Em03-C Joint Hawking-Packet Correlations and Certified Tests of Semiclassical Departures Finite-mode states, information-sensitive readout, and local-field matching obligations

We construct a two-packet operational interface for the stationary free-scalar Hawking model. The inherited outgoing quasifree prescription determines a normal joint Gaussian density, its joint count generating function, and its interpacket number covariance. Orthogonal packet modes are independent precisely when their off-diagonal occupation covariance vanishes.

The apparatus construction uses two memories coupled to one effective temporal port. Compact temporal tests are corrected at zero frequency, jointly orthonormalized, and compared with the desired packet plane. An explicit matrix-valued passive interaction captures both tests during the same switching interval. This avoids treating overlapping windows as independent apparatuses. Joint preparation errors, memory and register coupling, and correlated assignment noise enter one complete, reference-stable budget.

After coherent extraction, bounded single-qubit rotations and destructive pointer readout implement an informationally complete Pauli measurement family. We prove its effects, its complete-record implementation bound, an independent-trial statistical estimate, and a trace-distance reconstruction certificate. Destructive count tails and coherent state-approximation tails are distinguished.

A bounded-effect discrimination theorem translates a specified state departure into a certifiable record departure. A separate input-information theorem proves that an input-independent radiation channel cannot recover an unknown microscopic label or preserve its reference entanglement. The results concern a declared effective apparatus and a fixed-background baseline. They do not construct a local Schwarzschild detector, a collapse packet limit, a quantum-gravity correction, or microscopic evaporation dynamics.

The progression is

The supplied sources Em03A,Em03B do not contain an unknown black-hole microstate as a dynamical input. This manuscript does not insert one implicitly. A correction state, when discussed, must be supplied as additional model data.

Three levels remain separate:

1. Radiation-state interface: a specified algebraic field state and its normal finite-mode restrictions;

2. Effective apparatus model: a temporal port, memories, register controls, preparations, and complete readout instruments;

3. Physical correspondence: a local field–probe coupling, its relation to the effective port, and a calibrated implementation.

Only the first two levels are constructed here.

Figure fig:dependency records the dependency order used below. Solid arrows are channels or restrictions defined on their stated input spaces. Dashed boxes are comparisons that use a named state, not channels. The third level is drawn only as an open obligation.

The finite-mode Gaussian construction is standard in structure Weedbrook. The supplied companion account attributes a general version to En02-E, at :gaussian, :outcov, and :K En02EAccount. The full En02-E source is not supplied. We therefore give the needed specialization and its proof explicitly rather than importing an unseen theorem or claiming novelty for Gaussian diagonalization.

The remaining constructions are ordinary channel calculations. Landmark I is optional semantic input for calling finite destructive readout $\mathsf{Enter}$; it is not a field-state or apparatus premise.

boundary. The source supplies analytical arguments. No compilation, machine checking, numerical experiment, statistical calibration, or physical realization is asserted. The term “certified” refers to explicit mathematical estimates under stated model data.

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

Channels are normal CPTP maps on trace-class operators. A finite spectator is denoted $S$; an arbitrary finite-dimensional external reference is denoted $R$. Their tensor placements are retained throughout. The normal state identities also hold with separable references.

We write $\Gamma (U)$ for bosonic second quantization of a one-particle unitary. This notation is distinct from the greybody probability ${\Gamma }_{\ell }(\omega )$.

Fix one real angular harmonic of degree $\ell$ in the massless scalar model of Em03-A. The outgoing radiation state is centered, gauge-invariant, regular, and quasifree, with

where

Annihilation tests are conjugate-linear:

This input uses Em03-A's boundary-state prescription, :horizonoccupation, :quantumscatter, :numberdensity, and :packetlaw. Its bulk Hadamard identification remains the declared external Dappiaggi–Moretti–Pinamonti input DMP. Neither that global proof nor Em03-A's manuscript proofs are independently verified here.

Choose orthonormal $h_1,h_2\in {C_c}^{\infty }((0,\infty ))$, supported in a common compact band

Set $b_i=a(h_i)$ and $N_i={b_i}^{†}{b}_i$. Then $[b_i,{b_j}^{†}]={\delta }_{ij}$. Define

Thus

The transposition is important. Equation (eq:A) is not the formula for $⟨{b_i}^{†}{b}_j⟩$ with the same indices under the predecessor's test convention.

The matrix $A$ is positive semidefinite and satisfies

Figure fig:covariance fixes the index convention that the following characteristic function, passive transformation, and Wick formula all use. A complex off-diagonal entry is the control that exposes an accidental transpose.

For $z\in {\mathbb{C}}^2$, define

Under Assumption ass:radiation, the two-packet state is the unique normal density on ${\mathcal{H}}_{12}={\mathcal{F}}_{\mathrm{s}}({\mathbb{C}}^2)$ with characteristic function

If $A=W\mathrm{diag}({\lambda }_1,{\lambda }_2)W^{†}$, then

A zero eigenvalue means vacuum in that diagonal mode. No density operator for the entire stationary continuum is required.

Proof. Restriction and complex-linear scattering preserve the inherited centered quasifree prescription. Its zero anomalous covariance and number matrix (eq:A) give (eq:characteristic).

