En07-C Hawking Radiation under Destructive Enter

Hawking's 1975 calculation identifies late-time particle emission for quantum fields on a classical collapse background, with a thermal factor modified by absorption probabilities. It does not construct a microscopic quantum-gravitational evaporation channel through the final Planck-scale regime. This paper supplies a self-contained finite-interface comparison under the Philosophy of Enter. A declared code is embedded isometrically into accessible and complete residual outputs. Exact reference-preserving recovery from the accessible output is equivalent to constancy of the complete complement, with an explicit deterministic decoder and a residual-rank dimension bound. Destructive basis readout followed by forgetting equals partial trace, whereas its complete coherent dilation remains isometric. Nearly constant accessible channels give recovery lower bounds; nearly constant complete complements give recovery upper bounds, with the approximate statement explicitly importing the Kretschmann–Schlingemann–Werner theorem. Finite counterexamples separate thermal or spectral data, selected entropy, conserved energy, and termination from quantum return. An access-preserving endpoint theorem combines convergence, complement, and receiver allowances under four same-model obligations. Neither those gravitational obligations nor an outgoing gravitational receiver is constructed here. Companion manuscripts are documentary comparisons, and no inherited gravitational quantum coefficient is admitted.

Keywords: Hawking radiation; quantum error correction; complementary channel; diamond distance; destructive measurement; Landauer principle; conditional evaporation endpoint.

The governing distinction is between preservation of a complete quantum output and recovery from a particular accessible output. A subsequent classical observation is a third operation, and thermodynamic reset is a fourth question. This distinction is documented in Cheung's En07-A, En07-B, and Hawk04-B CheungEn07A,CheungEn07B,CheungHawk04B.

The finite recovery structure used below is standard quantum error correction and no-hiding structure KnillLaflamme,BraunsteinPati. The approximate theorem uses the explicitly stated external input of Kretschmann, Schlingemann, and Werner KSW. The contribution here is a complete, typed comparison: the proofs, normalization, counterexamples, access restrictions, and endpoint error budget are stated in one place. No new microscopic law of black-hole evaporation is claimed.

For a nonrotating uncharged black hole, Hawking's late-time packet calculation gives, in the natural-unit convention,

for bosons, with the corresponding plus sign in the denominator for fermions. Here ${\Gamma }_{\omega \ell m}$ is the absorption probability, not an additional information-recovery certificate. The temperature is

when $\kappa$ denotes the horizon redshift rate in inverse-time units. If surface gravity is instead expressed as an acceleration ${\kappa }_{\mathrm{acc}}$, then $\kappa ={\kappa }_{\mathrm{acc}}/c$. These are external semiclassical statements, not conclusions of the finite theorems below Hawking1975 (Section 2, especially Eq. (2.29)).

Hawking's backreaction discussion associates net positive outgoing energy flux with decreasing black-hole mass and an ingoing negative-energy flux in the semiclassical description. Net mass loss requires the outgoing flux to exceed incoming energy. The same paper explicitly states that the quasi-stationary classical-metric approximation breaks down at the Planck-scale endpoint Hawking1975 (Sections 1 and 4). Consequently, its final-disappearance discussion is not used here as a construction of a complete microscopic endpoint isometry.

The historical estimate that sufficiently light primordial black holes would have evaporated “by now” belongs to the cosmological age assumptions used in the 1975 paper. It is not a present-day numerical estimate in this manuscript.

The entropy expression relevant to the historical comparison is

The abbreviated expression $S+A/4$ assumes the corresponding natural units. Black-hole mechanics and the generalized second law are separate gravitational and thermodynamic subjects BardeenCarterHawking,Bekenstein1974,Hawking1975. Neither an area theorem nor a generalized-second-law proof is derived from a code dimension in this paper.

1. A thermal state, an occupation spectrum, or one prepared radiation density operator is not automatically a replacement channel on a quantum code. A channel statement must control all logical operators, including off-diagonal operators and external references.

2. A verified nearly constant complete accessible channel obstructs recovery from that channel. In a complete isometric interface it also constrains possible complement-decoupling allowances. It is therefore inaccurate to say that such a channel estimate has no consequence for the complement. Thermal data alone, however, do not establish that channel estimate.

3. Complete unitary evolution does not automatically conserve an independently chosen Hamiltonian. Energy conservation requires its own commutation or intertwining hypothesis.

4. Complete disappearance is not promoted from a semiclassical extrapolation to a proved quantum-gravitational endpoint.

5. The four obligations introduced in Definition def:four specify this paper's same-model endpoint certificate. They are not a universal necessity theorem about every conceivable description of evaporation.

