Hawk06-A Existence of Operational Islands in a Finite Enter Model Entanglement-assisted decoupling, deterministic decoding, and a positive-residual cutoff

We prove existence of finite operational islands, defined here as reference-preserving reconstruction certificates for a specified logical factor initially located in an inaccessible vault. The definition is deliberately not a gravitational definition of an island. The final physical residual remains a tensor factor and is not silently granted to the receiver.

The complete model contains an unknown $d$-dimensional diary, an $h$-dimensional old-vault register maximally entangled with accessible early radiation, and a pure blank register of dimension $f$. One known, input-independent unitary scrambles the vault; a nested schedule releases its cells. For Haar scrambling we derive the exact complementary Choi Hilbert–Schmidt moment and convert it to a worst-case diamond certificate. A single-cut probability estimate, together with nested access, certifies all subsequent retained cuts. A deterministic decoder is specified by a finite semidefinite program, and its quantum workspace and implementation allowance are stated.

The entangled preparation changes the entropy calculation: the late radiation is a reduction of a random rank-$h$ projector, not generally a Haar pure-state reduction. We derive its exact purity moments and rigorous entropy bounds. A concrete parameter family has positive Haar measure on which accurate whole-diary recovery and strict coherent entropy turnover both occur before a cutoff leaving one physical residual qubit. We also construct an exact entanglement-assisted example with a positive-dimensional residual and an explicit controlled-unitary decoder.

Append-only Enter records cannot turn over. Their record-only quantum recovery obstruction does not apply when a genuine quantum early register remains accessible; teleportation supplies a precise counterexample to that overextension. No quantum extremal surface, area term, replica wormhole, gravitational Hamiltonian, or spacetime emergence is constructed.

of the existence claim. The proved objects are finite encodings, channels, decoupling certificates, and quantum-return decoders. Calling their logical reconstruction regions operational islands is a definition of this manuscript, not a claim that Cheung's Philosophy of Enter has already defined or proved gravitational islands. A proof of the latter would require additional gravitational structure.

Cheung's Philosophy of Enter separates destructive readout from quantum recovery, reduced channels from thermal machines, and finite analogies from physical identifications. Enter consumes a measured quantum name at its declared interface and returns a classical record. A decoder instead returns a quantum target using specified accessible systems and preserving the required untouched reference correlations CheungPhilosophy.

We adopt ordinary finite Born instruments explicitly. Cheung's operational measurement characterization applies within its stated strategy class; it is not used here as a derivation of a gravitational instrument or of scrambling CheungEnter,CheungI. The operative questions are

The entanglement-assisted model is a finite version of the Hayden–Preskill recovery setting HaydenPreskill. Its ingredients are additionally declared. They are not consequences of Enter.

Several statements in the proposed plan would invalidate the proof if retained without qualification.

{}1.12

These repairs preserve the finite target while refusing an unsupported gravitational conclusion.

In gravitational island calculations, one supplies a gravitational state, admissible regions and surfaces, and a generalized entropy involving bulk entropy and a geometric term EngelhardtWall,IslandReview,Penington. None of those objects is supplied by declaring a finite register to be a vault.

Here an “island assignment” means only that a specified logical factor, initially outside the receiver's access, has certified representatives obtainable from the receiver's later output. It is not an equation asserting

not physical access to $B$, and not a minimization over surfaces.

Fix finite spaces

where $d\ge 2$ and $h,f\ge 1$. The diary $L$ is variable. The other input data are fixed.

Prepare

independently of the diary and of every external reference. A fixed embedding unitary $J$ may encode $LHF$ into the vault. A fixed unitary $U$ then acts on the complete vault, of dimension

For the Haar analysis, $UJ$ is one Haar unitary on this $D$-dimensional space. We absorb $J$ into $U$.

Choose in advance a tensor-cell presentation of the vault and a nested release schedule. At cut $k$, write

Later cuts only release additional cells. No emitted cell or early register is subsequently scrambled by the vault.

The complete prepared isometry is

with its factors regrouped into

Thus the declared coherent access is $A_k$, not $R_k$ alone.

Define

These channels preserve the mathematical meaning of every reference extension.

The early register has constant marginal $I_E/h$ for every input. At cut zero it does not by itself reveal the diary.

A retained schedule ends at a predeclared cut $K$ with $b_K\ge 2$. No later drain of $B_K$ is included. The endpoint is a finite interface, not evaporation to nothing.

For qubits, write

A cutoff retaining $q\ge 1$ vault qubits is $K=n-q$.

