Em03-B A Certified Apparatus-Model Readout of One Hawking Packet Passive capture, coherent finite extraction, and complete destructive records

We construct and analyze a specified apparatus-model realization of finite destructive number readout for one outgoing Hawking packet. The stationary field prescription is an explicit predecessor input. Its centered quasifree restriction gives a normal geometric one-mode density, proved here without assuming a density operator for the entire stationary radiation field.

The apparatus consists of a passive temporal-mode capture port, a memory oscillator, a binary register, and a complete readout instrument. A smooth, finitely switched capture interaction has an explicit transfer kernel. A zero-integral temporal-window construction controls packet mismatch and prevents an infrared integrability shortcut. Unselected positive-frequency radiation is not replaced by vacuum: its occupation and correlations enter a reference-stable mode-comparison bound. Vacuum auxiliary ports and independent apparatus preparations are separately declared model data.

Coherent oscillator–register extraction is generated by explicit number-selective swap pulses. Its finite control-domain error is exactly the probability above a second cutoff when the stated global block structure is present. Bounded control perturbations and a distinct support-only certificate are treated separately. A complete noisy readout instrument includes assignment errors, quantum readout errors, spectator coupling, imperfect preparation, and visible failure outcomes.

The resulting transfer and implementation budgets are derived from model parameters and substituted into finite-count, mean-bias, and record–spectator–reference bounds. A constructive tolerance schedule and an apparatus resource ledger are supplied. These are model-level results. A local Schwarzschild-field coupling, a justified laboratory effective model, and experimental implementation remain separate obligations.

The architecture is

Here $c$ is a memory oscillator, $Q$ is a finite binary register, and $F$ contains raw words, decoded values, and declared failures.

Three levels are kept separate.

{}1.15

The capture port is a one-dimensional Markov input–output model. It is not identified with a local detector coupled to the massless scalar on Schwarzschild. In particular, its two-sided temporal envelope includes auxiliary degrees of freedom that are not negative-energy modes of that scalar field.

Our model uses explicitly available number-selective control Hamiltonians. This is a declared control set, not a claim that a dispersive oscillator–qubit laboratory automatically implements those Hamiltonians. General oscillator-control constructions Krastanov do not supply a compiler or a leakage certificate for the pulses constructed below.

Local system–probe measurement theory requires a separate spacetime coupling and an induced-observable identification FewsterVerch. No theorem below supplies that missing correspondence.

boundary. The arguments are analytical. Compilation, formal verification, numerical pulse optimization, parameter calibration, and laboratory execution are not asserted. “Certified” means that the displayed implications have stated hypotheses and mathematical proofs.

and distances. We use $c=\hslash =k_B=1$, $G>0$, and

A channel on trace-class operators is understood as the predual of a normal completely positive unital map when its input is infinite dimensional. All operational comparisons retain finite spectators and arbitrary finite-dimensional external references. The mathematical identities also hold for normal extensions to separable reference spaces.

status. Em03-A and Landmark I are used through the supplied text Em03A,LandmarkI. Their proofs and registry status are not independently verified here. Other internal manuscript names are not formal premises of the apparatus construction.

Fix one angular harmonic of degree $\ell$, a mass parameter $M>0$, and

The normalized Killing time is the predecessor's time, and ${\beta }_{\mathrm{H}}=8\pi GM$.

The horizon and past-infinity input states are independent, centered, regular, and quasifree. Their positive-Killing-frequency covariances are

The infinity state is its positive-frequency vacuum, hence is centered with zero anomalous covariance. Cross-channel covariances vanish.

Stationary scattering is complex linear:

Write

The outgoing state is therefore centered, gauge-invariant, and quasifree, with number covariance $n(\omega )\delta (\omega -{\omega }^{\prime })$.

The components of this assumption have distinct source roles.

{}1.15

Hadamard regularity alone does not select these covariances.