For completeness, let ${\beta }_1,{\beta }_2$ be independent circular complex Gaussian random variables with variances ${\lambda }_1,{\lambda }_2$, with a zero-variance variable identically zero, and put $\alpha =W\beta$. The positive trace-class mixture

has trace one and characteristic function

For a single variance $\lambda >0$, angular and radial integration give the number probabilities

Consequently the mixture is (eq:jointdensity). Equality of Weyl characteristic functions identifies the abstract restricted state and uniquely determines its normal density. This proves normality by construction.

Figure fig:gaussian separates the two constructions of the same normal density: diagonal thermal modes, and a coherent-state mixture with covariance $A$. Neither arrow is a density of the whole continuum.

Write

For $0\le s_1,s_2\le 1$, with $S=\mathrm{diag}(s_1,s_2)$,

More explicitly,

Its Taylor coefficients at $(0,0)$ give the complete joint count law.

Each marginal is geometric with mean ${\nu }_i$, and

Moreover,

The packet factors are statistically independent if and only if $A_{12}=0$.

Proof. Conditional on a coherent amplitude $\alpha$, the two counts are independent Poisson variables of means $|{\alpha }_i{|}^2$. The coherent-state mixture in Theorem thm:gaussian gives

The circular Gaussian integral equals

This also covers singular $A$ by its diagonal Gaussian construction. Setting either $s_i=1$ gives its geometric marginal.

Gaussian contraction gives (eq:wick). For different modes, $N_1{N}_2={b_1}^{†}{b_2}^{†}{b}_1{b}_2$; hence $⟨N_1{N}_2⟩={\nu }_1{\nu }_2+|A_{12}{|}^2$. The same-mode calculation gives the stated variance. All polynomial moments are finite.

If $A_{12}=0$, the characteristic function factorizes, so the state is the product of its marginals. Conversely, a product centered state has zero cross covariance.

Let $u,v$ be normalized tests with disjoint supports and occupations ${\nu }_u\ne {\nu }_v$. Then

are orthonormal, but $A_{12}=({\nu }_u-{\nu }_v)/2\ne 0$. Disjoint supports of $h_1,h_2$ themselves would instead force $A_{12}=0$.

The joint number law depends on $A_{12}$ only through its modulus. A local phase transformation changes its phase without changing any number-diagonal probability. Number counts therefore do not determine the complete joint state.

Figure fig:counts records what the generating function determines and what it does not. Orthogonal packets are independent only when $A_{12}=0$. Equal moduli with different phases share the number law and are separated only by a phase-sensitive observable.

Choose reported saturation levels ${N_i}^{\mathrm{cut}}\ge 1$ and control-domain cutoffs $K_i\ge {N_i}^{\mathrm{cut}}$. To avoid confusing operators and cutoff values, abbreviate

Let $P_{i,a_i}={{\sum }_{n=0}}^{a_i}|n⟩⟨n|$.

The one-mode carrier map is

Its Kraus maps are $V_i{P}_{i,a_i}$ and $|a_i⟩⟨n|$, $n>a_i$, whose effects sum strongly to the identity.

Set

The binary embeddings $J_i$ send carrier integer $k$ to its fixed binary word. Define

For raw integers $j_i\in \{0,\dots ,d_i-1\}$, put

An ordinary record contains both raw words, both decoded integers, and an “ordinary” status:

Enlarge this alphabet $\mathcal{Y}$ by all declared failure records. Failure records retain their labels; their decoded pair is $(0,0)$. A different bounded assignment could be specified, but must be fixed before using a mean certificate.

The record can be embedded into $F_1{F}_2{F}_{\mathrm{flag}}$. We denote the whole classical output by $F$. For arbitrary record operators, the total decoder is

It is CPTP on the full alphabet.

For computational destructive readout, set

The measured register is absent from the output.

For ${\rho }_{12SR}$, define the positive blocks

Let

The ideal full record is

For every normal input,

Let ${\Omega }_{\mathbf{a},SR}$ be its decoded version and ${\Omega }_{\infty ,SR}$ the unsaturated joint number record. In the common countable record space,

For the baseline,

No independence assumption is needed.

Proof. Tensor the two Kraus families. Destructive readout selects the number-diagonal input blocks and sums exactly the blocks with the same clipped pair. Positive block traces sum to one, so the series converge in trace norm.

The two decoded records agree on the rectangular interior. Outside that rectangle, the unsaturated record has positive blocks that are absent from the clipped record. The clipped record adds their positive mass to boundary records inside the rectangle. These inside and outside differences occupy orthogonal classical records. Their total trace norm is twice the outside probability, proving (eq:jointtail). The union bound and the geometric marginal tails give the last assertion.

The marginal ideal mean deficit is exactly

This statement is unaffected by interpacket correlations.

Embed the binary register into the oscillator pair by

including unused binary words. Define

For any normal packet–reference input,

where

Proof. The embedded carrier output is

This follows by separating the low–low Kraus map from all other Kraus maps.

