Hawk04-B Black-Hole Information as an Access Problem: Enter, Quantum Recovery, and Complete Residuals

We distinguish complete unitary preservation, subsystem quantum recovery, destructive observation, and operational discard in a reference-preserving channel formulation. A finite acyclic typed fragment represents allocation, specified unitary transformations, destructive computational readout, and classical forgetting. Quantum recovery returns a quantum register; terminal readout returns a classical record and is a different task.

For a complete isometry $V:C\to R\otimes B$, we give an explicit global recovery channel and prove that exact recovery from $R$ is equivalent to constancy of the complete complementary channel to $B$. A process retaining only classical outcomes cannot reconstruct an arbitrary $d$-dimensional quantum code with half-diamond error below $1-d^{-1}$. Unobserved readout of $B$, followed by discarding its outcomes, is exactly partial trace over $B$ and leaves the radiation channel unchanged. Its coherent dilation retains the information omitted from the classical interface.

Independently, a finite-band quartic source–scalar-reservoir Hamiltonian is constructed and approximated by finite-dimensional proxies with explicit complete-channel allowances. Separate qubit dictionaries preserve the accessible–inaccessible partition. For a commuting four-Majorana control, the exact errors and complete residual leakage are

Thus the complete output is reversible although neither output alone need recover the diary.

Access-preserving endpoint continuity and fixed-proxy recurrence are proved separately from syntactic termination. The gravitational implication remains conditional on a same-model unitary dilation, outgoing access construction, and constant endpoint residual. No positive noncommuting recovery window, physical field-to-memory receiver, quantum-gravity unitarity theorem, or gravitational evaporation endpoint is established.

{{0.93} The governing distinctions. \[

complete preservation \ \ recovery from a chosen subsystem,
quantum recovery \ \ destructive classical observation,
operational discard \ \ proved physical annihilation,
finite-program termination \ \ a physical evaporation endpoint.

The residual is an output ledger, not a deletion mechanism. }

The black-hole information problem asks whether a proposed evaporation description is compatible with complete quantum evolution. Hawking's radiation calculation and information-loss proposal provide its historical setting Hawking1975,Hawking1976. A finite circuit semantics does not derive that physical evolution.

This paper supplies an operational formulation in which the following questions have distinct mathematical objects:

1. Is the finite quantum code preserved by the complete evolution?

2. Does the declared accessible subsystem admit a quantum decoder?

3. What instrument produces a specified classical record?

4. Which outputs are omitted from the observer's description?

5. Does the same physical model have an access-preserving endpoint?

Every code comparison retains arbitrary external references.

The exact correction and no-hiding principles are established results KnillLaflamme,BraunsteinPati. The approximate channel conversion used here is due to Kretschmann, Schlingemann, and Werner KSW. We prove the exact statements locally, identify the imported approximate theorem explicitly, and make no priority claim for these principles. Superoperator semantics for quantum languages also has established precedent Selinger.

The contribution is the joint accounting of typed destructive readout, complete complementary outputs, access-preserving regulator certificates, and an exact dynamical negative control. In particular, a nonconstant residual is not promoted to a recoverable residual-only copy of the entire unknown code.

Hawk04-A separates complete preservation from radiation-only recovery and supplies the predecessor restore map, exact correction structure, four-Majorana control, and endpoint distinctions CheungHawk04A (Theorems thm:global, thm:exact, thm:four-majorana, and thm:endpoint). Those results are proved or refined below.

Landmark I characterizes Born probabilities and positive-branch destructive residual rays within its declared strategy class CheungI (thm:main). En03-A specifies a minimal linear circuit interface and distinguishes that interface from assertions about a larger calculus CheungEn03A. Here the finite fragment and its channel semantics are defined independently. Neither a field-to-qubit preparation nor the full hypotheses of Landmark I's characterization are reconstructed from its excerpt.

Hawk02-E supplies the regulated quartic source–scalar architecture, finite-proxy comparison discipline, and separation of source energy from complete residual independence CheungHawk02E. Hawk02-D concerns a different quadratic fermionic Hamiltonian and its scattering endpoint CheungHawk02D. No quadratic survival law or scattering amplitude is spliced into the present scalar model.