6. “XV-B not opened” refers to the admission-dependent downstream quantum exit. It does not deny the existence of supplied technical revisions of XV-B that prove conditional lifting interfaces while leaving the inherited full map unresolved.

boundary The mathematical target is a declared finite quantum interface. No Bogoliubov calculation, gravitational factorization, bulk–boundary identification, coherent gravitational receiver, Page curve, spacetime-emergence theorem, or microscopic evaporation endpoint is constructed. No statement below supplies the missing inherited characteristic quantum product.

Let $C,R,B$ be finite-dimensional Hilbert spaces, with $d=\dim C\ge 2$. A complete code interface is an isometry

Its complete, accessible, and complementary channels are

The factor $R$ is the declared quantum access. The factor $B$ contains every other output of the complete prepared dilation, including preparation purifiers, apparatus environments, lost carriers, and retained memories.

The word “complete” is a modeling requirement, not a consequence of naming one environmental marginal. It is essential to Theorem thm:exact and Example ex:purifier.

One way to obtain (eq:interface) is to specify an encoding $J:C\to H_{\mathrm{payload}}$, a fixed normalized ancilla $|\Omega ⟩$ containing any necessary purifiers, and a complete unitary $U$, and then set

The ancilla is fixed independently of the unknown logical input. The construction is mathematically isometric; whether it describes a gravitational preparation and evolution is a separate obligation.

For a linear map $\Lambda$ between matrix spaces, its diamond norm is

For channels with a common input and output,

For channel differences, the diamond norm can be evaluated on density operators with a reference of dimension equal to the input dimension. Thus the errors used here are worst-case reference-preserving errors, not errors for one selected input.

The finite stabilized norm facts in this convention are standard channel-norm facts; see the preliminaries of KSW. All distances between channels in this paper use (eq:distances).

For a density operator $\sigma$ on a specified output space, define

Set

Write

A decoder is deterministic: it is a completely positive, trace-preserving map on all of $\mathsf{L}(R)$. Postselected success is not substituted for (eq:recovery-error).

The minima exist because finite-dimensional state spaces and CPTP-map sets are compact, and the objectives are continuous.

The following statements hold.

1. CPTP pre- and postprocessing do not increase $d_{\diamond }$.

2. For replacement channels with the same input,

   $$d_{\diamond }({\mathcal{T}}_{\sigma },{\mathcal{T}}_{\omega })=d_{\mathrm{tr}}(\sigma ,\omega ).$$

3. If $V,W:C\to R\otimes B$ are isometries, then

   $$d_{\diamond }({\mathrm{Ad}}_V,{\mathrm{Ad}}_W)\le \min \{1,{\Vert V-W\Vert }_{\infty }\}.$$

   The same bound holds for either of their two partial-trace channels.

4. If ${\mathcal{N}}_1,{\mathcal{N}}_2$ have common interfaces, then

   $$\left|{\epsilon }_{\mathrm{rec}}({\mathcal{N}}_1)-{\epsilon }_{\mathrm{rec}}({\mathcal{N}}_2)\right|\le d_{\diamond }({\mathcal{N}}_1,{\mathcal{N}}_2).$$

   The analogous inequality holds for ${\delta }_{\mathrm{leak}}$ of two channels with a common residual output.

Proof. A channel has diamond norm one, so submultiplicativity proves (i). For (ii), applying the replacement-channel difference to a referenced state ${\rho }_{QC}$ gives ${\rho }_Q\otimes (\sigma -\omega )$. Its trace norm is ${\Vert \sigma -\omega \Vert }_1$.

For (iii), after adjoining any reference, expand

The trace norm is at most $2{\Vert V-W\Vert }_{\infty }{\Vert X\Vert }_1$. Divide by two; channel distances are at most one. Part (i) gives the marginal statements.

For (iv), use any decoder for ${\mathcal{N}}_1$ also on ${\mathcal{N}}_2$, apply the triangle inequality and (i), minimize, and interchange the two channels. The leakage statement follows by using the same replacement state in both comparisons.

Let $a:R\to R^{\prime }$ and $b:B\to B^{\prime }$ be isometries, and define

Then

This statement does not authorize a joint relabeling that moves an inaccessible logical register into the accessible factor.

Proof. An isometry $a$ has a CPTP left inverse

where ${\tau }_R$ is any fixed state on $R$. A decoder on $R^{\prime }$ restricts to one on $R$ by composition with ${\mathrm{Ad}}_a$; a decoder on $R$ extends to $R^{\prime }$ by composition with $L_a$. These two operations give the recovery equality.

