Hawk02-B Hamiltonian-Derived Emission and Quantitative Recovery in a Finite Evaporation Model

We construct an excitation-conserving sequential-emission model and derive its complete emission isometry from finite-dimensional Hamiltonians. A $d$-dimensional code occupies a degenerate excited memory sector and exchanges its excitation with initially vacant radiation cells through prescribed pulses. After $N$ pulses the survival probability is $q_N={{\prod }_{j=1}}^N{\cos }^2{\theta }_j$. The optimal single-cell recovery error in half-diamond distance is $q_N(1-d^{-2})$, and the complete single-cell residual leakage has the same exact value. For $K$ emission mechanisms carrying an arbitrarily entangled $K$-cell code, the exact recovery error is $1-[1-q_N(1-d^{-2}){]}^K$, hence at most $K{q}_N$. A Duhamel estimate supplies a compatible finite-time implementation budget.

Emission-time dephasing alone does not destroy the logical code: time-resolved, label-coherent recovery remains optimal. Internal-label dephasing, destructive label-blind records, incomplete collection, and decoder-mode mismatch have different operational consequences. Entropy turnover is calculated but is not used to certify recovery.

A supplied execution record reports successful completion of all 32 predicates in the finite-matrix diagnostic script at $d=2$, $N=3$, ${\theta }_j=\pi /4$, with a memory-label detuning of strength $10^{-4}$ and numerical tolerance $2\times 10^{-10}$. The corrected exact targets are $q_N=1/8$, single-cell recovery and leakage $3/32$, and two-cell recovery $183/1024$. These reported checks support the selected matrix identities and Bell witnesses; they do not replace the analytical proofs or constitute diamond-norm optimization. The model is an externally scheduled, bare-energy-conserving evaporation analogue, not gravitational Hawking radiation or an autonomous microscopic black-hole model.

The construction follows \[

code and Hamiltonian complete emission isometry
accessible channel and complement recovery and perturbation certificate.

The transfer channel is calculated, not selected by matching a radiation occupation spectrum.

Repeated quantum interactions provide an established Hamiltonian framework AttalPautrat. Here the finite collision sequence is the specified dynamics itself: no continuous-time limit or master-equation approximation is used. The collision-model overview Ciccarello supplies background only.

After identifying the radiation code, the accessible channel is a quantum erasure channel. This mechanism is standard Erasure. The contribution is a complete finite Hamiltonian-to-certificate application, including exact recovery constants, complete residual accounting, many-cell release, finite-time robustness, and access controls. No new erasure-channel mechanism or capacity theorem is claimed.

Hawk02-A records a conditional certification interface rather than the Hamiltonian instantiated here CheungHawk02A ({thm:main, thm:sequential}). Em03-E is a different Hamiltonian-derived finite emission model, with its own key and environmental accounting CheungEm03E. No early key is generated or assumed here.

References attributed to Cheung, N., identify the manuscript versions and documentary excerpts supplied with the research programme. They are cited as comparison records, not as independently verified substitutes for the proofs below. The supplied corpus locator is d2c1e6b3, retrieval-3; it is not a public archive manifest or evidence of a complete corpus search.

The execution evidence is the supplied Hawk02-B Optional Diagnostic Script Execution Summary, filename summary (20).txt, including its transcription of manifest.json ExecutionRecord. It reports execution on October 10, 2026, at 16:28:41.703102+00:00 UTC.

This revision incorporates that report. It does not claim an independent rerun, direct inspection of the binary matrices.npz, or verification of the reported source hash. All mathematical conclusions remain established by the proofs. The computational report has a narrower, explicitly described role.

1. The execution-summary prose briefly suggests $q_N=1/512$ before correcting itself. For its $N=3$ run,

   $$q_N=({\cos }^2(\pi /4){)}^3=1/8.$$

   The value $1/512$ belongs to $N=9$, used separately in the analytical feasibility example.

2. The statement that the run confirms “all analytic claims” is too broad. The script checks selected finite matrices and witnesses for one configuration. It does not optimize over decoders, prove arbitrary-input bounds numerically, calculate all entropy claims, or test general perturbations.

3. Boolean success at a tolerance is not exact-arithmetic verification. The recorded manifest contains pass/fail predicates, not their individual residual magnitudes. An additional prose claim that an isometry residual is exactly zero is not used below.

4. The undefined normalized radiation mode at $q_N=1$ is avoided by introducing an unnormalized emission operator first. The retaining comparison uses this operator at all endpoints.