Define

where annihilation operators are conjugate-linear in their test: $a(h)=\int \overline{h(\omega )}a(\omega )\mathrm{d}\omega$. Set

Then

Under Assumption ass:field, the selected packet state is represented normally on ${\mathcal{H}}_h={\mathcal{F}}_{\mathrm{s}}(\mathbb{C})\cong {\ell }^2({\mathbb{N}}_0)$ by

For ${\nu }_h=0$, it is the vacuum. Its count law has mean ${\nu }_h$, variance ${\nu }_h(1+{\nu }_h)$, and generating function

when $q_h>0$.

Proof. Complex-linear scattering and restriction preserve centered quasifreeness. The packet has $⟨{b_h}^2⟩=0$ and $⟨{b_h}^{†}{b}_h⟩={\nu }_h$. Thus its displacement characteristic function is

For ${\nu }_h>0$, construct the trace-class coherent-state mixture

The measure has total mass one. Its characteristic function is the displayed Gaussian, since

Angular and radial integration give

This is (eq:rhoh). Equality of characteristic functions identifies the states on the Weyl algebra, because its dense linear span consists of Weyl operators. The vacuum case follows directly. Summing the geometric series proves the moment and generating-function statements.

This is a finite-mode identification, consistent with the Gaussian framework Weedbrook. It neither constructs nor requires a trace-class density operator for the entire stationary continuum.

The density $\frac{1}{2}|0⟩⟨0|+\frac{1}{2}|2⟩⟨2|$ is centered and gauge-invariant, has $⟨{b_h}^2⟩=0$, and has mean one. It is not the geometric state with mean one. Quasifreeness is essential to Theorem thm:marginal.

For positive-frequency tests, use the norm

The compact positive-frequency covariance extends by continuity to its completion. The resulting Gaussian characteristic functional remains positive: each finite matrix and each Weyl positivity condition is a limit of the corresponding compact-test condition.

For any finite-dimensional canonical subspace with finite occupation matrix $A⪰0$, the restricted state is normal. Indeed, diagonalize $A$ by a passive finite-mode unitary. The characteristic function becomes a product of one-mode characteristic functions; Theorem thm:marginal constructs their normal product density. Transforming it back constructs the normal finite-mode restriction.

This completion provides the radiation-mode calculations used below. It does not assert a new characteristic restriction theorem for local bulk fields.

Fix integers $N\ge 1$ and, later, $K\ge N$. The reported saturation level $N$ and the control-domain cutoff $K$ are different parameters.

Let

Let $V_N$ identify the number bases on $P_N{\mathcal{H}}_h$ and ${\mathcal{K}}_N$. Define

The map ${\mathcal{C}}_N$ is CPTP. Its adjoint is

For $D_N={{\sum }_{k=0}}^{N}k|k⟩⟨k{|}_C$,

The readout effects pulled back to the oscillator are

Proof. The Kraus maps are $K_0=V_N{P}_N$ and $K_n=|N{⟩}_C⟨n|$, $n>N$. Their effects sum strongly to $I$. The positive output series converges in trace norm on positive trace-class inputs. This proves the normal CPTP extension. The adjoint and its specializations follow by substitution.

Low-block coherences are preserved before readout. Saturation is not postselection.

Set

Identify carrier basis $k$ with binary integer $k$ in $Q$; write the isometry as $J_N$, and set

For every $0\le j<d_N$, including unused words, define

Let $\mathcal{Y}$ contain all distinct $y_j$ and any declared failure symbols. The full record space is ${\mathcal{H}}_F={\mathbb{C}}^{\mathcal{Y}}$. Ideal readout has zero weight on failures.

For a spectator $S$, define

The selective map is $X\mapsto M_{j}X{M}_j^{†}$, whereas its effect is $E_j$. Define

Assign every failure symbol a fixed decoded value in $\{0,\dots ,N\}$; below all failure values are zero. Write this total decoder as $d(y)$, with $d(y_j)=c_N(j)$. Its channel on arbitrary record operators is

This is CPTP on the full record space, not merely a prescription on its populated diagonal.