For leakage, embedding a replacement state by $b$ gives one inequality. Applying $L_b$ to an arbitrary candidate replacement state on $B^{\prime }$ gives the other inequality by contraction. A joint transformation mixing $R$ and $B$ is not of the stated product form and need not preserve either optimization problem.

For the interface (eq:interface), the complete channel $\mathcal{V}$ has the CPTP left inverse

Moreover, for every reference $Q$ and every operator $X$ on $Q\otimes C$,

The left inverse requires access to the joint output $RB$.

Proof. The compression term in (eq:global-decoder) is completely positive. The other term is a positive-effect measurement followed by preparation and is completely positive. Their traces sum to $\mathrm{Tr}Y$, so ${\mathcal{L}}_V$ is trace preserving. On $VX{V}^{†}$, the second term vanishes and the first returns $X$. An isometric embedding preserves all nonzero singular values, including after tensoring with a reference identity. This proves (eq:global-preservation).

This complete-output restoration is also documented in CheungEn07A,CheungEn07B,CheungEn04A,CheungHawk04A. It does not give a decoder on $R$ alone.

Fix a code basis $\{|i⟩{\}}_{i=1}^d$ and let

The normalized Choi state of a channel $\Phi$ with input $C$ is

For $V:C\to R\otimes B$, the following conditions are equivalent.

1. A deterministic CPTP decoder satisfies $\mathcal{DN}={\mathrm{id}}_C$.

2. There is one state ${\sigma }_B$ such that $\mathcal{E}={\mathcal{T}}_{{\sigma }_B}$.

3. For every logical matrix unit, $\mathcal{E}(E_{ij})={\delta }_{ij}{\sigma }_B$.

4. The normalized complementary Choi state satisfies

   $$J_{\mathcal{E}}=\frac{I_Q}{d}\otimes {\sigma }_B.$$

5. For every residual observable $O\in \mathsf{L}(B)$,

   $$V^{†}(I_R\otimes O)V=\mathrm{Tr}(O{\sigma }_B)I_C.$$

If these conditions hold and $r=\mathrm{rank}{\sigma }_B$, there are a Hilbert space $K$ of dimension $r$, a state ${\sigma }_K$, and an isometry $A:C\otimes K\to R$ such that

An exact decoder on all of $\mathsf{L}(R)$ is

Every external reference is preserved.

Proof. Matrix units span $\mathsf{L}(C)$, so (ii) and (iii) are equivalent. Expansion of (eq:choi-definition) gives equivalence with (iv). Trace duality gives

which proves equivalence with (v).

Assume (ii), and diagonalize

The residual support condition implies

The coefficients of $\mathcal{E}(E_{ij})$ give

Thus the rule $A(|i⟩\otimes |\alpha ⟩)=|r_{i\alpha }⟩$ defines an isometry, and ${\sigma }_K={\sum }_{\alpha }{p}_{\alpha }|\alpha ⟩⟨\alpha |$ gives (eq:normal-form). The two terms in (eq:exact-decoder) are completely positive; their traces sum to $\mathrm{Tr}Y$. Substitution of (eq:normal-form) proves (i).

Conversely, assume (i). A finite Stinespring dilation of the decoder has an isometry $L:R\to C\otimes G$ Stinespring. Define

The decoded output on input $|i⟩$ is pure, hence

for normalized ${\eta }_i$. Exact recovery of $E_{ij}$ implies $⟨{\eta }_j\mid {\eta }_i⟩=1$. All ${\eta }_i$ therefore equal one vector $\eta$. Tracing $C$ and $G$ gives

which is (iii).

Equality of channels on all logical matrix units persists after tensoring with any reference identity. This proves reference preservation.

Theorem thm:exact is the exact correction/no-hiding structure associated with KnillLaflamme,BraunsteinPati. The explicit complete-complement and decoder formulations occur in the supplied Cheung manuscripts CheungEn07A,CheungEn07B,CheungHawk04A,CheungHawk03A. Their physical models are not identified with one another here.

Exact recovery implies

If ${\sigma }_B$ is pure, $\mathcal{N}$ is an isometric encoding. Equation (eq:dimension-bound) is not an area law.

Proof. The $dr$ vectors $r_{i\alpha }$ constructed in the proof of Theorem thm:exact are orthonormal in $R$. If $r=1$, (eq:normal-form) reduces to conjugation by an isometry $C\to R$. No geometric area or gravitational dynamics enters this count.

If $\mathcal{N}$ is constant on the entire code, the complete output $B$ exactly recovers the code. If $\mathcal{E}$ is constant, $R$ exactly recovers it. For $d\ge 2$, both complementary outputs cannot separately recover the entire unknown code, and both cannot be constant.