5. The fixed-per-event feasibility criterion is stated with $\eta >0$ before division by $\eta$; the case $\eta =0$ is treated separately.

6. Historical “unexecuted” wording inside the version-1 source listing is retained as provenance, not as the evidence status of this revision. The reported hash belongs to the executed extracted file, not to this revised .tex file.

All Hilbert spaces are finite dimensional. We set $\hslash =1$. Entropy is in bits; logarithms in pulse-count estimates are natural. Write

For identified state and channel spaces, use

The diamond norm is the stabilized induced trace norm. For differences of channels with a $D$-dimensional input it may be evaluated on density operators with an untouched reference of dimension $D$. These distances contract under channel pre- and postcomposition Watrous (Chapter 3).

Define

for an isometry $V$ and a density operator $\tau$. For a channel $\mathcal{N}:C\to A$ and a specified complete complement $\mathcal{C}:C\to B$, define

where $\mathcal{D}$ is CPTP. Finite-dimensional compactness gives attainment of both infima.

The external reference is not available to the decoder. A half-diamond recovery bound controls every code–reference state, not merely a selected pure code input.

For a $d$-dimensional system, let

Then

Proof. Temporarily index a basis by $0,\dots ,d-1$. With $\zeta =e^{2\pi \mathrm{i}/d}$, define

where addition is modulo $d$. Summation on matrix units gives

Separating the identity term and using that the half-diamond distance between channels is at most one proves $d_{\diamond }(\mathrm{id},{\mathcal{T}}_{I/d})\le 1-d^{-2}$.

Let

For every density operator $\sigma$,

The Bell-projector test distinguishes ${\Phi }_d$ from the replaced output with probability difference $1-d^{-2}$. It supplies the matching lower bound for every $\sigma$.

Suppose ${\Phi }_j,{\hat{\Phi }}_j$ are channels on common retained spaces and $d_{\diamond }({\Phi }_j,{\hat{\Phi }}_j)\le {\eta }_j$. Then

Proof. Insert one channel replacement at a time:

Contractivity and the triangle inequality prove the bound.

This is the locally restated comparison architecture of Hawk02-A and VII-G CheungHawk02A,CheungVIIG. It is not a discretization error of the ideal finite sequence.

Fix $d\ge 2$, $N\ge 1$, and $\omega >0$. Let

The code ${\mathcal{H}}_C={\mathbb{C}}^d$ has embedding

Set

Each radiation cell begins in $|0⟩$. Every code state therefore has bare energy $\omega$.

Let $I_j$ be chronologically ordered disjoint intervals. The real integrable pulse $g_j$ is supported on $I_j$ and prescribed independently of the code and reference. In the interaction picture, define

where

Other cells are acted on by the identity. Let

Previously emitted radiation does not reinteract.

The interval propagator is

where

It is unitary on the full space and satisfies

In particular,

The sequence acts linearly on arbitrary reference-entangled inputs.

Proof. The mutually orthogonal spaces $\mathrm{span}\{|a,0⟩,|0,a⟩\}$ are invariant. On each, $S_j$ is the exchange Pauli matrix; it vanishes on their orthogonal complement. Thus ${S_j}^2=P_j$ and ${S_j}^3=S_j$, which gives (eq:Uj). All Hamiltonians within an interval are scalar multiples of $S_j$, so time ordering reduces to the pulse area. The expression solves the finite-dimensional Schrödinger equation almost everywhere for integrable pulses.

Both members of each coupled pair have energy $\omega$. This proves the energy commutator and (eq:rotation). Tensoring all propagators and the preparation with $I_Q$ proves the reference statement.

The Schrödinger-picture Hamiltonian is $H(t)=H_0+g_j(t)S_j$ during a pulse. The conservation claim concerns $H_0$, not a time-independent total Hamiltonian including a switching apparatus. No autonomous controller, clock, or work-storage system is constructed. On the prepared one-excitation sector, free evolution adds a common phase, so it does not change the derived channels. The decoder is likewise an allowed CPTP laboratory operation, not a separately energy-priced apparatus.

Write

and let $|a;j⟩$ denote excitation label $a$ in cell $j$ and vacuum elsewhere. Define

Introduce the unnormalized emission operator

The Hamiltonian sequence induces

Furthermore,

When $q_N<1$, the normalized embedding

has orthonormal columns and

With complete ideal outputs $A=R_1\cdots R_N$ and $B=M$, the two channels, for all $q_N$, are

For $q_N<1$ the accessible formula is equivalently

All formulas are identities on operators and on reference extensions.

