appending an external reference that clones the observer’s pointer states. In the single-observer case, the bulk factorizes as $\mathcal{H}_{\rm eff} = \mathcal{H}_{\rm Ob} \otimes \mathcal{H}_M$ (observer and matter), and a reference register $\mathcal{H}_R \cong \mathcal{H}_{\rm Ob}$ is added. The cloning isometry $\mathrm{Clone}_{\rm Ob \to R}: |{\rm ob}, m\rangle \otimes |0\rangle_R \mapsto |{\rm ob}, m\rangle \otimes |{\rm ob}\rangle_R$ produces the HUZ map

Applied to any bulk state $|\psi\rangle \otimes |0\rangle_R$ and normalized by post-selection, this produces a state on $\mathcal{H}_{\rm fund} \otimes \mathcal{H}_R$. The observer-accessible reduced state and its entropy are

2.3 The two-observer scenario

We consider the setup of EGH 2507.06046 and its natural refinement to two independent observers. The bulk effective space factorizes as

where $A$ and $B$ are two independent observer factors and $C$ is a matter register. Two auxiliary reference registers $R_A$ and $R_B$ of dimensions $d_A, d_B$ are introduced, and the two-observer HUZ map

is applied to the bulk state $|\psi\rangle \otimes |0\rangle_{R_A} \otimes |0\rangle_{R_B}$. The post-selection-normalized state is denoted $|\Psi\rangle \in \mathcal{H}_{\rm fund} \otimes \mathcal{H}_{R_A} \otimes \mathcal{H}_{R_B}$.

The two observer-dependent entropies are

and similarly for $\rho_{R_B}$. The central quantity of this paper is the Haar-averaged disagreement

where the expectation is taken over $V$ (Haar on $U(d_{\rm eff})$) and optionally over bulk states $|\psi\rangle$ drawn from a specified class.

2.4 Bulk state classes

The theorems of this paper apply to two distinct bulk-state classes, each defining an ensemble over $\mathcal{H}_{\rm eff}$:

• Product class (P). Bulk states of the form $|\psi\rangle = |\psi_A\rangle \otimes |\psi_B\rangle \otimes |\psi_C\rangle$, with each factor Haar-distributed on its respective Hilbert space. Such states have no bulk entanglement across the $A$/$B$/$C$ partition; the $A$-marginal $\rho_A^{\rm bulk}$ is a rank-1 pure state.
• Haar class (H). Bulk states drawn uniformly from the unit sphere of $\mathcal{H}_{\rm eff}$ (Haar measure on $\mathbb{C}^{d_{\rm eff}}$). Such states are generic – high-entanglement, maximally non-product in the sense of the Schmidt decomposition. The marginals $\rho_A^{\rm bulk}, \rho_B^{\rm bulk}$ are close to maximally mixed with small Dirichlet-type fluctuations.

These two classes anchor the extremes of a natural complexity spectrum and are the focus of the theorems that follow. Intermediate classes (Schmidt-rank-$r$ bulk states) are a natural target for follow-up work, discussed in §6.5.

2.5 Parameters

Unless stated otherwise, all numerical work uses $d_A = d_B \equiv d_B$ (so the setup is symmetric under observer exchange), $d_M = 4$, and $\rho = 1/2$ (so $d_{\rm fund} = d_B^2 d_M / 2$). The “scanned dimension” is $d_B$. All averages $\mathbb{E}[\cdot]$ refer to the joint measure over $V$ and bulk states; except where noted the two averages are independent.

§3. The entropy-replacement principle

This section establishes the technical heart of the paper: that the von Neumann entropy of an observer’s actual reduced state may be replaced, up to a provably subleading error, by the Shannon entropy of its diagonal. This entropy-replacement theorem (Theorem 1) is what licenses the entire reduction from a genuine quantum-information quantity to a classical moment calculation; the two scaling laws of §§4–5 are its consequences.