For a normal state ${\rho }_{hA}$, where $A=SR$ collects unmeasured systems, put

Define

For every normal packet–spectator–reference state,

Applying ${\mathcal{G}}_N$ gives ${\Omega }_{N,SR}(\rho )$. All positive diagonal block sums converge in trace norm.

Proof. The low-block Kraus map contributes the diagonal blocks at $k\le N$. Every Kraus index $n>N$ contributes its reference block to the same carrier state $N$. Readout adds those positive blocks to the $n=N$ block. Their traces sum to at most one, proving convergence. The total decoder gives the second identity.

Spectators undergo the identity operation in this ideal channel. Their conditional state can change when they are correlated with the carrier. No nondestructive residual field instrument is identified by this destructive record.

For an arbitrary normal oscillator state of finite mean $\mu$,

Let

The finite decoded record is embedded into this countable record space by assigning zero weight above $N$.

For every normal input,

Consequently, for the geometric marginal,

Proof. At record $N$, the difference is the positive block ${\sum }_{n>N}{{\rho }_A}^{(n)}$; at each larger record it is $-{{\rho }_A}^{(n)}$. Orthogonal record blocks have additive trace norm. The half trace norm is therefore the discarded probability. Equation (eq:generalbias) follows from $N_h-\min (N_h,N)=(N_h-N{)}_{+}$. Geometric summation gives $\mathrm{Pr}(N_h\ge j)={q_h}^j$ and

For $E>0$,

Indeed, $(1-E/L)|0⟩⟨0|+(E/L)|L⟩⟨L|$, with $L>\max (E,N)$, has bias $E(1-N/L)\to E$. A mean-only constraint gives a probability bound $\mathrm{Pr}(N_h>N)\le E/(N+1)$, not a uniformly vanishing mean bias.

Let

Its positive-frequency part carries the outgoing radiation covariance (eq:nomega). Its negative-frequency part is a new, independent auxiliary vacuum in the apparatus model. Define

The Fourier convention is

with $h$ extended by zero outside its band.

Annihilators on this port satisfy

as smeared identities. The two-sided envelope makes the Markov port canonical. It is not a claim that the physical scalar field contains independent negative-Killing-frequency annihilators.

All positive-frequency components outside the target packet retain their actual prescribed radiation covariance. In particular, no vacuum substitution is made for unselected Hawking radiation.

For a chosen capture envelope $f$, the memory output depends only on $a(\hat{f})$. Together with the target $a(h)$, it therefore requires at most two canonical input modes. Define their plane by

The normal state on its Fock space is justified by Section subsec:completion whenever its occupation matrix is finite.

The declared operational input is a normal state

Here $g$ will be the orthogonal participating mode. The ideal capture target identifies $h$ with $c$ and traces out $g$:

It does not assume ${\rho }_{hgSR}$ factorizes.

The Hawking input class used for the model theorem consists of all such normal extensions with the specified two-mode Gaussian marginal. External finite references are unrestricted. A general occupation-constrained class is treated separately below.

No normal density representation of the entire continuum is assumed. The capture model defines its reduced channel through its output Weyl operators on these finite participating modes. A physical continuum interaction with further participating degrees of freedom would require a larger input interface.

A compactly switched temporal capture cannot exactly select the nonzero band-limited packet $p$. We construct an explicit approximation, including a low-frequency integrability repair.

Choose real $\chi \in {C_c}^{\infty }((-2,2))$, with $0\le \chi \le 1$ and $\chi =1$ on $[-1,1]$. Choose $\psi \in {C_c}^{\infty }((-2,2))$ with $\int \psi (t)\mathrm{d}t=1$. For $L>0$, set

For sufficiently large $L$, $v_L\ne 0$. Normalize $v_L$ and choose its constant phase so that

Write

When ${\theta }_L=0$, omit $g_L$ and all mismatch terms.

For sufficiently large $L$:

1. $f_L\in {C_c}^{\infty }((-2L,2L))$, ${\Vert f_L\Vert }_2=1$, and $\int f_L=0$.