Proof. Interchange $R$ and $B$ in Theorem thm:exact. An exactly recoverable output has constant complement. A constant channel cannot recover two distinct logical states.

The measured factor in this section is explicitly $B$. A classical record is denoted by $Z$, so it is not confused with the quantum access factor $R$.

Choose an orthonormal basis $\{|z⟩\}$ of $B$, set

and define

The record $Z$ is classical: the output is diagonal in its declared basis. The measured quantum factor $B$ is not returned. Forgetting the record is ${\mathrm{Drop}}_Z={\mathrm{Tr}}_Z$.

This is the destructive primitive, not a nondestructive instrument that returns the measured register. The distinction is documented in CheungEn03A,CheungEn07A,CheungEn07B,CheungHawk04B.

On all operators, including all referenced operators,

A complete coherent dilation is the isometry

It satisfies

Proof. Write

Then

The same block expansion applies with an untouched reference. The vectors on the right side of (eq:enter-dilation) preserve inner products of basis vectors, so $W$ is an isometry. Tracing $F$ removes terms with $z\ne z^{\prime }$ and gives (eq:enter).

Equation (eq:enter-drop) does not assert physical annihilation of a state or its complete information. The complete dilation (eq:enter-dilation) preserves the unknown state and its references, whereas the restricted output after forgetting need not recover it. It also does not determine a heat, work, or memory-reset cost.

Proof. Theorem thm:global applies to $W$ and gives a complete-output left inverse. Restriction by partial trace changes the available output but not the isometry identity. No Hamiltonian, bath temperature, work reservoir, or resource-return condition occurs in (eq:enter-drop). The thermodynamic non-identification is made explicit in Proposition prop:no-landauer-from-channel.

For example, the coherent image of a superposition is

suppressing a spectator $R$. It is not two independent copies of that unknown superposition.

Let $\mathcal{M}:C\to Z$ be any channel with a purely classical output, and let $\mathcal{P}:Z\to C$ be any deterministic CPTP decoder. Then

The conclusion concerns record-only processing. It does not apply to an instrument that also retains an authorized quantum output.

Proof. A finite classical-output channel has the form

Only $\mathcal{P}(|z⟩⟨z|)={\rho }_z$ is relevant on this output. Consequently

is measure-and-prepare. Its Bell-input state is separable across $Q\mid C$.

For a normalized product vector $|a{⟩}_Q|b{⟩}_C$,

by Cauchy–Schwarz. Convexity gives the same Bell overlap bound for every separable state. Testing the projector $|{\Phi }_d⟩⟨{\Phi }_d|$ therefore gives trace distance at least $1-d^{-1}$ from the ideal Bell output. This test is one admissible referenced channel test, proving (eq:classical-bound).

A measurement of outgoing Hawking modes is thus a channel applied to quantum access; it is not itself a quantum-return decoder. A useful record may coexist with quantum return when the authorized quantum factor is retained. A record-only replacement cannot preserve arbitrary reference entanglement.

For every replacement channel with input dimension $d$,

For every channel with input $C$, ${\epsilon }_{\mathrm{rec}}\le q_d$.

Proof. Any decoder applied to a replacement channel produces another replacement channel ${\mathcal{T}}_{\tau }$ on $C$. Its Bell-input output is $I_Q/d\otimes \tau$, whose Bell overlap is $d^{-2}$. The Bell-projector test therefore gives error at least $q_d$.

For attainment, use the decoder that always prepares ${\tau }_C$. Let $U_{ab}$, $0\le a,b<d$, be the Weyl unitaries. Their uniform twirl is ${\mathcal{T}}_{{\tau }_C}$, so

The triangle inequality and diamond norm one of each channel give half-diamond norm at most $q_d$. The Bell test gives equality. The same constant decoder can be used after any channel, proving the final assertion.

Suppose

Every deterministic decoder obeys

This is a recovery lower bound.

Proof. Contraction gives $d_{\diamond }(\mathcal{DN},\mathcal{D}{\mathcal{T}}_{\omega })\le \eta$. The triangle inequality and the replacement-channel lower bound in Lemma lem:constant-error give

Distances are nonnegative.

The corresponding comparisons in CheungEn07A,CheungEn07B,CheungHawk04A must be read in this direction. Closeness of an accessible channel to a replacement is not a recovery upper certificate.