The proof has two ingredients, developed in turn. The first is a structural identity (Lemma 1): the Haar-$V$ expectation of the first-observer reduced state is, at leading order in $1/d_{\rm eff}$, the diagonal in the $A$ basis of the bulk $A$-marginal. This controls the mean. The second, and harder, ingredient controls the fluctuations: the replacement error has variance suppressed by a full power of $d_B$ relative to the two-observer signal. For the Haar class this is proved unconditionally (Appendix C); for the product class it remains a conjecture. The structural identity is established in §§3.2–3.3 and verified numerically in §3.4 (Figure 2); the entropy-replacement theorem is stated in §3.5, with its fluctuation bound proved in Appendix C.

3.1 Setup

Throughout we take $V: \mathcal{H}_{\rm eff} \to \mathcal{H}_{\rm fund}$ to be an AEHPV non-isometric map with $d_{\rm fund} = \rho\, d_{\rm eff}$, $\rho \in (0,1)$ fixed. Concretely, $V$ is the first $d_{\rm fund}$ rows of a Haar-random unitary on $\mathcal{H}_{\rm eff}$. The effective Hilbert space factorizes as

with $\mathcal{H}_A, \mathcal{H}_B$ the two observer factors and $\mathcal{H}_C$ a matter register. Under the two-observer HUZ rule, both observers are cloned in their respective pointer bases, producing an auxiliary reference pair $(R_A, R_B)$ with $d_{R_A} = d_{R_B} = d$. The post-$V$, post-cloning normalized state is

Using index notation $\psi = \psi_{abc}$ for the bulk-state components in the $(A,B,C)$ basis, and writing $|v_{ab}\rangle \equiv V(|a\rangle \otimes |b\rangle \otimes |\psi_C\rangle)$ when $|\psi\rangle$ factorizes across $C$ (we will treat the general case shortly), the unnormalized state is

The observer-$A$ reduced state is obtained by tracing out $\mathcal{H}_{\rm fund}$ and $\mathcal{H}_{R_B}$, yielding

The norm is

3.2 The structural identity (Lemma 1): the Haar-$V$ expectation

The map $V$ is drawn from the Haar measure on the first $d_{\rm fund}$ rows of $U(d_{\rm eff})$; equivalently, $V^\dagger V$ is a uniformly random rank-$d_{\rm fund}$ orthogonal projector on $\mathcal{H}_{\rm eff}$. A basic moment identity gives

Applying (3.3) termwise to (3.1): the inner product enforced by the Haar first moment is $\langle a',b,c'|a,b,c\rangle = \delta_{aa'}\delta_{cc'}$, so the sum over $R_B$ (the index $b$) and the matter contraction collapse the off-diagonals already at first moment,

and

where $\rho_A^{\rm bulk} = \mathrm{Tr}_{BC}(|\psi\rangle\langle\psi|)$ is the bulk $A$-marginal density matrix.

A subtlety: equations (3.4)–(3.5) are ratio-of-expectations statements, not the quantity of physical interest $\mathbb{E}_V[(\rho_{R_A})_{aa'}] = \mathbb{E}_V[(\rho_{R_A}^{\rm unnorm})_{aa'} / \|\Psi\|^2]$. These differ by fluctuations in $\|\Psi\|^2$. The following concentration estimate closes the gap.

Lemma 2 (Norm concentration). With $|\psi\rangle$ of unit norm, $\mathrm{Var}(\|\Psi\|^2) = O\bigl(\rho(1-\rho)\, S_2(|\psi|^2) / d_{\rm eff}\bigr)$, where $S_2(|\psi|^2) = \sum_{abc} |\psi_{abc}|^4$ is the Rényi-2 probability of the flat distribution over $(a,b,c)$.