Each released cell can be transferred to a receiver cell by SWAP. The reset source placeholders are fixed pure factors and may be included in the complete residual without changing the channels. All physical qubits can remain inside one laboratory. No gravitational energy law is assumed.

and entropy conventions. We use

Diamond distance includes arbitrary external references WatrousBook. Set

All entropies below are in bits.

For an isometry $V:L\to A\otimes B$, an $e$-accurate operational island certificate for the initially interior logical factor $L$ consists of:

1. the specified noncommutative target algebra $\mathcal{B}(L)$, with $d\ge 2$;

2. the accessible interface $A$, excluding physical $B$;

3. one deterministic CPTP quantum-return map $\mathcal{D}:A\to L_{\mathrm{out}}$;

4. the reference-preserving bound

   $$d_{\diamond }(\mathcal{DN},{\mathrm{id}}_L)\le e.$$

Exact certificates have $e=0$. The term “island” refers to this logical reconstruction assignment. It does not designate an independently variable physical residual factor.

For a finite complete isometry $V:L\to A\otimes B$, the following are equivalent:

If $s=\mathrm{rank}{\tau }_B$, then $\dim A\ge ds$. There are an isometry $T:L\otimes {\mathbb{C}}^s\to A$ and a state $\sigma$ such that

An exact decoder on the entire accessible space is

for any fixed state ${\omega }_L$.

Proof. Suppose a decoder is exact and dilate it by $W:A\to LG$. Then $(W\otimes I_B)V$ dilates the identity channel. Its Choi operator has rank one, so every Kraus operator is a scalar multiple of identity. The composite therefore has the form $|\psi ⟩\mapsto |\psi ⟩|\chi {⟩}_{GB}$, with fixed $\chi$. Its residual is constant.

Conversely, diagonalize ${\tau }_B={{\sum }_{\alpha =1}}^s{\lambda }_{\alpha }|\alpha ⟩⟨\alpha |$, where ${\lambda }_{\alpha }>0$, and write

Constancy on every matrix unit gives

These $ds$ vectors define $T$. Equations (eq:normalform) and (eq:exactdecoder) follow directly. The decoder is CP, its traces sum to $\mathrm{Tr}Y$, and it returns every logical matrix unit.

This is the established correction criterion KnillLaflamme. Comparable locally scoped statements occur in Cheung's En07-B, Hawk01-B, and Oai280-C CheungEn07B,CheungHawk01B,CheungOai280C. Their model-specific channels are not premises here.

The finite information–disturbance theorem implies

A nearly constant accessible channel instead gives

Proof. The imported information–disturbance inequality, in full diamond normalization, is

Substitution of the half-diamond convention yields (eq:KSW). A constant decoder supplies the $a_d$ cap.

For a replacement accessible channel, any decoded Bell input has Bell overlap $d^{-2}$, hence error at least $a_d$. Preparing $I_L/d$ attains it: depolarization is the uniform average of $d^2$ Weyl conjugations, one of which is identity. The nearly constant lower bound follows by CPTP contraction and the reverse triangle inequality.

An $e$-accurate operational island certificate gives a unital CP map

such that, for every Hermitian $X$,

At $e=0$, a faithful star representation of the logical algebra exists on the correctable accessible support:

For $e>0$, $\mathcal{K}$ need not be multiplicative.

Proof. The left side of (eq:Heisenberg) is ${\Vert (\mathcal{DN}{)}^{\ast }(X)-X\Vert }_{\infty }$. Trace/operator-norm duality and the channel bound give the result. The exact representation follows from (eq:normalform); its unit is the support projection $T{T}^{†}$. An arbitrary CP recovery adjoint is not a star homomorphism.

Thus approximate recovery does not automatically construct an exact radiation subalgebra or a second logical tensor factor.

Let $Q\simeq L$ be an inaccessible test reference, and prepare

For a cut $RB$, omit the cut index and write $D=rb$. Define

The state ${{\sigma }_B}^U$ is a feasible replacement state. It may depend on the known device $U$, not on the variable input.

On a $D$-dimensional space with $D\ge 2$,

where

For $O=I_{R{R}^{\prime }}\otimes {\mathsf{F}}_{B{B}^{\prime }}$,

Proof. The averaged operator commutes with every $U\otimes U$. The symmetric and antisymmetric subspaces give its two scalar blocks, equivalently its expression as $\alpha I+\beta \mathsf{F}$. Preservation of $\mathrm{Tr}O$ and $\mathrm{Tr}(O\mathsf{F})$ solves the two linear equations. For the specified $O$, those traces are $r^{2}b$ and $r{b}^2$.