Proof. For one interval, (eq:Wunnormalized) is (eq:rotation). After emission at an earlier interval the memory is in $|0⟩$, so a later pulse with a fresh vacant cell acts trivially on that branch. The surviving branch acquires $\cos {\theta }_j$, or emits $-\mathrm{i}\sin {\theta }_j{\prod }_{k<j}\cos {\theta }_k$. Induction proves the complete formula.

Let $q_0=1$ and $q_j=q_{j-1}{\cos }^2{\theta }_j$. Then

Orthogonality of the time and label basis vectors proves ${L_N}^{†}{L}_N=(1-q_N)I_C$. The memory ground and excited sectors are orthogonal, as are radiation vacuum and excitation sectors, so ${W_N}^{†}{W}_N=I_C$.

Tracing memory kills the cross terms because $⟨0|J_C=0$. Tracing radiation kills them because $⟨\mathrm{vac}|L_N=0$. The respective remaining traces give (eq:Nall) and (eq:C).

If $q_N=1$, $L_N=0$, the accessible channel is constant, and the complement is the memory code embedding. If $q_N=0$, the final memory is the fixed state $|0⟩$ and radiation is an isometric code.

Every added mixed auxiliary is purified. Its purifier, inaccessible controller registers, detector environments, discarded radiation, and other inaccessible dilation outputs belong to the complete residual unless explicitly made accessible. The ideal assignment $B=M$ does not justify ignoring new outputs.

This agrees with the complete-output accounting recorded in Hawk02-A and Em03-E CheungHawk02A,CheungEm03E.

For $q_N<1$, let $P_N=J_N{J}_N^{†}$ and define

For $q_N=1$, define ${\mathcal{D}}_N={\mathcal{T}}_{I_C/d}$. The unused-subspace rule is fixed and trace preserving.

The coherent-mode decoder receives all radiation cells, their internal labels, and knowledge of $J_N$. It receives neither memory nor reference, uses no early key, and does not postselect. Section sec:controls gives a less demanding time-resolved decoder with the same optimum.

The declared decoder is CPTP, and

For all $q_N\in [0,1]$,

Proof. For $q_N<1$, compression and complement measure-and-prepare are completely positive and their traces sum to $\mathrm{Tr}Y$. If $\{e_b\}$ is an orthonormal basis of ${P_N}^{\perp }$, a Kraus family is

The Kraus effects sum to $I_R$. Direct substitution proves the decoded formula. Lemma lem:replacement yields error $q_N{c}_d$. The same conclusion holds for the constant endpoint.

For an arbitrary CPTP decoder, use a Bell input ${\Phi }_d$. The success branch contributes at most $1-q_N$ to decoded Bell overlap. The vacuum branch is independent of the input and has reference marginal $I_Q/d$; its contribution is exactly $q_N/d^2$. The Bell test therefore gives error at least $q_N(1-d^{-2})$.

The argument restates the Bell-witness architecture reported in Hawk01-B without importing its constants CheungHawk01B. At complete discharge it agrees with the constant-complement criterion recorded in Em05-B CheungEm05B ({thm:constantComplement}).

For the complete ideal complement,

An optimal comparison state is

For every ${\rho }_{CQ}$,

Comparison with the discharged state alone has distance $q_N$.

Proof. For the stated comparison,

Isometric output embedding preserves the diamond norm. Lemma lem:replacement proves the upper bound.

For any alternative ${\tau }_M$, set $P_{\mathrm{ex}}=J_C{J}_C^{†}$ and $t=\mathrm{Tr}(P_{\mathrm{ex}}{\tau }_M)$. On a Bell input the actual complement output is

Pinch the output difference by the projections

Their block trace norms sum to

For $t\le q_N$, half this is $q_N-t/d^2\ge q_N{c}_d$. For $t\ge q_N$, using $|q_N-t/d^2|\ge q_N-t/d^2$ gives at least $t{c}_d\ge q_N{c}_d$. Pinching contracts trace norm, so this is a lower bound on the original distance for every ${\tau }_M$.

The reference estimate follows from the diamond bound. For comparison with $|0⟩⟨0|$, the difference is $q_N$ times two channels with orthogonal output supports. Their half-diamond distance is one.

General information–disturbance results relate recovery and complementary decoupling KSW. Here the constants are proved directly in the stated half-diamond convention.