The apparatus and process distinctions in Em03-C, En02-B, En02-C, and En02-E are maintained CheungEm03C,CheungEn02B,CheungEn02C,CheungEn02E. In particular, the existence of a decoder is distinct from its execution under a particular resource budget; consuming syndrome qubits is distinct from consuming the recovered quantum output.

The analytic and OpenAI catalogue comparisons are confined to Appendices app:analytic and app:catalogue. They are documentary separations, not premises of the channel or Hamiltonian proofs.

The phrase “radiation recovery, then $\mathsf{Enter}$ on $R$” requires a correction. Applying $\mathsf{Enter}$ directly to the radiation register produces a record, not an arbitrary reference-entangled diary. The correct recovery interface is

with quantum return. A prescribed observation may subsequently consume $C_{\mathrm{out}}$. These two target channels are ${\mathrm{id}}_C$ and the chosen measurement channel, respectively.

Likewise, a linear program cannot silently leave a live residual wire out of its return statement. Section sec:programs therefore distinguishes:

• the complete program returning $RB$;

• the local decoder program $R\to C_{\mathrm{out}}$;

• a joint program returning $C_{\mathrm{out}}B$;

• the observer's radiation-only marginal;

• an explicitly added readout-and-drop implementation of that marginal in a finite qubit encoding.

The last implementation is not inferred from the autonomous Hamiltonian.

Let $C\simeq {\mathbb{C}}^d$, with $d\ge 2$, be the logical code. All output Hilbert spaces are separable. States are normal density operators, and channels are normal completely positive trace-preserving maps on trace-class operators. We abbreviate “completely positive and trace preserving” as CPTP.

We use

The diamond norm is the reference-stable trace-norm distance. For a finite input of dimension $d$, a reference of dimension $d$ suffices for channel discrimination; the usual finite-input conventions are as in Watrous.

For an isometry

define

The inaccessible output $B$ is complete: it includes uncollected modes, retained source degrees of freedom, displaced carriers, preparation purifiers, and inaccessible apparatus factors. A factor is not removed from this ledger merely because an observer does not control it.

For a state $\sigma$, the replacement channel is

Set

The infimum in (eq:erec) is over deterministic decoders. A heralded success branch, by itself, is not such a decoder.

Quantum recovery returns a register carrying the logical code, including its correlations with an arbitrary untouched reference.

A classical observation returns a state of a commutative algebra, possibly together with explicitly declared surviving quantum registers. Its target is the specified observation channel, not ${\mathrm{id}}_C$.

For a fixed code basis, write $E_{ij}=|i⟩⟨j|$. Every logical channel is determined by its action on all $E_{ij}$; agreement on populations $E_{ii}$ alone does not establish quantum preservation.

Let $\Phi ,\Psi$ have input dimension $d$, and define

Then

For common-output isometries,

The same allowance holds after a common partial trace.

CPTP pre- and postprocessing contract these distances. On fixed output spaces,

and

Proof. A normalized pure referenced input can, after an isometry on its reference support, be written as $(L\otimes I)|{\Phi }_d⟩$, where $\mathrm{Tr}(L^{†}L)=d$ and ${\Vert L\Vert }^2\le d$. Multiplication of the normalized Choi difference by $L$ and $L^{†}$ bounds its trace norm by $d$ times the Choi trace norm. Convexity handles mixed inputs. This proves (eq:choibound).

For normalized vectors $u,v$,

Apply this to referenced isometric outputs and then use convexity and partial-trace contraction.

The contraction statements follow from the definition of the diamond norm. For each fixed decoder, the triangle inequality changes its error by at most the channel comparison allowance. Taking infima in both directions proves recovery stability. Distance to the replacement-channel set is likewise one-Lipschitz.

Separate isometric embeddings of $R$ and $B$ preserve ${\epsilon }_{\mathrm{rec}}$ and ${\delta }_{\mathrm{full}}$. They do not exchange accessible and inaccessible outputs.

Proof. If $J:H\to H^{\prime }$ is an isometry, its channel has the CPTP retraction

where ${\tau }_H$ is any fixed state. Embedding candidate decoders or replacement states gives one inequality. Applying the retraction and contraction gives the reverse inequality. The two embeddings act on their respective local factors only.

Suppose unitary channels ${\mathrm{Ad}}_{U_j}$ have orthogonal Bell outputs. For real coefficients $a_j$,

In particular,

Every channel from $C$ has recovery error at most $a_d$.