In the complete preparation of Section sec:model,

The formula includes the blank dimension $f$ through $D=dhf$.

Proof. Tracing the accessible early register before the calculation gives the initial reference–vault state

Its squared trace is $1/h$, whereas its vault marginal has squared trace $1/(dh)$, and its reference marginal has squared trace $1/d$.

The swap test and Lemma lem:twirl give

Since ${\mathrm{Tr}}_Q{{\omega }_{QB}}^U={{\sigma }_B}^U$,

The $\alpha$ terms cancel. Insert $\beta$ from (eq:twirlcoeff).

For channels with input dimension $d$,

Consequently, with

Proof. The lower Choi inequality uses the Bell input. A pure referenced input is $(T\otimes I)|{\Phi }_d⟩$, with $\mathrm{Tr}(T^{†}T)=d$ and ${\Vert T\Vert }_{\infty }^2\le d$. Multiplying the Choi difference by $T\otimes I$ on both sides bounds its trace norm by $d$ times the Choi trace norm. Convexity handles mixed inputs WatrousBook.

Since ${X_{QB}}^U$ acts on dimension $db$,

Take expectations and use (eq:gamma). For the last bound,

The factor $d$ in the Choi-to-diamond conversion matters. Discarding it changes a Bell-input statement into an unjustified whole-code statement.

Fix $\zeta >0$, $0<p<1$, and a positive-residual cutoff $K$. If a retained cut $k_{\ast }\le K$ satisfies

then, with Haar probability at least $1-p$, one sampled unitary has, simultaneously for every $k_{\ast }\le k\le K$,

A sufficient dimensional threshold is

In the qubit model, a sufficient cut is

provided $k_{\ast }\le n-q$.

Proof. Markov's inequality and (eq:K) imply

For a later cut, the complete complement is obtained by tracing the additional released cells:

The same trace maps ${{\sigma }_{B_{k_{\ast }}}}^U$ to ${{\sigma }_{B_k}}^U$. CPTP contraction proves (eq:leakall) on the single event; no independence or repeated Haar sampling is used. Lemma lem:approx gives (eq:recall). Equation (eq:thresholdsimple) follows from the final bound in (eq:K).

The theorem proves a high-probability statement and, whenever its hypotheses hold, existence of fixed unitaries with the certificate. It does not certify every unitary or guarantee a qualifying cut before every chosen cutoff. The sufficient threshold is not claimed optimal.

The old-vault entropy is ${\log }_{2}h$. Its purification is accessible and explicitly included. The resulting sufficient bound has no additional $h$ factor. Losing that purification changes the complete complement and the task.

For a sharper device-specific certificate, compute

These are finite-matrix quantities. One may use the best earlier certificate at a later cut. An ordinary floating-point evaluation without a rigorous norm enclosure is not a certified upper bound.

We choose a finite optimization-defined decoder, not an unproved Petz estimate.

For a linear map $\Phi$, use the unnormalized Choi operator

Then $J_{\mathrm{n}}(\Phi )=J(\Phi )/d$ for a diary-input channel.

At cut $k$, let $A=A_k$, $a_A=\dim A=r_{k}h$, and let $C_{\mathcal{D}}$ be a decoder Choi variable on $A^{\prime }\otimes L_{\mathrm{out}}$. It defines

Set

Choose a minimizing solution of the finite semidefinite program

If several decoder Choi matrices minimize it, choose the unique minimum-Frobenius-norm matrix among those matrices. Denote the resulting CPTP map by ${{\mathcal{D}}_k}^{\mathrm{opt}}$.

Program (eq:SDP) specifies a decoder on the whole accessible space, including inputs outside the encoded output support. Its optimal value is ${\epsilon }_{\mathrm{rec}}({{\mathcal{N}}_k}^U)$. On the event of Theorem thm:threshold,

Equivalently, the full diamond norm is at most $2\min \{a_d,\sqrt{2\zeta }\}$.

Proof. The Choi constraints are precisely complete positivity and trace preservation. Composition with the fixed channel is linear in $C_{\mathcal{D}}$.

For a Hermiticity-preserving map, the finite diamond-norm dual is

Applying it to the decoded difference gives (eq:SDP).

The upper-bound direction can also be checked directly: $Z\pm J(\Delta )\ge 0$ define two CP maps whose difference is $\Delta$, while their sum has Choi matrix $Z$. On any referenced state, its output trace norm is at most the trace of that CP sum, hence at most $t$.