For a prescribed finite Hamiltonian $H_R$ and temperature $T>0$, one may define

Proposition prop:featureless applies if a model proves $d_{\diamond }(\mathcal{N},{\mathcal{T}}_{{\omega }_T})\le \eta$. Defining (eq:finite-gibbs), or matching a selected occupation law to it, does not prove this hypothesis.

In particular, (eq:hawking-occupation) is an occupation statement for specified semiclassical preparation and mode data. It is not a uniform half-diamond estimate on an independently declared microscopic code. Absorption factors, packet selection, and correlations also prevent replacing the entire outgoing continuum by an unqualified finite Gibbs density.

A finite selection of bosonic modes still has an infinite occupation space. Applying the finite theorems requires a specified finite encoding or cutoff and a reference-preserving allowance for the discarded sector. Such an allowance must not be silently omitted from the complete residual. Finite-packet and regulated comparison interfaces are documented in CheungEn02E,CheungHawk02E; they are not imported as gravitational constructions.

Let $R=C$, choose any full-rank Gibbs state $\omega$ on $C$, and compare ${\mathcal{N}}_1={\mathrm{id}}_C$ with ${\mathcal{N}}_2={\mathcal{T}}_{\omega }$. For the prepared input $\omega$,

Nevertheless

Both channels admit complete finite isometric dilations. Thus equality of the entire accessible state for one preparation, not merely equality of a spectrum, fails to determine recovery.

Proof. The recovery values follow from identity recovery and Lemma lem:constant-error. A dilation of ${\mathrm{id}}_C$ is $|\psi ⟩\mapsto |\psi {⟩}_R|0{⟩}_B$. A dilation of ${\mathcal{T}}_{\omega }$ is $|\psi ⟩\mapsto |{\chi }_{\omega }{⟩}_{RP}|\psi {⟩}_L$ with $B=P\otimes L$, where ${\chi }_{\omega }$ purifies $\omega$.

A proposed uniform thermal replacement comparison must specify which logical inputs are compared. If their macroscopic mass, angular momentum, charge, preparation, or access changes, a common replacement state is an additional claim. Holding those data fixed is a possible model specification, not a conclusion of the finite-interface theorem.

We import the finite-dimensional information–disturbance consequence of the Stinespring continuity theorem KSW (Section IV). For an isometric channel $\mathcal{N}$ with its complete complement $\mathcal{E}$, define full-diamond quantities

The imported estimates are

Ancilla enlargement in a Stinespring proof is mathematical dilation bookkeeping. It is not permission to give inaccessible physical factors to the decoder.

No companion Cheung manuscript is needed as a premise for this external theorem. Its half-diamond use is also documented in CheungEn07B,CheungHawk01B,CheungHawk03A.

Under Convention conv:ksw,

In particular, if a specified state satisfies $d_{\diamond }(\mathcal{E},{\mathcal{T}}_{\sigma })\le \zeta$, then a deterministic decoder exists with error at most

Also,

Proof. By the definitions, $a=2{\epsilon }_{\mathrm{rec}}(\mathcal{N})$ and $b=2{\delta }_{\mathrm{leak}}(\mathcal{E})$. The first inequality of (eq:ksw-full) becomes ${{\epsilon }_{\mathrm{rec}}}^2\le 2{\delta }_{\mathrm{leak}}$. The second becomes ${\delta }_{\mathrm{leak}}\le \sqrt{2{\epsilon }_{\mathrm{rec}}}$. Lemma lem:constant-error gives the independent upper cap ${\epsilon }_{\mathrm{rec}}\le q_d$, and channel distances give ${\delta }_{\mathrm{leak}}\le 1$. A specified replacement allowance upper-bounds the minimized leakage. Squaring (eq:approx-reverse) proves (eq:leak-lower).

A bound on $d_{\diamond }(\mathcal{E},{\mathcal{T}}_{\sigma })$ is a recovery upper certificate. A bound on $d_{\diamond }(\mathcal{N},{\mathcal{T}}_{\omega })$ is a recovery lower certificate. The output factor, the replacement state, the quantifier over code inputs, and the factor of one half in (eq:distances) are part of the statement.

If one complete interface satisfies

then

Equivalently,

In particular, exact accessible constancy and exact complement constancy cannot coexist for $d\ge 2$.

Proof. Proposition prop:featureless lower-bounds the optimized recovery error, while Theorem thm:approx upper-bounds it. If $\eta <q_d$, combine the two inequalities; if $\eta \ge q_d$, (eq:compatibility) is immediate. The exact conclusion also follows without KSW from Corollary cor:nohiding.

This is the precise correction to the phrase “thermal implies information loss.” Thermal data alone do not give (eq:featureless-hypothesis). If the complete channel hypothesis is actually proved, it gives a rigorous obstruction and a complement-allowance constraint.