2. $f_L\to p$ in the Schwartz topology and ${\hat{f}}_L\to h$ in that topology. In particular, ${\theta }_L\to 0$.

3. The occupations

   $$E_f={{\int }_0}^{\infty }n(\omega )|{\hat{f}}_L(\omega ){|}^2\mathrm{d}\omega ,E_g={{\int }_0}^{\infty }n(\omega )|g_L(\omega ){|}^2\mathrm{d}\omega$$

   are finite for every nondegenerate participating plane.

4. With

   $$w_L={\hat{f}}_L-{\alpha }_{L}h,\mathcal{N}(w)={{\int }_0}^{\infty }n(\omega )|w(\omega ){|}^2\mathrm{d}\omega ,$$

   one has $\mathcal{N}(w_L)\to 0$, $E_f\to {\nu }_h$, and

   $${{\theta }_L}^2{E}_g⟶0.$$

Proof. The support and zero integral follow from (eq:windowconstruction). The inverse Fourier transform $p$ is Schwartz. Also $\int p=0$, because $h$ vanishes near zero. Hence

which decays faster than every inverse power of $L$. Differentiating the window shows that $(1-{\chi }_L)p\to 0$ in every Schwartz seminorm. The scaled bump $m_L{\psi }_L$ has the same property. Normalization and the phase alignment therefore preserve Schwartz convergence.

For positive frequencies,

The zero integral gives ${\hat{f}}_L(0)=0$. Thus ${\hat{f}}_L(\omega )=O(\omega )$ at zero; the same holds for $w_L$ and, for fixed $L$, $g_L$. Their weighted integrals are finite near zero. At high frequencies, the occupation is bounded and the tests are square integrable.

More quantitatively, for any differentiable $w$ with $w(0)=0$,

On $[1,\infty )$, $n$ is bounded. Schwartz convergence proves $\mathcal{N}(w_L)\to 0$ and $E_f\to {\nu }_h$. Finally,

Without the subtraction in (eq:windowconstruction), a compact temporal window generally has nonzero Fourier value at zero. The predecessor bound ${\Gamma }_{\ell }\le 1$ alone would then not prove integrability against $n(\omega )\lesssim 1/\omega$. We do not import an unproved low-frequency greybody asymptotic to bypass this issue.

For simplicity write $\theta ={\theta }_L$, $g=g_L$. Let

Up to the choice of rotation sign, $V_{\theta }=e^{-\mathrm{i}\theta J}$ identifies mode $h$ with $\cos \theta h+\sin \theta g$. The generator is self-adjoint as the direct sum of finite total-number sectors.

Suppose the $h,g$ marginal is centered, gauge-invariant, and quasifree, with occupations $E_h,E_g$. For every normal spectator–reference extension,

For the radiation marginal, $E_h={\nu }_h$, and ${\mu }_L\to 0$. No independence between the two modes is required.

Proof. Purify the joint input, retaining its external systems. The unitary comparison and partial-trace contractivity give

The last inequality follows from the scalar spectral inequality $|e^{-\mathrm{i}\theta x}-1|\le |\theta x|$.

Put $m=⟨a(g{)}^{†}a(h)⟩$. Wick's rule gives

Cauchy–Schwarz gives $|m{|}^2\le E_h{E}_g$, hence $⟨J^2⟩\le E_h+E_g+6E_h{E}_g$. All these moments depend only on the two-mode marginal, so the bound is unchanged by its reference extension.

For the radiation input,

by Lemma lem:window. This proves convergence.

The background contribution in (eq:modeerror) is computed from $n(\omega )$ and the actual orthogonal mode $g_L$. It includes its possible correlation $m$ with $h$, through a uniform Wick bound. There is no separate “background vacuum” assumption or overlapping additive background-error label.