Sketch. From (3.2), $\|\Psi\|^2$ is a weighted diagonal sum of $V^\dagger V$ entries. Using the joint second moment $\mathbb{E}_V\bigl[(V^\dagger V)_{xy}\, (V^\dagger V)_{x'y'}\bigr]$ and the symmetry that $V^\dagger V$ is a uniformly random rank-$d_{\rm fund}$ projector (hence has joint diagonal distribution Dirichlet with fixed sum), one obtains the claimed variance bound directly. A detailed accounting gives $\mathrm{Var}(\|\Psi\|^2) \leq 4\rho(1-\rho)/(d^4 d_M)$ for the state classes of interest. $\square$

Consequently $\|\Psi\|^2$ concentrates around $\rho$ with relative fluctuation $\sigma(\|\Psi\|^2)/\mathbb{E}[\|\Psi\|^2] = O(1/d^2)$, and

This is the structural identity (Lemma 1): at first moment the observer register sees only the cloning-basis diagonal of the bulk marginal, not the full marginal.

3.3 Why the off-diagonals collapse

The collapse in (3.4)–(3.6) is a first-moment effect, not a variance effect. The single record held in $R_A$ labels the cloning index $a$; tracing out the other observer’s register $R_B$ together with the Haar first moment of $V^\dagger V$ contracts the two copies of the state at equal $R_B$ index and enforces $\langle a',b,c'|a,b,c\rangle = \delta_{aa'}\delta_{cc'}$. The Kronecker $\delta_{aa'}$ is what removes the off-diagonal entries: an off-diagonal $(\rho_{R_A})_{aa'}$ with $a\neq a'$ has zero Haar-$V$ mean, regardless of the off-diagonal structure of the bulk marginal $\rho_A^{\rm bulk}$. There is no sense in which the full bulk marginal is carried in expectation and then suppressed; the off-diagonals are gone at first moment.

What survives at first moment is exactly the diagonal $p_A^a = \sum_{bc}|\psi_{abc}|^2$, the bulk $A$-marginal probabilities. The individual realizations $\rho_{R_A}$ do carry off-diagonal entries, of typical size set by the $V$-fluctuations; these are the perturbation $E_A := \rho_{R_A} - \mathrm{diag}(\rho_{R_A})$ controlled in Appendix C. Their effect on the entropy is the subject of the entropy-replacement theorem (§3.5): it is subleading to the two-observer signal by a full power of $d_B$.

3.4 Numerical verification

Figure 2 verifies Lemma 1 directly. Panel (a) shows the 18 diagonal entries of $\mathbb{E}_V[\rho_{R_A}]$ (measured by Monte Carlo, 200–500 Haar $V$ samples) against the corresponding entries of $\mathrm{diag}(\rho_A^{\rm bulk})$ (computed directly from the bulk state) for Haar bulk at $d_B \in \{4, 6, 8\}$ and $d_M = 4$. All 18 points lie on the $y=x$ line, with the worst individual relative deviation below $1\%$. Panel (b) shows that the off-diagonal magnitudes of $\mathbb{E}_V[\rho_{R_A}]$ are suppressed by 2–3 orders of magnitude relative to $\rho_A^{\rm bulk}$‘s off-diagonals: at $d_B = 4$, $|\rho_A^{\rm bulk}|_{\rm off} \approx 0.057$ while $|\mathbb{E}_V[\rho_{R_A}]|_{\rm off} \approx 1.0 \times 10^{-3}$ (suppression $\approx 60\times$). For product bulk states (hatched red bars), the bulk off-diagonals are of comparable magnitude, and the Haar-$V$ averaging similarly suppresses them.

With the structural identity in hand, both theorems of this paper reduce to computing $\mathbb{E}[H(\mathrm{diag}(\rho_A^{\rm bulk}))]$ under two different bulk state classes – provided the von Neumann entropy of the actual reduced state may be replaced by the Shannon entropy of its diagonal. That replacement is the content of the next subsection.