For channels $\Phi ,\Psi$ with common input dimension $d$,

If ${\Vert \Phi (E_{ij})-\Psi (E_{ij})\Vert }_1\le r$ for every matrix unit, then

Proof. The Bell state is an admissible referenced input, proving the first inequality. A pure referenced input can be written, after a reference isometry, as $(A\otimes I_C)|{\Phi }_d⟩$ with ${\Vert A\Vert }_{\infty }^2\le d$. Applying $\Phi -\Psi$ and multiplying its Choi difference on the reference side gives a trace norm at most $d{\Vert J_{\Phi }-J_{\Psi }\Vert }_1$. Convexity and the channel-difference state optimization prove the upper bound.

Alternatively, write a pure referenced input vector as ${\sum }_i|v_i⟩\otimes |i⟩$, with ${\sum }_i{\Vert v_i\Vert }^2=1$. The matrix-unit hypothesis bounds the output difference by

Divide by two and use convexity.

These conversions are finite-input estimates, not dimension-independent replacements for a channel certificate. Related fixed-model certificates are documented in CheungEn07B,CheungEm03H,CheungHawk02E.

Let $\sigma \in \mathsf{D}(B)$ and suppose

Then

If a justified model approximation satisfies $d_{\diamond }({\mathcal{E}}_{\mathrm{phys}},\mathcal{E})\le \lambda$ on the same complete residual interface, then

A physical recovery conclusion additionally requires ${\mathcal{E}}_{\mathrm{phys}}$ to be the complete complement of the declared physical accessible channel.

Proof. Use Lemma lem:choi with $J_{{\mathcal{T}}_{\sigma }}=I_Q/d\otimes \sigma$, then apply Theorem thm:approx. For the physical comparison, use the triangle inequality. The final sentence retains the hypothesis needed to apply that theorem to the physical channel.

If two logical states satisfy

then

If $O=O^{†}$, ${\Vert O\Vert }_{\infty }\le 1$, and

then

Proof. For every replacement state $\sigma$,

Minimize and use (eq:leak-lower). Trace duality gives $w\le 2x$, proving the remaining statements.

A positive residual witness establishes dependence on some logical input. It does not establish residual-only recovery of the entire code.

Let $C=R={\mathbb{C}}^2$, and let $E$ and $P$ have bases indexed by $a=(a_1,a_2)\in \{0,1{\}}^2$. Prepare

Let $U_a=X^{a_1}{Z}^{a_2}$ be the four Pauli representatives and apply the controlled unitary to $C$ and $E$. The complete isometry is

Then

The selected residual $E$ is constant, but the complete residual $EP$ is not. The complete residual exactly recovers the unknown qubit.

Proof. Orthogonality of the key labels and unitarity of $U_a$ show $V^{†}V=I_C$. Tracing $EP$ gives the Pauli twirl, which is (eq:purifier-access). Tracing $P$ first removes off-diagonal key terms; tracing $R$ then gives (eq:selected-residual). Tracing only $R$ gives (eq:complete-purifier-complement). For example, an off-diagonal key coefficient with ${U_b}^{†}{U}_a=Z$ distinguishes the two $Z$ eigenstates. Finally, (eq:purifier-access) and Corollary cor:nohiding give exact recovery from $EP$.

The preparation is a purification of a mixed key. Removing $P$ from the ledger is precisely the error that would falsely suggest two constant outputs in a complete pure dilation. This example does not identify $P$ with a black-hole interior, a holographic boundary, or a gravitational partner algebra.

All von Neumann entropies in the finite arguments use natural logarithms and are dimensionless: $S(\rho )=-\mathrm{Tr}(\rho \log \rho )$. A thermodynamic entropy in conventional units includes the factor $k_{\mathrm{B}}$.

Choose any finite state ${\sigma }_K$, a purification $|\chi {⟩}_{KP}$, and a logical register $L\simeq C$. Let

Define two complete isometries, with tensor factors ordered as $R$ and $B$, by

Their accessible channels are

For every pure logical input, the two accessible outputs have the same eigenvalues and the same entropy $S({\sigma }_K)$. For the input $|0⟩⟨0|$, the entire accessible density operators are equal. Nevertheless

If $H_R=I_{C_R}\otimes H_K$, every energy-measurement distribution is also equal in the two models for all logical inputs.