If only $⟨N_h+N_g⟩\le E_{\mathrm{pl}}$ is known, then for every integer $D\ge 0$ the same mode comparison obeys

Proof. Project onto total occupation at most $D$. Its discarded probability is at most $E_{\mathrm{pl}}/(D+1)$. The distance to the normalized projected state is at most the square root of that probability, as follows by purification. On this invariant sector, $\Vert J\Vert \le D$, so the unitary comparison is at most $D|\theta |$. Two projection comparisons and contractivity give the result.

The constraint is packet-plane occupation, not a separately identified packet Hamiltonian. Occupation-constrained comparisons are a specialization of infinite-dimensional constrained-channel methods Shirokov.

Fix a normalized $f=f_L$, supported in $(a,b)=(-2L,2L)$, and a capture transmissivity $0<\tau <1$. Set

Outside $[a,b]$, the coupling is zero.

The effective interaction is defined first on one-particle amplitudes $z(t)$ of the memory and incoming amplitudes $u(t)$:

Its quantum evolution is the bosonic second quantization of this one-particle input–output map.

The memory is initially independent of the radiation, spectators, and references. Its specified preparation is the geometric state of mean $m_0\ge 0$. A subsequent storage interval of duration $T_s$ has vacuum attenuation rate ${\kappa }_s\ge 0$, implemented by an independent auxiliary loss port. Define

All phases are defined in a fixed interaction picture. Its clock is the predecessor's asymptotic retarded-time coordinate, up to a fixed translation.

This is a declared passive model in the input–output tradition GardinerCollett,YamamotoJames. We give its transfer calculation directly, rather than assuming an exact pulse-capture claim from those references.

Equation (eq:passiveODE) defines a unitary map

from initial memory and incoming port to final memory and outgoing port. Its memory annihilator is

The switching is smooth and finite, and

Proof. The energy identity

follows by substitution. Integration proves that the map is isometric. Given final memory and outgoing amplitude, solve backwards the equation obtained by substituting $u=u_{\mathrm{out}}+\overline{\lambda }z$; bounded coefficients give a unique initial memory and incoming amplitude. Thus the isometry is onto.

The memory solution is

Since $D^{\prime }=\kappa D$, $D(a)=1-\tau$, and $D(b)=1$,

The transfer kernel is therefore $\sqrt{\tau }\overline{f(t)}$. Second quantization gives (eq:captureann). The integral and peak bound follow from the same identities. Because $f$ is smooth and supported away from the endpoints and $D\ge 1-\tau >0$, the switched coefficient $\lambda$ is smooth.

The formal white-noise interaction has not been treated as a bounded Hamiltonian. Its well-defined unitary model is the one-particle map just proved and its second quantization. Accordingly, no finite-dimensional Hamiltonian norm estimate is applied to this unbounded bosonic capture evolution.

The storage attenuation is an ordinary vacuum beam-splitter channel with transmissivity $e^{-{\kappa }_s{T}_s}$; it can be defined by the same passive dilation. With the initial memory replaced by vacuum, capture and storage together are vacuum pure loss of transmissivity ${\tau }_{\ast }$ on the selected port mode.

Let ${\mathcal{L}}_t$ be the vacuum pure-loss channel of transmissivity $0\le t\le 1$. If $\mathrm{tr}(\rho N)\le E$, then, including any normal reference,

Proof. Purify $\rho$, append a vacuum loss oscillator, and apply the beam splitter. The overlap with the initial purified state is

This is nonnegative. The inequality $1-s^n\le n(1-s)$, $0\le s\le 1$, gives $1-A\le E(1-\sqrt{t})$. The pure-state trace distance is $\sqrt{1-A^2}\le \sqrt{2(1-A)}$. Partial trace proves (eq:lossbound).

This lemma is used only after the independent vacuum-environment reduction has been established by Specification spec:capture. It is not applied directly to unselected radiation.

The initial memory's distance from vacuum is exactly

This preparation error is priced once, before the common capture evolution.

For the radiation two-mode marginal and every admitted spectator–reference extension,

Here $\mathcal{M}$ is the capture-plus-storage channel obtained from Specification spec:capture.