Because the residual memory space is fixed, (eq:uniformdecoupling) is uniform as $q_N\to 0$. It is not an assertion that arbitrarily long implemented evolution remains accurate.

Take $K$ memory cells with distinct radiation cells $R_{\ell ,j}$. The code is

Allow arbitrary entanglement between cells and with a reference.

Ideal interactions act on distinct cell mechanisms. Their complete map factorizes, up to output ordering:

This factorization of maps does not assume a product input. Write $q_{\ell }$ for the individual survival probabilities.

The product of the declared single-cell decoders is optimal:

For identical mechanisms,

The complete residual has the sufficient bound

No exact joint leakage optimization is required or asserted.

For identical mechanisms, the initial bare energy is $K\omega$, the probability of full discharge is $(1-q_N{)}^K$, and

The emitted-excitation count is binomial with parameters $K$ and $1-q_N$, for every input code state.

Proof. The decoded map is

Each factor is a Weyl mixture with identity weight $1-c_d{q}_{\ell }$. The product mixture has identity weight ${\prod }_{\ell }(1-c_d{q}_{\ell })$. Separating that term gives the upper bound (eq:manyexact).

For the lower bound, use a product of $K$ Bell states between code cells and reference cells. Let $F$ be the erased-cell subset. Its probability is

On that branch the erased references are independent and maximally mixed. After any decoder the normalized state has form

Its overlap with the full Bell projector is

The total overlap is at most

The Bell test proves the matching unrestricted-decoder lower bound.

The conservative inequalities follow from the union bound.

For leakage, compare with the product of the single-cell comparison states. The orthogonal memory sector $F$ has weight $p_F$ and contains an embedded copy of ${\mathrm{Tr}}_{F^c}X$. Its distance from the corresponding replacement is at most $p_F(1-d^{-2|F|})$, by Lemma lem:replacement in dimension $d^{|F|}$ and partial-trace contractivity. Summing gives (eq:manyleak). The distance to this particular product comparison is attained by the product Bell input, but alternative joint comparison states are not optimized here.

Finally, each emitted/vacuum flag effect is a scalar multiple of the cell-code identity. The product flag effects therefore have probabilities $p_F$ for every input, including entangled inputs. The branch emits $K-|F|$ excitations, giving the count and energy laws.

The all-discharged-event decoder supplies a valid $1-(1-q_N{)}^K$ upper bound. Partial-discharge branches also retain useful quantum information; their inclusion explains the smaller exact optimum (eq:manycommon).

For each event $e$, on common finite retained spaces, let

where $E_e(t)$ is Hermitian and integrable.

The event propagators satisfy

The allowance also holds after common preparation isometries, with arbitrary references.

Proof. Duhamel's formula is

Unitarity bounds the integral by $h_e$.

For every amplified operator $Z$, insert one unitary replacement on each side. The trace norm of the output difference is at most

Take the stabilized supremum, divide by two, and apply preparation contractivity.

With $\hslash$ restored, $h_e$ has an additional ${\hslash }^{-1}$. This channel allowance is not Em06-B's classical vector-stability estimate CheungEm06B ({thm:input-stability}).

Let the ideal preparation embed the code, append all vacant radiation, and append any declared fixed auxiliary states. For complete-output claims, auxiliaries are purified and retained.

The implementation specifies:

1. ideal and actual complete preparation maps $\mathcal{P},\tilde{\mathcal{P}}$ with $d_{\diamond }(\tilde{\mathcal{P}},\mathcal{P})\le e_{\mathrm{prep}}$;

2. event Hamiltonian deviations with integrals $h_e$;

3. a collected-register channel $\tilde{\mathcal{L}}:R\to R$ with $d_{\diamond }(\tilde{\mathcal{L}},{\mathrm{id}}_R)\le e_{\mathrm{collection}}$;

4. a decoder $\tilde{\mathcal{D}}:R\to C^{(K)}$ with $d_{\diamond }(\tilde{\mathcal{D}},{\mathcal{D}}^{(K)})\le e_{\mathrm{decoder}}$.

All allowances are nonnegative. Space identifications and padding are fixed before comparison. Every inaccessible dilation output is included for complete residual claims.

A preparation estimate for one selected input is not (I1). For an otherwise identical preparation using different fixed auxiliary states, their trace distance gives an accessible preparation-channel allowance. Comparing chosen purifications may require a separate complete allowance.