Proof. Both maps preserve logical inner products. The partial traces give the displayed channels. For a pure $|\psi ⟩$, both accessible states are a rank-one logical state tensored with ${\sigma }_K$; therefore their nonzero spectra and entropies agree. The chosen input $|0⟩$ makes the density operators identical. Tracing $K$ exactly decodes the return model. The retain model is a replacement channel, so Lemma lem:constant-error gives its error. A spectral effect of $H_R=I\otimes H_K$ acts as the identity on the logical factor and has the same expectation in both outputs.

Taking ${\sigma }_K$ to be a Gibbs state makes the selected energy statistics thermal. It does not make the entire return channel constant. Thus selected entropy, occupation calibration, and logical recovery answer different questions. Sequential versions of this separation are documented in CheungHawk01B; finite occupation and source-channel comparisons are documented in CheungEm03E,CheungEm03F,CheungEm03G.

For a complete unitary $U$ on a finite system with declared Hamiltonian $H_{\mathrm{tot}}$, exact conservation means

For an isometry between declared input and output energy spaces, the stronger intertwining form is

These are additional hypotheses.

Equation (eq:unitary-energy) preserves the expectation of $H_{\mathrm{tot}}$ for every input state. Equation (eq:isometric-energy) preserves the corresponding input energy expectation. Neither property follows merely from $U^{†}U=I$ or $V^{†}V=I$. Neither property by itself implies outgoing recovery.

Proof. The first two statements follow by trace duality. For the nonimplication, a qubit bit flip is unitary but does not commute with $H=ϵ|1⟩⟨1|$ when $ϵ>0$. Failure of recovery despite exact conserving evolution is proved in Proposition prop:insufficiency.

Free-Hamiltonian conservation, as in the finite comparison documented in CheungEm03E, is not automatically a ledger for switching work, interaction energy, or reusable apparatus reset.

A thermodynamic cost claim must specify at least:

1. the physical memory and its initial state or input ensemble;

2. all relevant Hamiltonians and interaction/switching conventions;

3. reservoirs, their initial states and temperatures;

4. work storage, controllers, and usable side information;

5. the complete dynamics and final correlations;

6. definitions of heat and work, and conditions for returning reusable resources.

A channel on selected visible factors is not this package.

Landauer's original work concerns physical implementation Landauer1961. A precise finite thermal equality is available under additional assumptions ReebWolf; it is not inferred from (eq:enter-drop).

Let a memory $M$ and reservoir $L$ initially have the product state ${\rho }_M\otimes {\gamma }_L$, where

After a specified unitary on $ML$, write the state as ${{\rho }^{\prime }}_{ML}$ and define

Then

Here $D$ is quantum relative entropy. This proposition neither constructs a reset operation nor optimizes its work or cyclic resource cost.

Proof. Unitary entropy preservation and the product preparation give

Hence $S({{\rho }^{\prime }}_L)-S({\gamma }_L)=\Delta S_M+I(M:L{)}_{{\rho }^{\prime }}$. Using $\log {\gamma }_L=-\beta H_L-\log Z$ gives

Combining the identities proves the equality. Mutual information and relative entropy are nonnegative. The operation and its resource constraints are hypotheses, not outputs of the calculation.

Equation (eq:landauer-equality) is the finite entropy-balance identity of ReebWolf, included with its assumptions and a local derivation. It is not a gravitational generalized second law.

Equality of a selected visible channel does not determine a unique reservoir heat, work-storage change, or reusable-memory reset ledger, even when the appended complete dynamics conserves a specified total energy.

Proof. Take any fixed visible implementation and append independent qubits $W,L$ with

Prepare $W$ in $|1⟩$ and $L$ in its Gibbs state ${\gamma }_L=\mathrm{diag}(1-p,p)$, with $0<p<1$. Compare doing nothing on $WL$ with swapping $W$ and $L$. Both spectator operations conserve $H_W+H_L$, and neither changes the selected visible channel.

The first operation has $Q_L=0$. The swap leaves $L$ excited and has $Q_L=ϵ(1-p)>0$. The battery's energy decreases by the same amount. The two implementations therefore have different heat and resource-return ledgers despite the same visible channel. The swap consumes a resource; it is not claimed to be a cyclic cost optimum. Precisely that missing resource-restoration specification prevents a reset-cost inference from channel equality.

three ledgers{ Complete energy conservation is a statement about specified Hamiltonians and complete dynamics. Thermodynamic erasure cost is a statement about a specified implementation and resource restoration. Residual leakage is the dimensionless channel distance (eq:leakage-error). None is identified with either of the others. In particular, leakage is not heat, stress–energy, horizon area, or geometric backreaction. }

The endpoint statements below compare maps on the same code and retain the distinction between accessible and residual factors. They do not obtain an endpoint by merely declaring that a mass ledger has reached zero.