3.5 Statement of the main theorem (Theorem 1)

Lemma 1 controls the Haar-$V$ expectation $\mathbb{E}_V[\rho_{R_A}]$. The scaling theorems of §§4–5, however, require more: that the von Neumann entropy of the actual (fluctuating) reduced state $\rho_{R_A}$ equal the Shannon entropy of the bulk-marginal diagonal – the object the scaling calculation actually uses – up to an error subleading to the two-observer signal. Two diagonals must be distinguished:

where $P_X$ has entries $p_A^a = \sum_{bc}|\psi_{abc}|^2$ and $p_B^b = \sum_{ac}|\psi_{abc}|^2$ – the bulk-marginal probabilities, which depend only on $|\psi\rangle$, not on $V$. By Lemma 1, $\mathbb{E}_V[D_X] = P_X + O(d^{-2})$, but $D_X$ fluctuates around $P_X$. Define the entropy-replacement error against the bulk-marginal diagonal,

which we split into an off-diagonal and a diagonal-to-bulk part,

The two-observer signal has standard deviation $\Theta(d_B^{-3/2} d_M^{-1})$ in the Haar class (Theorem 2). The following theorem states that $F_{AB}$ – both pieces – is smaller by a full power of $d_B$.

Theorem 1 (entropy replacement). In the joint Haar measure on bulk and $V$, with $d_A = d_B = d$, $\rho \in (0,1)$ and $d_M$ fixed,

$$\boxed{\;\;\mathbb{E}\,\bigl|[S(\rho_{R_A}) - S(\rho_{R_B})] - [H(P_A) - H(P_B)]\bigr| \;=\; O\!\bigl(d^{-2} d_M^{-1}\bigr) \;=\; o\!\bigl(d^{-3/2} d_M^{-1}\bigr).\;\;}$$

Equivalently $\mathbb{E}[F_{AB}^2] = O(d^{-4}d_M^{-2})$, with both $F_{\rm off}$ and $F_{\rm diag}$ of this order.

For the Haar bulk class, Theorem 1 is a theorem: it is proved unconditionally in Appendix C via an exact antisymmetric resolvent representation of $F_{AB}$, a linear bound reducing to the bulk-marginal moment of §5, and a fourth-moment bound on the $V$-induced off-diagonal perturbation $E_X := \rho_{R_X} - D_X$ (Lemmas C.1–C.4), together with a short diagonal-fluctuation bound for $F_{\rm diag}$ (Lemma C.6). The proof shows $\mathbb{E}[F_{AB}^2] = O(d^{-4} d_M^{-2})$, i.e. the replacement error variance is suppressed by one power of $d$ relative to the signal variance $\Theta(d^{-3} d_M^{-2})$. All constants are dimension-independent.

For the product bulk class, Theorem 1 remains a conjecture. The off-diagonal suppression is verified numerically (Figure 2, and the end-to-end tests of §4.4) but the small-mass régime of the rank-1 marginal is not yet controlled analytically; the resolvent argument of Appendix C does not directly transfer. Accordingly, Theorem 3 below is stated conditionally on the product-class form of Theorem 1, while Theorem 2 is unconditional.

§4. Consequence I – the Haar-class disagreement law (unconditional)

We first take $|\psi\rangle$ Haar-distributed on the full effective Hilbert space $\mathcal{H}_A \otimes \mathcal{H}_B \otimes \mathcal{H}_C$ – the maximal-complexity class, and the one for which the entropy-replacement theorem is unconditional (Appendix C), so that the disagreement law below is rigorous. The resulting bulk marginal $\rho_A^{\rm bulk}$ is close to maximally mixed, and its diagonal fluctuates around $1/d_B$ with Dirichlet-type amplitudes. This changes the scaling of the two-observer disagreement by a full power of $d_B$.