Finite decoder Choi matrices form a compact set. Diamond error is continuous, so an optimum is attained. A norm-dual variable is attained as well; bounded $t$ bounds $\mathrm{Tr}Z$, and $Z\ge 0$, so a minimizing sequence has a convergent subsequence. Strict convexity of the Frobenius norm makes the stated secondary choice unique on the convex optimal decoder set.

Finally, Lemma lem:approx bounds this optimum on the decoupling event.

This is an explicit finite decoder prescription given the matrix of $U$. It is not a reported numerical execution or an efficient decoder-complexity theorem.

The decoder has Kraus rank

It has a Stinespring isometry

For a qubit interface, a batch implementation using fresh output and environment registers has peak quantum occupancy at most

The received early and late registers are included in this count. The inaccessible reference and residual are outside the laboratory decoder's holdings.

Proof. Diagonalize the decoder Choi matrix to obtain at most $d{a}_A$ Kraus operators. Their column stack is a Stinespring isometry. Allocate the output and a padded environment, and extend

from its orthonormal prepared subspace to a unitary. The displayed register count prices this deliberately nonoptimal construction. A finite decomposition into elementary gates exists with unrestricted continuous gate parameters Barenco. No efficient gate-count bound is asserted.

A restricted streaming observer need not realize this batch decoder. Cheung's En02-B and En02-C provide benchmark-specific examples of that distinction CheungEn02B,CheungEn02C. Their one-live-qubit bounds are not universal bounds for the present interface.

For a full-rank input state $\sigma$, write $s=\mathcal{N}(\sigma )$, $P=\mathrm{supp}s$, and invert $s$ only on $P$. A completed Petz candidate is

It is CP, and the traces of its two terms are $\mathrm{Tr}(PY)$ and $\mathrm{Tr}((I-P)Y)$, so it is TP on the entire accessible space. This alone does not establish (eq:decerror). Petz and Yoshida–Kitaev decoders have additional theorems under their own hypotheses ChenPetz,YoshidaKitaev.

Any specified candidate can instead be tested by

which gives half-diamond error at most $d{r}_{\mathrm{dec}}$ by Lemma lem:Choi.

Suppose actual and proxy complete channels are compared on the same code after access-preserving identifications, with distance at most $ϵ$. If a proxy decoder has error $e$, and its implemented accessible decoder differs from it by at most $\iota$, then

A proxy complementary replacement allowance $\zeta$ gives

Proof. Partial trace contracts the complete comparison. Insert the same fixed decoder and use contraction and the triangle inequality. Do the same for the proxy replacement state.

The transfer requires a common accessible interface. An output unitary mixing a still-inaccessible $B$ into $A$ is not an access-preserving comparison. Neither a power-law error nor a selected entropy error is automatically a channel allowance CheungHawk04D,Hawk05C.

Fix a pure diary input $|\psi ⟩$, independent of the sampled unitary. After tracing $E$, the vault state is

Its total entropy is ${\log }_{2}h$. The late-radiation state is its reduction to $R$.

At a cut $D=rb$, define

Then

For computational-basis Enter on the released cells, with prefix probabilities ${p_y}^U$,

Writing

one has

Proof. For the quantum purity, apply Lemma lem:twirl to ${\mathsf{F}}_R\otimes I_{B{B}^{\prime }}$. Its coefficients are

Since the initial vault state has trace one and purity $1/h$, the expected purity is ${\alpha }_R+{\beta }_R/h$.

For the record, $p_y=\mathrm{Tr}({\Pi }_y{{\rho }_{\mathrm{vault}}}^U)$, where ${\Pi }_y=|y⟩⟨y{|}_R\otimes I_B$ has rank $b$. The averaged squared state is

Therefore

Sum over the $r$ outcomes to obtain (eq:cpurity).

The rank of ${{\rho }_R}^U$ is at most $\min (r,bh)$, proving its entropy upper bound. The record has $r$ outcomes. For either a state or probability distribution, entropy is at least minus the logarithm of its purity. Jensen's inequality then proves the two lower bounds. Finally, direct comparison in (eq:qpurity) gives

which proves (eq:ranktent).

These are computed mean bounds for the same preparation used in the recovery theorem. They are not a substitution of a different pure-state ensemble.

When $h=1$, (eq:randomprojection) is Haar pure. With $m_0=\min (r,b)$, $n_0=\max (r,b)$, the established Page–Sen evaluation gives

The second identity follows by differentiating the Dirichlet moment of the prefix probabilities; the first is the Page–Sen integral PageAverage,Sen. These are the pure-preparation comparisons of Hawk05-B and Hawk05-C Hawk05B,Hawk05C.