A same-model endpoint package for the declared code must supply:

1. Complete preparation. A justified code and a complete pure prepared isometry $P_0:C\to H_{\mathrm{tot}}$. Mixed ancillary preparations are purified and their purifiers remain in the complete system.

2. Complete evolution. A specified logical-input-independent unitary family $U_t$ on the complete system, on a justified time domain. Any energy-conservation claim has the separate hypotheses of Definition def:energy.

3. Justified access and endpoint comparison. Complete output identifications $F_t:H_{\mathrm{tot}}\to R_t\otimes B_t$ and factorwise isometries

   $$a_t:R_t\to R_{\ast },b_t:B_t\to B_{\ast }$$

   into common finite comparison spaces, preserving the declared access roles. Define

   $${\hat{V}}_t=(a_t\otimes b_t)F_t{U}_t{P}_0.$$

   There must be an endpoint isometry $V_{\ast }:C\to R_{\ast }\otimes B_{\ast }$ and a proved allowance

   $${\nu }_t=\min \{1,{\Vert {\hat{V}}_t-V_{\ast }\Vert }_{\infty }\}⟶0.$$

   The comparison maps are not allowed to move inaccessible logical registers into $R_{\ast }$.

4. Complete complement allowance. For ${\mathcal{E}}_{\ast }={\mathrm{Tr}}_{R_{\ast }}{\mathrm{Ad}}_{V_{\ast }}$, a state ${\sigma }_{\ast }\in \mathsf{D}(B_{\ast })$ and a proved bound

   $$d_{\diamond }({\mathcal{E}}_{\ast },{\mathcal{T}}_{{\sigma }_{\ast }})\le \zeta .$$

For a physical use of the package, all identifications and estimates must be justified within one microscopic model, on compatible domains and in a common regime of validity.

Definition def:four gives sufficient data for the theorem. It does not derive any of them from the outgoing occupation law. It is the finite form of the conditional endpoint framing documented in CheungEn07B,CheungHawk04B.

The raw and embedded access channels are

Similarly, ${\hat{\mathcal{E}}}_t={\mathrm{Tr}}_{R_{\ast }}{\mathrm{Ad}}_{{\hat{V}}_t}$.

Suppose Definition def:four is satisfied. Then

Under the KSW input,

There is an endpoint-optimal deterministic decoder ${\mathcal{D}}_{\ast }$ such that the fixed decoder ${\mathcal{D}}_{\ast }{\mathrm{Ad}}_{a_t}$ on the raw access has error at most

If $\zeta =0$, the endpoint recovery is exact and can use (eq:exact-decoder). If ${\sigma }_{\ast }$ is pure in that exact case, the endpoint radiation channel is an isometric encoding.

Proof. Every map ${\hat{V}}_t$ is isometric because it is a composition of isometries and a unitary. The first two inequalities follow from Lemma lem:stability. The triangle inequality with (eq:endpoint-complement) gives (eq:finite-leakage).

Choose an optimal endpoint decoder, which exists by compactness. Contraction and the triangle inequality give

proving (eq:fixed-endpoint-decoder). Theorem thm:approx gives ${\epsilon }_{\mathrm{rec}}({\mathcal{N}}_{\ast })\le \sqrt{2\zeta }$. Alternatively apply that theorem directly to (eq:finite-leakage). The constant decoder gives the independent cap $q_d$. Proposition prop:access-embedding identifies the optimized raw and embedded recovery errors and proves (eq:finite-endpoint-recovery).

For $\zeta =0$, (eq:endpoint-complement) is exact channel equality. Theorem thm:exact and Corollary cor:dimension give the remaining conclusions.

Suppose an actual quantum-output receiver gives a channel ${{\mathcal{N}}_t}^{\mathrm{recv}}:C\to R_{\ast }$ with a proved allowance

Then

A specified fixed endpoint decoder has error at most $\min \{1,{\alpha }_t+{\nu }_t+{\epsilon }_{\mathrm{rec}}({\mathcal{N}}_{\ast })\}$. The receiver assumption is not a consequence of an observable probability comparison or a classical count record.

Proof. Apply contraction and the triangle inequality once more to the fixed decoder in Theorem thm:endpoint. Use the endpoint upper bound and the independent constant-decoder cap. The final sentence distinguishes the stated quantum-output channel hypothesis from weaker data.

The coherent-access distinction is important in the documentary comparisons CheungEm03D,CheungEm03G. A decoder on an abstract packet factor does not construct a local field-to-memory receiver.

If, in addition to the endpoint package,