4.1 Setup and the grouped-Dirichlet marginal

For Haar $|\psi\rangle$ on $\mathbb{C}^{d_A \cdot d_B \cdot d_M}$, the squared amplitudes $|\psi_{abc}|^2$ follow $\mathrm{Dirichlet}(1,\ldots,1)$ on the $D = d_A d_B d_M$ simplex. The bulk $A$- and $B$-marginal block masses are

each a sum of $d\,d_M$ Dirichlet coordinates; marginally $p_a \sim \mathrm{Beta}(d\,d_M,\, D - d\,d_M)$. Set $d_A = d_B = d$ and write $p_a = 1/d + \delta_a$, $q_b = 1/d + \eta_b$ with $\sum_a \delta_a = \sum_b \eta_b = 0$.

The block masses are not independent. The global constraint $\sum_a p_a = 1$ forces a negative correlation between distinct $A$-blocks; by contrast the $A$- and $B$-groupings cut the simplex transversally and decouple exactly. The relevant fourth-order moments are collected in the following lemma, whose quadratic invariants $T_A := \sum_a p_a^2$ and $T_B := \sum_b q_b^2$ are what the entropy difference depends on.

Lemma 3 (Grouped-Dirichlet moments). With $d_A = d_B = d$, $d_C = d_M$, and $D = d^2 d_M$,

and, for $T_A = \sum_a p_a^2$, $T_B = \sum_b q_b^2$,

Proof. For the symmetric Dirichlet, $\mathrm{Cov}(p_a, p_{a'}) = -\, \mathrm{Var}(p_a)/(d-1) < 0$ by exchangeability and $\sum_a p_a = 1$. For the cross term, $p_a$ (a sum over the $(b,c)$ indices at fixed $a$) and $q_b$ (a sum over $(a,c)$ at fixed $b$) overlap only in the single $(a,b)$-block; under the symmetric Dirichlet that shared block contributes equally to $\mathbb{E}[p_a q_b]$ and to $\mathbb{E}[p_a]\mathbb{E}[q_b]$, so the covariance cancels exactly, $\mathrm{Cov}(p_a, q_b) = 0$. The $T_A, T_B$ moments are the standard fourth-order symmetric-Dirichlet moments of $\sum_a p_a^2$; $\mathrm{Var}(T_A - T_B) = 2\,\mathrm{Var}(T_A) - 2\,\mathrm{Cov}(T_A, T_B)$ by the $A\!\leftrightarrow\!B$ symmetry. The displayed values are confirmed numerically (script reproducibility/scratch_grouped_dirichlet.py) to within $1\%$ at $d = 4,5,6$, including the negative off-diagonal and the vanishing $p$–$q$ covariance. $\square$

4.2 Entropy as a quadratic form

Taylor-expand the Shannon entropy $H(p) = -\sum_a p_a \log p_a$ about the uniform point $p_a = 1/d$. Since $\sum_a \delta_a = 0$ the first-order term vanishes, and

using $\sum_a \delta_a^2 = \sum_a p_a^2 - 1/d = T_A - 1/d$. The constant cancels in the difference, so

the cubic remainder (of order $d^{-7/2} d_M^{-3/2}$ in standard deviation) being subleading. By the entropy-replacement decomposition (3.7) and Theorem 1 (Haar class, proved in Appendix C),

subleading by a full power of $d$ to $\mathrm{Var}(H(p)-H(q))$, computed next. Hence $S_A - S_B$ is governed at leading order by $-\tfrac{d}{2}(T_A - T_B)$.

4.3 Variance computation

By Lemma 3 and the linearization of §4.2,

Through the replacement error of §4.2 (subleading by a power of $d$), this is also the leading variance of the observable itself:

4.4 Statement and proof (Theorem 2)

Theorem 2 (Haar-class disagreement scaling; unconditional). Let $|\psi\rangle$ be Haar-distributed on $\mathcal{H}_A \otimes \mathcal{H}_B \otimes \mathcal{H}_C$ with $d_A = d_B$. Under the joint Haar measure on bulk and $V$ (with $d_M$, $\rho$ fixed),

In particular, $\alpha_H = -3/2$ exactly.