Proof. A Bell input gives a diagonal output difference with eigenvalues $a_j$, proving the lower bound. Each unitary channel has diamond norm one, so the triangle inequality gives the upper bound.

For any replacement state $\sigma$, the Bell output is $I_Q/d\otimes \sigma$. Its overlap with the target Bell projector is $d^{-2}$. That projector therefore gives the lower bound $1-d^{-2}$.

The $d^2$ Weyl unitaries $X^p{Z}^q$ have orthogonal Bell outputs, and their uniform twirl is ${\mathcal{T}}_{I_C/d}$. The preceding equality gives the matching upper bound. For an arbitrary channel, the decoder which ignores its input and prepares $I_C/d$ attains this same allowance.

Let $J:C\to H_{\mathrm{in}}$ be an isometry, let $|\Omega ⟩$ be a fixed normalized ancillary state, and let $U$ be unitary on the complete physical input space. Then

is an isometry. Its complete channel has the CPTP left inverse

For every trace-class $X$, including operators on a code–reference space,

For the Bell input, the complete output satisfies

Proof. The composition gives $V^{†}V=I_C$. The two terms in (eq:restore) are completely positive. Their traces sum to $\mathrm{Tr}Y$, and $\mathcal{L}(VX{V}^{†})=X$.

An isometric embedding preserves nonzero singular values, proving the trace-norm identity. For the Bell input, the complete output is pure with $d$ equal Schmidt coefficients. Its reduced entropies are both ${\log }_{2}d$, and the joint entropy is zero.

If preparation is mixed, a fixed purification is included in the complete ancillary state. If preparation depends on the diary, the code-to-prepared-state map must itself be specified; it is not covered by a fixed-ancilla assumption.

The restore map is a complete-access inverse. Its existence does not imply an efficient inverse or a radiation-only inverse. It is the same global structure used in CheungHawk04A (thm:global).

For an isometry $V:C\to R\otimes B$, the following are equivalent:

1. one CPTP decoder on $R$ exactly recovers the entire code;

2. $\mathcal{C}={\mathcal{T}}_{\sigma }$ for some state $\sigma$;

3. $\mathcal{C}(E_{ij})={\delta }_{ij}\sigma$;

4. $$J_{\mathrm{n}}(\mathcal{C})=\frac{I_Q}{d}\otimes \sigma .$$

When these conditions hold, there are a separable Hilbert space $K$, a state ${\sigma }_K$, and an isometry $A:C\otimes K\to R$ such that

An exact decoder, defined on all trace-class operators on $R$, is

Proof. Linearity proves (ii) equivalent to (iii). Expansion of the Bell projector proves (iii) equivalent to (iv).

Assume (ii), and write the nonzero spectral decomposition

The index set is finite or countable. Constancy of the residual implies that every code vector is supported in $R\otimes \mathrm{supp}\sigma$. Write

The matrix-unit condition gives

Hence

extends to an isometry, and ${\sigma }_K={\sum }_{\alpha }{p}_{\alpha }|\alpha ⟩⟨\alpha |$ gives (eq:normalform). Both terms of (eq:decoder) are CP; their traces sum to $\mathrm{Tr}Y$. The decoder recovers every $E_{ij}$.

Conversely, let $L:R\to C\otimes G$ be a Stinespring isometry for an exact decoder. The normal Stinespring structure is the standard external dilation theorem KSW,Watrous. Set $Z=(L\otimes I_B)V$. For basis input $|i⟩$, the decoded marginal is the pure state $|i⟩⟨i|$, so

Exact recovery of $E_{ij}$ requires $⟨{\eta }_j,{\eta }_i⟩=1$. All ${\eta }_i$ are therefore one common normalized vector. Tracing the decoded register and $G$ yields $\mathcal{C}(E_{ij})={\delta }_{ij}\sigma$.

This is the exact correction/no-hiding structure associated with KnillLaflamme,BraunsteinPati and the normal form in CheungHawk04A (thm:exact). The local proof specifies the complete residual and the decoder's off-support completion.

If $\mathcal{N}$ is constant on the entire code, $B$ exactly recovers that code. If $\mathcal{C}$ is constant, $R$ exactly recovers it. For $d\ge 2$, both complementary outputs cannot independently recover the entire unknown quantum code.

Proof. Interchange $R$ and $B$ in Theorem thm:exact. An exactly recoverable output has constant complement, which cannot recover two distinct inputs.