For $h>1$, the exact means are the finite random-projector averages specified by (eq:randomprojection); the proved quantitative evaluations used here are (eq:qpurity)–(eq:ranktent). No claim of a closed Page–Sen formula for those mixed-state means is made.

At complete release $b=1$, included only as a comparison beyond a positive-residual cutoff,

for every unitary and pure diary input. For $D>h$,

Thus the coherent endpoint is not zero in the early-entangled model.

For each fixed unitary and each prescribed append-only destructive record process,

For fixed-basis Enter on $R_k$,

where ${\Delta }_{R_k}$ dephases that basis. This gap is not whole-code recovery error.

Proof. The first assertion is the classical chain rule and does not require Haar scrambling. For the second, the diagonal probabilities of $\Delta \rho$ are the measurement probabilities, and

This gives the relative-entropy identity and its nonnegativity. A Bell input to an identity channel has maximally mixed accessible marginal and hence zero marginal coherence gap, but destructive basis readout of that accessible system loses its Bell entanglement.

The gap is relative-entropy coherence in a specified basis Baumgratz. It is not a geometric island contribution or a thermal cost.

Take

Release vault qubits in order, with

Then

Choose any $\zeta >0$ satisfying

There is a set of Haar unitaries of measure at least $1/64$ for which, simultaneously:

1. the cutoff complement is within $\zeta$ of replacement;

2. the specified deterministic decoder has half-diamond error at most $\min \{a_{16},\sqrt{2\zeta }\}$;

3. for the fixed pure diary input,

   $$S({R_{k_{\mathrm{peak}}}}^U)>{\log }_2(2h)\ge S({R_K}^U);$$

4. every unedited cumulative Enter record is nondecreasing.

In particular, a nontrivial operational island and strict coherent entropy turnover coexist while one physical vault qubit remains.

Proof. At the intermediate cut, (eq:qpurity) gives

Markov's inequality implies

Thus the event $S({R_{k_{\mathrm{peak}}}}^U)>{\log }_2(2h)$ has probability greater than $1/32$. At the cutoff, the rank bound gives $S({R_K}^U)\le {\log }_2(2h)$ for every unitary.

The cutoff leakage estimate is

By (eq:jointsize),

Subtracting this possible failure set from the turnover event leaves measure at least $1/64$. The decoder theorem and Proposition prop:record finish the proof.

Take $m=15$, so $h=32768$, and $\zeta =1/200$. Condition (eq:jointsize) holds because $32768\ge 25600$. The vault has $19$ qubits and the cutoff emits $18$, leaving one. The early receiver register has $15$ qubits. On the positive-measure set of Theorem thm:joint, the decoder half-diamond error is at most $0.1$, equivalently the full diamond error is at most $0.2$.

This is an analytic existence example. It is not a sampled unitary, an executed decoder, or a hardware result.

The theorem proves a strict decrease somewhere in the retained history. It does not assert an exactly unimodal curve or identify that entropy event with the earliest successful recovery cut.

The Haar theorem is not the only finite existence result. The following construction supplies an exact channel and decoder.

Let $d=2^{\ell }$, $h=d^2$, $f=1$, and label $H,E$ by $d^2$ Pauli keys $z$. Let $\{W_z\}$ be the $\ell$-qubit Pauli operators, ignoring their irrelevant overall phases. Prepare

Apply the controlled unitary

and emit the $\ell$ ciphertext qubits, stopping before emitting any key qubit.

The endpoint complete isometry is

For (eq:padV),

The controlled inverse

exactly recovers the whole diary, including references.

At every ciphertext prefix $0\le k\le \ell$, for every diary input,

Thus the retained coherent curve reaches a plateau of $\ell$ bits at the cutoff, while the complete residual stays positive dimensional. Late radiation alone has recovery error $a_d$; augmented coherent radiation has error zero.

Proof. Orthogonality of the key labels makes $V$ an isometry. Tracing $B$ gives the accessible block channel, and tracing $RE$ gives the constant complete complement. The Pauli twirl gives the late-only replacement channel. Controlled inversion returns $X$ in every block, so the complete decoder is exact.

The late ciphertext marginal is $I_R/d$ for every input. Its $k$-qubit prefix is $I/2^k$. Both displayed entropy formulas follow, and Lemma lem:approx gives the late-only benchmark.