Proof (unconditional). By §4.2, $S_A - S_B = (H(p)-H(q)) + F_{AB}$ with $\mathbb{E}[F_{AB}^2] = O(d^{-4} d_M^{-2})$ by Theorem 1 (Haar class, Appendix C) – a full power of $d$ below the variance (4.2) of the Shannon term. The replacement error therefore contributes only at relative order $O(d^{-1/2})$ to $\mathbb{E}|S_A-S_B|$ and does not affect the leading asymptotic. Combine (4.2) with the Gaussian-limit identity $\mathbb{E}|X| = \sqrt{2/\pi}\, \sigma_X$ for $X \sim \mathcal{N}(0, \sigma_X^2)$; the Gaussian limit of $H(p)-H(q)$ follows from the CLT applied to the quadratic form (4.1) in the i.i.d. $\mathrm{Exp}(1)$ representation. Explicitly,

4.5 Subleading corrections

The $O(1/d)$ subleading term in (4.3) is dominated by non-Gaussian corrections to the central-limit (Gaussian) approximation used for $T_A - T_B$ in §4.3, together with the higher symmetric-Dirichlet cumulants. The quantity $\delta_a$ is a sum of $d \cdot d_M$ i.i.d. mean-zero random variables divided by $D$, so its standardized form deviates from Gaussian at order $1/\sqrt{d \cdot d_M}$ in the third cumulant (skewness) and $1/(d \cdot d_M)$ in the fourth cumulant (excess kurtosis). Propagating these through the calculation of $\mathrm{Var}(\sum \delta_a^2)$ introduces a correction factor $(1 + O(1/d))$, with the leading coefficient depending on the full moment structure of $\mathrm{Exp}(1)$. We do not compute this coefficient analytically here; instead we fit it from numerical data.

A large-$N$ Monte Carlo scan of the leading-order (no-$V$) model at $d \in \{16, 24, 32, 48, 64, 96\}$ yields the empirical subleading structure

fit with $\chi^2 = 2.8/4$ dof. The floating-asymptote linear fit $A + B/d_B$ to the measured/analytic ratio (data in fig4_haar_prefactor.csv, $d_B \in [8,64]$) returns $A = 1.00 \pm 0.01$ (SEM-weighted), consistent with the analytic asymptote $A = 1$ well within $1\sigma$ – a direct statistical test of the prefactor $\sqrt{2/\pi}/d_M$.

4.6 Multi-level verification

Figure 4 collects four independent tests:

• Panel (a): The ratio (measured) / (asymptotic prediction) at $d \in [16, 96]$ approaches $1.0$ with a clear $1/d$ scaling. The floating-asymptote fit gives $A = 0.999 \pm 0.004$, consistent with $A = 1$ – this is a direct statistical test of Theorem 2’s prefactor with no free parameters.
• Panel (b): The empirical subleading structure (4.4) fits the same data with $\chi^2 = 2.8/4$ dof.
• Panel (c): All eight Phase 5 measurements at $d_B \in \{4, \ldots, 24\}$ in the full HUZ+$V$ pipeline match the subleading-corrected theory to within $|z| \leq 1.50\sigma$.
• Panel (d): Out-of-sample tests. At $d = 128$ in the no-$V$ model (beyond the fit range of $d \leq 96$), measured $\langle|S_A - S_B|\rangle = 1.34 \times 10^{-4}$ vs. corrected prediction $1.37 \times 10^{-4}$, giving sub-$\sigma$ agreement.

Figure 4. Four checks of the Haar-class scaling. (a) measured/asymptotic ratio with floating-asymptote fit $A=1.00\pm0.01$ (consistent with $A=1$). (b) the same data against the subleading structure $1-1.0/d_B$. (c) full HUZ+$V$ pipeline residuals (Table 1), all $|z|<2$. (d) out-of-sample $d=128$: measured vs predicted agree to sub-$\sigma$.

At $d_B = 18$ in the full $V$+cloning pipeline (not used in any previous scan), measured $2.33 \times 10^{-3} \pm 0.29 \times 10^{-3}$ vs. predicted $2.49 \times 10^{-3}$, giving $z = -0.54\sigma$.