A nonconstant $\mathcal{C}$ means that some logical inputs are distinguished by the residual channel. It does not mean that $B$ contains an independently recoverable copy of the entire code. Recovery may require joint access to $RB$. The exact control in Section sec:control demonstrates this.

A fixed pure residual gives $V|\psi ⟩=A|\psi ⟩|\omega {⟩}_B$. A fixed mixed residual gives the syndrome form (eq:normalform); radiation may be mixed for a pure diary and still be exactly recoverable.

Define the fragment

with observable algebras

Hybrid states are block diagonal on their classical registers. A bit is not an unrestricted qubit under a different name.

A judgement $P:T\to T^{\prime }$ consumes its input interface and returns its output interface. Sequential and tensor composition require matching interfaces. A live quantum wire is never duplicated. Every live quantum output is either returned or explicitly consumed by an admitted operation. Classical control acts blockwise.

The primitive channels are

Return commands designate the retained interface; they are not additional dynamics.

Only specified unitary constants are admitted. A semantic constant is not a claim of exact compilation over a fixed finite gate set, efficient synthesis, or physical realization. This independently defined ownership discipline agrees with the minimal interface supplied in CheungEn03A.

Let $S$ be $s$ measured qubits, and let $T$ be the surviving quantum register. Set

The observable instrument is

The measured names in $S$ are consumed. The surviving quantum output is $T$, not a renamed copy of $S$.

For a pure input,

No normalized residual is assigned to a zero-probability branch. Unnormalized branches suffice for complete-channel calculations.

These formulas are the destructive Born residuals in the supplied Landmark I characterization CheungI (thm:main). The instrument is defined here directly. This does not reproduce that manuscript's uniqueness theorem without its full strategy-class hypotheses.

For every finite acyclic instruction tree, and each fixed classical input block:

1. the denotation is a CPTP channel to the declared hybrid output;

2. there are quantum path maps $K_h$ satisfying

   $$\sum _h{K}_h^{†}{K}_h=I;$$

3. each reported output block is the sum of its unnormalized maps $X\mapsto K_{h}X{K}_h^{†}$;

4. the same denotation holds with an arbitrary untouched quantum reference;

5. every evaluation branch terminates within the finite height of the tree.

Proof. Allocation is an isometry, and a unitary gate is trace preserving. Destructive readout satisfies ${\sum }_x{M}_x^{†}{M}_x=I$. Classical control selects a complete channel on each classical block. Classical discard sums blocks without conditional normalization.

Induction composes path maps. Backward induction, using completeness at each readout node, gives ${\sum }_h{K}_h^{†}{K}_h=I$. Grouping histories with the same reported output gives the corresponding hybrid block. Tensoring each quantum path map with the reference identity proves complete positivity and the reference statement.

A finite acyclic tree has a finite maximum branch length.

The theorem is about the declared finite fragment. It is neither a complexity theorem nor a normalization theorem for an unprovided larger calculus. It uses the complete branch accounting emphasized in CheungEn02C.

A finite-dimensional isometry has a qubit encoding and a unitary extension after pure allocation.

A finite-dimensional CPTP channel can be represented by pure allocation, a specified unitary, destructive readout of its unused work register, discard of those work outcomes, and return of its quantum output.

Proof. Choose orthonormal basis dictionaries into sufficiently large qubit registers. Complete the columns of the encoded isometry to an orthonormal basis, obtaining a unitary extension.

For Kraus operators $D_j$, the map

is an isometry because ${\sum }_j{D}_j^{†}{D}_j=I$. A unitary extension implements this coherent map. Readout of $G$ and discard of the outcome give

The designated logical output remains quantum.

When a logical dimension is not a power of two, the logical space is an encoded subspace of its qubit register. Every physical channel is completed on invalid padded inputs, for example by fixed-state preparation. Logical recovery is tested on $C$, including references; it is not an assertion about arbitrary invalid padded inputs.

An infinite-dimensional decoder or syndrome space need not admit a finite program. Theorem thm:exact still supplies a normal CPTP decoder under its hypotheses.

For a finite proxy, let $Q_C,Q_R,Q_B$ denote qubit dictionaries for the logical code and the two outputs. These are register abbreviations, with encoded-subspace conditions understood. Let $U_{\mathrm{body},t}$ be a specified padded unitary whose restriction implements the prepared proxy isometry.