Let ${\tilde{\mathcal{N}}}^{(K)}$ be the actual precollection radiation channel. Define

Under Assumption ass:implementation, put

and

Then

For identical mechanisms,

For one cell the ideal term is $q_N{c}_d$.

If ideal and actual complete outputs are identified before collection, their residual channels satisfy

An accessible collection allowance alone does not compare particular complete collection dilations with the same linear error.

Proof. Lemmas lem:duhamel and lem:telescoping bound the complete dynamics by $e_{\mathrm{comp}}$. Partial trace yields the same accessible comparison. Insert collection and decoder replacements; contractivity gives

Add Corollary thm:B5.

The residual comparison follows by partial trace of the complete dynamical comparison, followed by comparison with the ideal product replacement state.

A distance between accessible collection channels does not, by itself, compare their chosen isometries in operator norm. That requires a specified complete comparison or an appropriate Stinespring continuity estimate KSW.

Recovery permits arbitrary integrable $E_e$. Actual bare-energy conservation additionally requires $[E_e(t),H_0]=0$ and an appropriate initial sector. It is not implied by a small norm error.

For $0\le H\le H_{\max }I$, a trace-distance comparison of size $e_{\mathrm{comp}}$ bounds the expectation discrepancy by $H_{\max }{e}_{\mathrm{comp}}$. The full $KN$-cell radiation Hamiltonian has maximum energy $KN\omega$, not $K\omega$, unless conservation of the initial sector is separately established.

For a constant $0<\theta <\pi /2$, set

Fix $K\ge 1$ and $0<\epsilon <1$. Let

Use $K{N}_{\ast }$ serial events of duration $\tau >0$. If

then ${\epsilon }_{\mathrm{implemented}}\le \epsilon$.

The allowed set contains nonzero bare-energy-conserving perturbations, for example constant event perturbations

Proof. $K{r}^{N_{\ast }}\le \epsilon /2$, and the accumulated implementation allowance is at most $\epsilon /2$. Apply (eq:robustrequested). Since ${\Vert S_e\Vert }_{\infty }=1$ and $[S_e,H_0]=0$, the displayed perturbations have the claimed norm and conservation.

Suppose $h_e\le \eta$ for each of $KN$ events and the remaining allowances sum to $e_0<\epsilon /2$. For $\eta >0$, the sufficient conditions

are simultaneously feasible if

For $\eta =0$, $N=N_{\ast }$ is sufficient.

Proof. The first condition holds for integers $N\ge N_{\ast }$. For $\eta >0$, the second holds up to the displayed integer upper bound. For $\eta =0$ it is independent of $N$.

This supplies a finite accuracy regime, not a claim that residual information can be reduced indefinitely at fixed positive event error.

{Analytical $N=9$ example; not the reported run.} For $d=2$, $K=2$, $\theta =\pi /4$, and $\epsilon =1/100$, take $N_{\ast }=9$. Then $q_N=1/512$ and $K{q}_N=1/256$. Fixed allowances at most $1/400$ and $18$ event allowances each at most $1/7200$ give $e_{\mathrm{tot}}\le 1/200$. Thus

The ideal exact error is

These parameter values are analytical and were not the configuration executed in the supplied report.

For a single-cell code state ${\rho }_C$,

For a pure unreferenced input, $S(R)=h_2(q_N)$. For a reference-entangled pure input, radiation need not become pure at discharge.

For identical $K$-cell mechanisms and arbitrary reduced code state ${\rho }_{C^{(K)}}$,

where the empty subsystem has entropy zero. For independent pure cell states this reduces to $K{h}_2(q_N)$.

Proof. The accessible output has orthogonal vacuum and code blocks. Their eigenvalues are $q_N$ and $(1-q_N)$ times the eigenvalues of ${\rho }_C$. The complement has the reversed block weights. The entropy formulas follow, including endpoints by continuity.

For many cells, the emitted subset $S$ is an orthogonal flag with the displayed probability. Its conditional state is an isometric copy of ${\rho }_{C_S}$. The flag entropy is $K{h}_2(q_N)$. The entropy of orthogonal blocks gives (eq:manyentropy). Tracing an external reference first gives the same formulas for the reduced code state.

As $q$ decreases from one to $1/2$, $h_2(q)$ increases; as $q$ decreases from $1/2$ to zero, it decreases. A discrete schedule may skip its continuous maximum. These are finite-model entropy histories, not gravitational Page-curve calculations.

Small-corrections results constrain evolution under stated near-horizon pair and locality hypotheses Mathur. Hawk02-A records a corresponding pair-entropy interface CheungHawk02A ({thm:pairentropy}).