Proof. Replacing the independently prepared memory by vacuum costs $p_c$, including references, by tensor-product equality and contractivity. For that preparation, capture and storage reduce to ${\mathcal{L}}_{{\tau }_{\ast }}$ on the $f$-mode, so Lemma lem:loss costs $\ell (E_f,{\tau }_{\ast })$. The comparison between the $f$-mode and the target $h$-mode costs ${\mu }_L$, by Lemma lem:modeerror. The ideal target traces out the other participating mode; no independence between that mode and $hSR$ is used. Apply the triangle inequality.

The three terms have different comparisons: preparation, mode selection including the nonvacuum background, and vacuum loss. There is no additional overlapping “window” or “background” term.

For this Gaussian input, the actual memory marginal is geometric with mean

This follows directly from (eq:captureann), independence of the memory preparation, and gauge-invariant quasifreeness. It is an apparatus-model prediction, not a replacement for the target value ${\nu }_h$.

Use the binary register $Q$ itself, initially in $|0{⟩}_Q$. For $1\le n\le K$, define the disjoint pairs

These pairs are mutually disjoint. Let $U_{N,K}$ swap each $A_n,B_n$ and act as the identity on the remaining basis states.

Tracing out $c$ after $U_{N,K}$, with initial register $|0{⟩}_Q$, induces

For every normal memory–reference state,

For the ideal target packet this error is ${q_h}^{K+1}$.

Proof. For $0\le n\le N$, the output memory is the same vacuum. Consequently all low-block coherences survive in $Q$. For $N<n\le K$, memory state $n$ labels the environment and the register is $N$. For $n>K$, the untouched input has memory state $n$ and register zero. Distinct such memory states are orthogonal, proving (eq:extractionchannel).

The channel difference on a referenced input is exactly

Because $N\ge 1$, the two register states are orthogonal. Its half trace norm is $\mathrm{tr}H$.

This is coherent extraction, not copying all number labels while leaving a distinguishable low-number environment. Finite output does not imply a finite environment on the whole oscillator: an isometry from an infinite-dimensional input cannot have finite-dimensional output and environment simultaneously.

Define the orthogonal rank-one projections

The available extraction controls in our effective model are

These are the specified number-selective matrix elements; their availability is part of the model.

For a common duration $T_E>0$, choose

The pulses are zero outside this interval. They have area $\pi$, maximum $2\pi /T_E$, and derivative bounded by $2{\pi }^2/{T_E}^2$.

The $K$ simultaneous pulses (eq:swapPulse) generate $U_{N,K}$ exactly, with no relative swap phases. If the envelope limits are ${\Lambda }_E>0$ and $B_E>0$, it suffices to take

For input supported at occupation at most $K$, the ideal sequence never exceeds occupation $K$.

Proof. The projections commute and are mutually orthogonal. On each pair,

is the swap matrix, including its phase. Outside the pairs the evolution is the identity. The bounds follow by differentiating (eq:swapPulse). Its invariant subspace contains only memory occupations $0,\dots ,K$.

The constant-pulse version is the Hamiltonian

on each pair, as a special case.

-energy convention. A memory Hamiltonian ${\Omega }_c{N}_c$, ${\Omega }_c>0$, may be specified, with degenerate computational register levels. The displayed controls are in its interaction picture. Their laboratory-frame version contains the corresponding number-dependent phase modulation. The required frequency range grows with $K{\Omega }_c$; it is not hidden in the envelope derivative bound. No field packet Hamiltonian is identified with ${\Omega }_c{N}_h$, and extraction is not asserted to conserve the memory's free energy without work exchange.

Suppose the implemented interaction-picture generator is

Here $G_E(t)$ is explicitly bounded and self-adjoint. One permitted leakage model takes it supported on

with finite-dimensional matrix elements. This allows temporary occupation up to $J$, rather than silently identifying the initial cutoff $K$ with the maximum populated level.

Put