The complete program is

Its body is

a := new; (C,a) := EncodeAndPrepare(C,a); (R,B) := Evolve_t(C,a); returnQ(R,B);

Here new abbreviates the required finite pure allocation. The preparation and evolution are specified unitary extensions. Together they realize $U_{\mathrm{body},t}$ on the encoded input.

There is no $\mathsf{Enter}$ in this coherent body. Its logical denotation is ${\hat{\mathcal{V}}}_t$. The global restore map is (eq:restore); in finite dimension it is realizable by Proposition prop:finite-realization. This is the typed reading of complete preservation in CheungHawk04A,CheungEn03A.

The program begins after the carrier dictionary and preparation map have been defined. It constructs neither a quantum field nor a metric.

For a specified finite CPTP decoder, the local program is

with body

w := new; (Cout,G) := DecoderUnitary(R,w); y := Enter(G); dropBit(y); returnQ(Cout);

All unused work wires are included in $G$. The program consumes syndrome/work qubits, not the diary output. This interface is consistent with the quantum-return decoder programs in CheungEn02B.

Applying this program locally to the coherent output gives

Its body returns (Cout,B). The observer who controls only the first returned factor sees

Thus radiation-only access is a declared marginal of a complete output, not an implicit rule deleting a live quantum wire.

If one additionally wants a finite program whose observable output is only ${Q_C}^{\mathrm{out}}$, the residual must be consumed explicitly:

(R,B) := CoherentBody_t(C); x := Enter(B); dropBit(x); Cout := Decode(R); returnQ(Cout);

Its channel is again $\mathcal{D}{\hat{\mathcal{N}}}_t$, by Theorem thm:drop below.

This readout-and-drop is an external interface implementation. It is not a claim that the autonomous Hamiltonian measures $B$, nor that the omitted physical apparatus has disappeared. The complete dilation retains its apparatus outputs.

Directly measuring $R$ denotes a channel

not a quantum decoder.

For observation after recovery, let $F$ be a specified basis-selection unitary on the encoded output and append

Cout := F(Cout); x := Enter(Cout); returnBit(x);

The observable channel is ${\mathcal{M}}_F\mathcal{D}{\mathcal{N}}_t$. Its intended target is ${\mathcal{M}}_F$, not ${\mathrm{id}}_C$. For non-power-of-two codes, ${\mathcal{M}}_F$ includes the declared padded implementation and logical restriction.

If

then every prescribed terminal observation satisfies

The converse for one chosen observation is false.

Proof. The upper bound is CPTP postprocessing contraction. Computational dephasing preserves the computational-basis observation exactly but annihilates off-diagonal matrix units. That observation equality does not imply quantum recovery.

Evaluating pre-written coherent code and executing the particular command $\mathsf{Enter}$ are therefore different operations. A coherent finite program may terminate without any destructive readout.

A measure-and-prepare channel has the form

where $\{F_x\}$ is a POVM and the ${\sigma }_x$ are states. Such channels are entanglement breaking HorodeckiShorRuskai. The following proof gives the quantitative obstruction needed here.

Let $\mathcal{E}$ be a measure-and-prepare channel from $C$ to an arbitrary output, and let $\mathcal{D}$ be any CPTP decoder to $C$. Then

In particular, the bound applies to every finite adaptive process whose only retained output is classical.

Computational readout followed by basis-state preparation attains the bound.

Proof. The Bell-input output of $\mathcal{D}\mathcal{E}$ is separable across the reference–output partition. For a normalized product vector,

by Cauchy–Schwarz. Convexity gives Bell overlap at most $1/d$ for every separable state. Testing the Bell projector proves (eq:classical-obstruction).

A channel to a finite classical register has the form

A decoder from that register prepares a state for each outcome. Adaptive processing before the final classical output does not change this form.

For attainability, define

Computational dephasing is

Its $d$ Bell outputs are orthogonal. Lemma lem:unitary-mixture gives $d_{\diamond }({\Delta }_d,{\mathrm{id}}_C)=1-1/d$.

For a qubit, every record-only reconstruction has error at least $1/2$. The hypothesis is that only the classical register is retained: a surviving quantum memory, or quantum side information supplied to the receiver, is a different access task.