This model:

1. transfers an existing excitation instead of creating a fresh approximately Bell-entangled outgoing/interior pair at every step;

2. reuses one finite memory instead of introducing independently initialized interior partners;

3. supplies no smooth horizon, interior field algebra, gravitational dressing, or controlled near-horizon approximation;

4. retains emitted radiation without reinteraction;

5. does not stipulate a thermal outgoing state.

The absence of the pair-production premises explains why entropy can decrease. It is not an evasion or contradiction of a small-corrections theorem.

A second bare-energy-conserving sequential model has the same radiation entropy $h_2(q_N)$ for every pure code input and every prefix, but constant radiation and recovery error $c_d$.

Proof. Store the variable code in a zero-energy degenerate spectator $V$. Prepare another source memory $D$ in the fixed excited state $|1{⟩}_D$, of energy $\omega$, and apply the same exchange pulses to $D$ and radiation. All Hamiltonians act as identity on $V$.

The radiation channel is

The unnormalized expression is valid even when $q_N=1$. Its two nonzero block eigenvalues are $q_N$ and $1-q_N$, so its entropy is $h_2(q_N)$. Any decoder produces a fixed state; Lemma lem:replacement gives optimal error $c_d$. The complete residual retains the original code in $V$.

The comparison explicitly changes the source preparation and enlarges the residual ledger. It does not give two channels for one fixed Hamiltonian and preparation. It illustrates the entropy-history distinction in Hawk01-B CheungHawk01B ({thm:entropyseparation}).

Write

Let $J_j|a⟩=|a;j⟩$. When $p>0$, set $v_j={\alpha }_j/\sqrt{p}$.

Collect only cells in $\mathcal{S}\subseteq \{1,\dots ,N\}$. Set

The collected channel is an isometric erasure channel with

The complete residual includes memory and uncollected radiation.

Proof. In (eq:Wunnormalized), all collected emission amplitudes share a vacuum residual. The remaining amplitudes share collected vacuum and have residual support orthogonal to that vacuum. Tracing the residual gives the erasure-channel form with success weight $p_{\mathcal{S}}$. Apply Theorem thm:B3. If $p_{\mathcal{S}}=0$, the channel is constant.

Pinch radiation into vacuum, the label-preserving time sectors $J_j{\mathcal{H}}_C$, and the unused sector.

Time dephasing produces

Its optimal recovery error remains $q{c}_d$. A schedule-independent CPTP decoder is

It requires internal-label coherence within a time sector, not coherence between different time sectors.

Proof. Pinching removes only inter-time cross terms. The orthogonal sector embeddings and complement replacement make (eq:timedecoder) CPTP. Its composition is $p\mathrm{id}+q{\mathcal{T}}_{I/d}$. The Bell lower bound of Theorem thm:B3 still applies.

The measurement environment must be included in a complete postmeasurement ledger. Nevertheless, measuring time alone does not make the quantum code unrecoverable.

For $p>0$, applying the original coherent-mode decoder after time dephasing instead gives

Indeed ${J_N}^{†}{J}_j={\overline{v}}_j{I}_C$. Its error is $(1-a)c_d$. This is decoder-mode failure, not an optimal-recovery obstruction. At $p=0$, use a constant decoder without defining $v$ or $a$.

For $p>0$, let $\tilde{v}$ be normalized and $\tilde{J}|a⟩={\sum }_j{\tilde{v}}_j|a;j⟩$. The coherent-mode decoder using $\tilde{J}$ instead of $J_N$ has exact error

At $p=0$ its error is $c_d$.

Proof. ${\tilde{J}}^{†}{J}_N=⟨\tilde{v},v⟩I_C$. Compression transfers the stated weight and replaces its complement by $I_C/d$. Apply Lemma lem:replacement.

Let

Internal-label dephasing in radiation gives ${\mathcal{N}}_N{\Delta }_C$, whether or not time is additionally dephased.

After internal-label dephasing,

A label-blind occupation measurement that retains only the record “vacuum” or “occupied cell $j$” and discards all quantum carriers gives a constant channel and recovery error $c_d$.

Proof. The coherent-mode decoder gives $(1-q){\Delta }_C+q{\mathcal{T}}_{I/d}$. Since ${\Delta }_C$ is the uniform mixture of the $d$ phase unitaries, this is a Weyl mixture with identity weight $(1-q)/d+q/d^2$. Separating that term gives the upper bound.