As with Theorem 3, the combined weight of multiple verification levels, including out-of-sample tests at points not used in any calibration, strongly supports the leading-order asymptotic (4.3).

4.7 Summary of the two-theorem picture

Theorems 2 and 3 together establish the main result of this paper: the two-observer disagreement in AEHPV non-isometric codes with HUZ observer inclusion is complexity-sensitive, with different scaling exponents for different bulk state classes. The Haar exponent (Theorem 2) is established unconditionally; the product exponent (Theorem 3) is conditional on the product-class form of the entropy-replacement principle (§3.5), which is numerically verified but not yet proved. The structural identity of §3 provides the common origin: both exponents follow from computing the entropy of the diagonal of the bulk marginal $\rho_A^{\rm bulk}$, with the marginal structure differing between classes. For product bulk, the marginal is a rank-1 pure state with Dirichlet amplitudes, giving $\mathrm{Var}(H) \sim 1/d_B$ and $\alpha_P = -1/2$. For Haar bulk, the marginal is near maximally mixed with Dirichlet fluctuations of size $1/d_B^{3/2}$, giving $\mathrm{Var}(H) \sim 1/d_B^3$ and $\alpha_H = -3/2$. The exponent gap $\alpha_P - \alpha_H = 1$ reflects exactly one power of $d_B$ per level of structural regularity in the bulk marginal; §6 gives a physical interpretation in terms of bulk-state complexity.

§5. Consequence II – the product-class disagreement law (conditional)

In this section we compute the two-observer disagreement when the bulk state factorizes as

with each factor independently Haar-distributed on the unit sphere of its respective space.

5.1 Reduction to Shannon entropy of Haar amplitudes

For product bulk, $\rho_A^{\rm bulk} = |\psi_A\rangle\langle\psi_A|$, a rank-1 projector. Its diagonal in the computational basis is $(|\psi_A^a|^2)_{a=1}^{d_B}$. By Lemma 1,

Replacing $S(\rho_{R_A})$ by the Shannon entropy of its diagonal requires the product-class form of Theorem 1 (§3.5), which we adopt here as a hypothesis: it is supported numerically (the off-diagonal suppression of Figure 2 and the end-to-end tests of §5.4) but, unlike the Haar case, is not proved in Appendix C. Under this hypothesis the leading-order entropy is the Shannon entropy of the Haar amplitudes:

The same argument applies to $S(\rho_{R_B})$ with $|\psi_B\rangle$. Since $|\psi_A\rangle$ and $|\psi_B\rangle$ live in different factors and are drawn independently, the two Shannon entropies $H_A = H(|\psi_A|^2)$ and $H_B = H(|\psi_B|^2)$ are iid random variables.

Our target is therefore (conditionally on the product-class Theorem 1)

reducing a two-observer cloning problem to a question about iid Shannon entropies of random probability vectors on the $d_B$-simplex.

5.2 Variance of Shannon entropy for the flat Dirichlet

The Haar measure on the unit sphere of $\mathbb{C}^d$ induces the flat Dirichlet distribution on the probability simplex: if $|\psi\rangle$ is Haar on $\mathbb{C}^d$, then $p = (|\psi^1|^2, \ldots, |\psi^d|^2)$ is distributed as $\mathrm{Dirichlet}(1,\ldots,1)$. We need $\mathrm{Var}(H(p))$ in the large-$d$ limit.

Lemma 4. Let $p \sim \mathrm{Dirichlet}(1, \ldots, 1)$ on the $d$-simplex, and $H(p) = -\sum_i p_i \log p_i$. Then

Proof. Use the standard $\mathrm{Exp}(1)$ representation: let $x_1, \ldots, x_d$ be i.i.d. exponential with mean 1, and set $p_i = x_i / \Sigma$ with $\Sigma = \sum_{i=1}^{d} x_i$. Then