A classical record may identify a basis label or support repeated-preparation tomography. It does not preserve an arbitrary unknown quantum state and its reference correlations in one run.

For a finite encoded residual, define

by

x := Enter(B); return(R,x);

and define

by

x := Enter(B); dropBit(x); returnQ(R);

Neither program follows merely from the autonomous Hamiltonian.

The derived notation

denotes this finite destructive-readout-and-forget implementation; it is not a primitive assertion of physical annihilation.

For every joint state, including any external reference,

Consequently,

Its radiation-only recovery error is exactly the pre-existing ${\epsilon }_{\mathrm{rec}}({\mathcal{N}}_t)$.

More generally, every trace-preserving operation on $B$, with all its outputs ignored, leaves the state on the untouched reference and $R$ unchanged.

Proof. Partial trace in the computational basis is the sum of the diagonal $B$-blocks, which proves the first equality and (eq:forget-equality).

For the general statement, let ${\Lambda }_B$ be the local channel. Its Heisenberg adjoint is unital. For every bounded observable $A$ on the untouched factors,

All expectations on those factors are unchanged. This argument applies to normal channels on separable spaces; equivalently, in a Kraus representation, ${\sum }_j{L}_j^{†}{L}_j=I_B$.

This is no-signalling, not a statement that conditional radiation states cannot change. Conditioning on a supplied measurement outcome is a different operation.

For an infinite residual, partial trace remains well defined. The theorem does not assert a finite program that measures infinitely many modes.

Suppose residual readout produces the hybrid channel

A deterministic decoder on its classical interface has the form

where each ${\mathcal{D}}_x$ is CPTP on $R$. Moreover,

The inequality can be strict.

Proof. A channel on a direct sum of classical blocks acts by a CPTP channel on each block and adds the outputs. A decoder can ignore $K$, so every radiation-only strategy is available with $RK$ access. Example ex:assistance proves strictness.

For the destructive instrument (eq:Enter), define

This is an isometry. Tracing $E$ gives the observable hybrid readout instrument. The complete output $KET$ preserves the input globally, whereas the classical interface $K$, by itself, generally does not.

Proof. The identities $⟨x,y⟩={\delta }_{xy}$ and ${\sum }_x{M}_x^{†}{M}_x=I$ give ${W_{\mathrm{read}}}^{†}{W}_{\mathrm{read}}=I$. Tracing the orthogonal $E$-labels removes cross-outcome terms and yields (eq:Enter). Complete recoverability follows from Theorem thm:global.

The physical carrier $K$ in the coherent dilation is a quantum system. It becomes the observable $\mathsf{Bit}$ interface only after the inaccessible factor $E$ is omitted. A coherent superposition across outcome labels is not a state of the classical algebra.

For a fully measured qubit,

Neither marginal retains the unknown phase; joint $KE$ access does. There is no conflict with no-hiding, because neither marginal channel is constant on the entire qubit code.

Classical discard may be dilated by moving the record carrier into another inaccessible register. Under complete unitary implementation, that register and the other apparatus factors remain in the ledger. The observable trace does not prove physical disappearance.

This accounting is compatible with the joint-apparatus and coherent comparison discipline in CheungEm03C,CheungEn02C. No apparatus model from those manuscripts is silently substituted for the dilation specified here.

For

radiation is constant and the complete residual exactly recovers the diary. Computational readout of $B$ retains only basis populations; its bits cannot replace the diary by Theorem thm:classical. Readout-and-drop leaves radiation constant. A complete unitary measurement dilation remains globally reversible.

For a qubit, consider

The radiation channel is ${\Delta }_Z=(\mathrm{id}+{\mathrm{Ad}}_Z)/2$, with optimal recovery error $1/2$. Computational readout of $B$ gives a uniform bit $x$. Supplying $x$ to the observer and applying $Z^x$ to $R$ gives exact quantum recovery. The bit alone contains no diary information.

Proof. The unnormalized branch is

Correction by $Z^x$, followed by summation, returns $\rho$. Without correction, summation returns ${\Delta }_Z(\rho )$. Each outcome has probability $1/2$, independently of the input.

The two inputs $|+⟩$ and $|-⟩$ have identical dephased outputs. If a decoder has error $\epsilon$, the triangle inequality and contraction imply $1\le 2\epsilon$. The identity decoder attains $1/2$ by Lemma lem:unitary-mixture.