The decoder can be implemented with controlled Pauli gates on the received ciphertext and early key register. The batch input occupancy is $3\ell$ qubits; the ciphertext register itself can be returned as the diary output. The key and complete decoder environment are then discarded. No unpriced quantum output is used.

This example proves exact operational-island existence with a positive residual. It is a designed code, not evidence that Haar scrambling or gravitational dynamics automatically implement the controlled decoder.

For a measured factor $R$, define

The quantum name $R$ is consumed. The output is $YE$, not necessarily a purely classical system.

Every purely classical returned channel on a $d$-dimensional diary has recovery error at least $b_d=1-1/d$. The statement applies if all receiver outputs, including $E$, are measured and only their records are returned. It does not follow merely from measuring $R$ while keeping $E$ quantum.

Proof. Every decoder after a classical-output channel is measure-and-prepare, hence entanglement breaking EntanglementBreaking. Its Bell output is separable, and every separable state has Bell overlap at most $1/d$. Testing the Bell projector gives half-diamond error at least $1-1/d$. The last assertion is established constructively below.

Take $h=d$ and the fixed resource $|{\Phi }_d{⟩}_{HE}$. A Bell-basis transform on $LH$, followed by release of those two registers, has the expansion

with appropriate fixed Pauli phase conventions. One may include an untouched two-dimensional spectator $B$ prepared in $|0⟩$, so the residual space is positive dimensional.

Enter on $R$ gives

Reading $z$ and applying ${W_z}^{†}$ to the surviving quantum $E$ exactly returns $X$, including external references. The classical record $Y$ alone is independent of $X$ and cannot recover it.

This is the standard teleportation mechanism Teleportation. It does not reconstruct quantum information from a classical album alone. The quantum carrier is $E$, and the fixed entanglement resource has been included in the complete preparation.

For the exact encrypted construction of Section sec:exactexample, measuring the ciphertext does produce an entanglement-breaking channel even with the accessible key register: that register is already classical in the reduced channel. This is a property of that construction, not of all entanglement-assisted interfaces.

Enter records outcomes rather than returning the identity channel on the consumed measured factor. A legal continuation can nevertheless act on other surviving quantum factors and return the diary. The obstruction and the teleportation example concern different complete returned interfaces.

If $\mathcal{C}={\mathcal{T}}_{{\tau }_B}$, then for every residual observable $Z$,

Therefore complement constancy does not imply faithful reconstruction of an independently variable full matrix algebra $\mathcal{B}(B)$ from $A$.

Proof. Evaluate the residual expectation on every logical matrix unit. The constant-channel equation gives the displayed compression. It is scalar on the logical code. Recovering all states of an additional variable $B$ would be a different input and a different task.

A fixed residual state can of course be prepared without consulting the radiation. That trivial state preparation is not recovery of an unknown residual input.

Let $V:L\to \mathcal{H}$ be an isometry. There cannot be two commuting star representations of $\mathcal{B}(L)$ on $\mathcal{H}$ that preserve the code and both satisfy

for every logical $X$, when $d\ge 2$.

Proof. Commutation would give

Injectivity of $V$ would make every pair of logical matrices commute, which is false.

Independent commuting matrix algebras can exist on independent factors. The obstruction concerns two actions representing the same unknown logical input. The partner-algebra comparisons of Hawk03-A retain this distinction CheungHawk03A. Nothing here resolves the gravitational firewall assumptions.

Readout and forgetting of a residual factor equal its partial trace, including every untouched reference:

More generally, any unreported TP local operation on $B$ leaves the already-returned $A$ channel unchanged. For the nested release schedule, ${\epsilon }_{\mathrm{rec}}({{\mathcal{N}}_k}^U)$ cannot increase with $k$.

Proof. Expand an operator as ${\sum }_{i,j}{X}_{ij}\otimes |i⟩⟨j{|}_B$. Readout and discard return ${\sum }_i{X}_{ii}$. For an arbitrary TP residual channel, the adjoint fixes the identity, giving the same accessible reduction.

A later accessible channel has the earlier channel as postprocessing by discard of the additional cells. An earlier decoder can therefore be used at the later cut.

Neither operation moves $B$ into the receiver's laboratory. Changing the book label is not an additional access channel. Conditional postselection on a residual outcome would be a different instrument and must include its success and failure branches.

The main theorem has no nontrivial charge restriction. The following statement specifies one extension without inferring sector structure from a label.

Suppose a prepared interface is

with input-independent probabilities $p_{\alpha }$, and orthogonal sector supports in both accessible and residual outputs. If each sector satisfies