This example separates classical side information supporting a quantum correction from a classical replacement for the quantum diary.

The programming semantics does not construct the Hamiltonian below. The model follows the regulated quartic source–scalar architecture in CheungHawk02E,CheungHawk04A. We use units $\hslash =1$.

Let $N=2n\ge 4$, and let

act on ${\mathcal{H}}_S={\mathcal{F}}_{\mathrm{f}}({\mathbb{C}}^n)$. For frozen real coefficients $J_{ijkl}$, set

Each quartic monomial is self-adjoint with norm one, hence

Define

where ${s^{\prime }}_j\in \{-1,1\}$ and $\alpha ,\zeta \in \mathbb{R}$. For the estimates, allow any fixed finite family of even self-adjoint source operators $A_a$, $1\le a\le r$.

Fix

On ${\mathcal{F}}_{\mathrm{s}}(\mathfrak{h})$, let

Annihilation and the first slot of the one-particle inner product are conjugate-linear.

For frozen real $g_a$, set

Identity factors are suppressed. Define

The form-factor norms are

The Hamiltonian (eq:H) is self-adjoint on ${\mathcal{H}}_S\otimes D(H_B)$, is bounded below, and satisfies the form inequality

It generates $U(t)=e^{-\mathrm{i}tH}$. Every isometric diary loading with a fixed pure reservoir preparation is globally preserved at each finite time.

Proof. On the finite-particle core,

and the CCR imply

Consequently,

The square-root term has arbitrarily small relative bound with respect to $H_B$. Since $H_S$ is bounded, the Kato–Rellich theorem gives self-adjointness on the stated domain Kato. These field estimates are also standard in spin–boson Hamiltonian constructions HaslerHerbst.

For normalized form-domain vectors,

Together with $H_S\ge 0$, this proves (eq:formlower). The spectral theorem supplies the unitary group, and Theorem thm:global gives complete code preservation.

No ground-state, scattering, or evaporation theorem from HaslerHerbst is imported into this many-level model.

Choose a normalized simultaneous spin eigenvector $|B_s⟩$ of fixed source parity and let $\beta \ge 0$. Prepare

The bounds

prove that the preparation is well defined. This is motivated by Euclidean-filtered SYK preparations KourkoulouMaldacena, without importing a gravitational dictionary.

The microscopic diary is a qubit encoded in the even-parity sector of two carrier modes:

With a fixed occupation ordering, exchange those modes with two declared source modes by a fixed second-quantized fermionic swap:

The map $W:C\to {\mathcal{H}}_S\otimes {\mathcal{H}}_D$ is an isometry. It uses one diary-independent unitary. The displaced source state and its correlations remain in the carrier; that carrier is retained in the residual.

Choose orthonormal packet functions $p_1,\dots ,p_q\in \mathfrak{h}$, and set

With vacuum bath preparation and a fixed regrouping isometry $\mathcal{R}$,

All logical matrix units determine ${\mathcal{N}}_t$ and ${\mathcal{C}}_t$.

The source is prepared before the diary arrives. A heralded preparation may be conditioned on a diary-independent success event. A deterministic attempted protocol must retain failure outputs and compare its complete channel.

The channel theorems apply to arbitrary finite $d$. The specific two-mode diary construction above has $d=2$.

Let $P_m$ be the projection onto total bath occupation at most $m$. Put

At fixed $\mu ,\Lambda$, fixed packet family, and $T<\infty$, there is a finite-dimensional proxy with prepared isometry ${\hat{V}}_t$, embedded in the same complete output space, such that uniformly for $0\le t\le T$,

The same allowance holds for both reduced channels.

For $T>0$ and $0<b<1$, sufficient choices are

These give $b_{\mathrm{reg}}\le b$.

Proof. Use the interaction picture for $H_S+H_B$:

On sectors of occupation at most $k$,

An $\ell$-fold product starting from vacuum therefore has norm at most $(2G{)}^{\ell }\sqrt{\ell !}$. The ordered simplex has volume $t^{\ell }/\ell !$, so the $\ell$-th Dyson term is bounded by

These bounds are uniform over logical inputs and references.

Multiplication by any fixed power of the occupation number adds at most a polynomial in $\ell$, so the series converges in every fixed occupation-number graph norm on finite intervals. Since $h_B$ is bounded by $\Lambda$, this also controls the prepared energy domains. Termwise differentiation on these prepared vectors gives the Schrödinger equation; uniqueness for the self-adjoint Hamiltonian identifies the series with $U(t)$.

For the compressed Hamiltonian $P_{m}H{P}_m$, the Dyson terms agree with the exact terms through order $m$. Each additional projection is a contraction, and the same term bound applies to the compressed series. The prepared-isometry difference is therefore at most $2Z_m(2GT)$.

Partition $[\mu ,\Lambda ]$ into finitely many intervals $I_b$ of width at most $\Delta$, and replace $\omega$ on each interval by its midpoint, obtaining ${\tilde{h}}_B$. Then

On the truncated space both Hamiltonians are bounded. Duhamel's formula adds at most $Tm\Delta /2$ to the prepared-isometry comparison.

The active one-particle space

is finite dimensional, contains $\mathfrak{r}$ and all coupling vectors, and is invariant under ${\tilde{h}}_B$. Its orthogonal complement remains vacuum. Consequently the active proxy with total occupation cutoff $m$ is finite dimensional.

Its total-cutoff space embeds in

This is an embedding, not an equality: the ambient tensor product also contains states with total occupation greater than $m$. Embedding the remaining inactive modes as vacuum gives the common complete output space. The source and carrier stay in $B$.

Lemma lem:bounds converts the prepared-isometry comparison to (eq:breg), including references and reduced channels.

Finally, let $x=2GT$. The factorial bound gives

Later term ratios are at most $1/2$, whence $Z_m(x)\le 2^{-m}$. The occupation term $2Z_m(x)$ and the mesh term are each at most $b/2$ under the stated choices.

The theorem is a finite-time comparison at fixed positive infrared and finite ultraviolet cutoffs. Its constants do not provide uniform control under removal of either cutoff.

Let $p=\dim \mathfrak{f}$. Finite ambient output dimensions are

The factor $2^{n+2}$ accounts for the source and two-mode carrier. The total-cutoff proxy is a subspace of this ambient tensor product.

There are separate basis isometries

where

Their tensor product preserves the output partition, recovery error, and residual leakage. The encoded prepared proxy admits the coherent finite program of Section subsec:coherent-program.

Proof. Choose separate orthonormal basis dictionaries. Lemma lem:dictionaries proves invariance of the two tasks.

The proxy Hamiltonian acts on its invariant finite total-cutoff subspace. Its unitary extends to the finite ambient tensor product and then to the padded qubit space. The preparation isometry has a unitary extension after allocation by Proposition prop:finite-realization.

This is an abstract dictionary and semantic realization. Relabelling packet modes as qubits does not construct a compact local detector or a coherent field-to-memory receiver. The separate receiver obligations in CheungEm03C are not discharged by this proposition.

For the vacuum preparation above, define the source, bath, and interaction expectations by $E_S(t),E_B(t),E_I(t)$. On every finite interval,

where

Moreover,

and

Proof. The initial bath energy and interaction expectation vanish. Autonomous evolution conserves total energy. Taking expectations in (eq:formlower) gives $E_B(t)\le 2E_0+4{G_{\omega }}^2$.

The Dyson graph-norm estimates in Theorem thm:regulator give all fixed occupation moments on finite intervals. Since $h_B$ is bounded and the source is finite, the field commutators in the displayed currents are controlled by those moments. Finite-particle approximation therefore justifies differentiation of their expectations. The Heisenberg equation gives the source current, and total energy conservation gives the second identity.

An externally imposed switching schedule would introduce a work term. No such schedule, measurement, or discard is hidden inside (eq:H). Microscopic energy accounting is not a geometric evaporation law.

We use the finite-dimensional KSW information–disturbance theorem KSW (Theorem 3). In Schrödinger-picture full-diamond normalization, its recovery direction gives, for any replacement state $\sigma$, a CPTP decoder satisfying

Taking infima and converting to the present convention gives

Only finite complementary channels are used in this approximate step. The theorem supplies a channel conversion, not an estimate of interacting dynamics or an executable receiver.

Suppose a finite complete proxy has comparison allowance

on the common, access-preserving output space. For a specified finite residual state $\hat{\sigma }$, set

Then

There exists a finite proxy decoder whose full CPTP extension satisfies

If a specified finite CPTP decoder has

then its full extension satisfies