Purify $\rho$ to $|\Psi ⟩$, and set $|\Phi ⟩=(P_{\mathbf{a}}\otimes I)|\Psi ⟩$. Writing $p=p_{\mathbf{a}}$, one has $\Vert \Phi {\Vert }^2=⟨\Psi ,\Phi ⟩=1-p$. The rank-two calculation gives

Partial trace contracts this norm. Adding $T$ costs at most $p$ in trace norm. Divide by two and cap at one.

The exact probability error (eq:jointtail) concerns destructive number records. It is not the coherent state certificate (eq:coherenttail).

Figure fig:tails separates the two cutoff errors. The destructive distance is exactly the outside probability. The coherent comparison pays an additional square-root term because the inside and outside amplitudes need not occupy orthogonal record labels.

As in Em03-B, the effective port has one-particle space $L^2(\mathbb{R},\mathrm{d}\omega )$. Its positive-frequency sector carries (eq:radiationdensity); its negative-frequency sector is a new independent auxiliary vacuum.

This two-sided Markov envelope is model data. It is not a physical negative-energy sector of the Schwarzschild scalar. Unselected positive-frequency radiation is not replaced by vacuum.

Use

The input annihilator $a(\hat{f})$ is the temporal smearing $\int \overline{f(t)}{b}_{\mathrm{in}}(t)\mathrm{d}t$.

Finite restrictions are defined on the weighted domain

where $n_{\mathrm{port}}=n$ on positive frequencies and zero on negative frequencies. Compact positive-frequency tests are dense in the positive-frequency weighted space. Covariances and Gaussian Weyl positivity extend by continuity. Every finite canonical subspace with finite occupation matrix has a normal density by the same diagonalization proof as Theorem thm:gaussian.

This extension is a radiation-test construction, not a new local bulk characteristic-restriction theorem.

Choose real $\chi \in {C_c}^{\infty }((-2,2))$, equal to one on $[-1,1]$, and choose $\psi \in {C_c}^{\infty }((-2,2))$ with $\int \psi =1$. For $L>0$, put

Let $V_L:{\mathbb{C}}^2\to L^2(\mathbb{R},\mathrm{d}t)$ have columns $v_{i,L}$. For sufficiently large $L$, $G_L={V_L}^{†}{V}_L$ is positive definite. First orthonormalize:

Let $H:{\mathbb{C}}^2\to L^2(\mathbb{R},\mathrm{d}\omega )$ have columns $h_1,h_2$. Write the left polar decomposition

and define

The columns of $F_L$ are denoted ${\hat{f}}_{1,L},{\hat{f}}_{2,L}$.

For sufficiently large $L$:

1. $f_{1,L},f_{2,L}$ are orthonormal smooth tests supported in $(-2L,2L)$, and each has zero integral;

2. their Fourier transforms converge columnwise to $h_1,h_2$ in the Schwartz topology and in the weighted occupation norm;

3. the participating plane

   $${\mathfrak{v}}_L=\mathrm{ran}H+\mathrm{ran}{F}_L$$

   has dimension at most four and a finite occupation matrix.

Proof. Each $p_i$ is Schwartz, and $\int p_i=0$ because $h_i$ vanishes near zero. Hence

decays faster than every inverse power of $L$. The windowed error and the correcting bump tend to zero in every Schwartz seminorm. Thus $G_L\to I$; the inverse square root and the polar factors tend to $I$. Their finite matrix operations preserve supports and zero integrals.

For $\omega >0$,

A Fourier test $w$ with $w(0)=0$ satisfies

On $[1,\infty )$, $n$ is bounded. Schwartz convergence therefore implies weighted convergence. It also proves finite occupation for every vector in ${\mathfrak{v}}_L$.

Ordinary temporal windowing alone would not establish integrability at zero using only ${\Gamma }_{\ell }\le 1$. The zero-integral correction avoids importing a low-frequency greybody asymptotic that has not been proved here.

Figure fig:windows is the temporal-mode interface used before capture. The zero-integral correction is what makes the Fourier test vanish at $\omega =0$. Ordinary compact windowing, without that correction, does not by itself give finite occupation from $n(\omega )\le 1/({\beta }_{\mathrm{H}}\omega )$.

Diagonalize

The principal target modes ${h^{\prime }}_r=(H{R}_L{)}_r$ and captured modes ${f^{\prime }}_r=(F_L{R}_L{)}_r$ obey

For nonzero ${\theta }_r$, the $g_r$ are orthonormal and orthogonal to the target plane. Zero-angle terms are omitted.

Let $v_L$ be the passive one-particle rotation sending ${h^{\prime }}_r$ to ${f^{\prime }}_r$ in each principal plane and acting identically on its orthogonal complement. Let ${\mathcal{I}}_H$ select the target pair and discard the other participating modes. Then the captured-pair selector is

with both selected pairs identified with the same two memory coordinates. The operational input is the normal joint state ${\rho }_{{\mathfrak{v}}_{L}SR}$, not merely its target marginal.

For the inherited Gaussian marginal on ${\mathfrak{v}}_L$, and any spectator–reference extension,

Here each occupation is computed from the actual port covariance. Moreover, ${\mu }_L\to 0$.

Proof. The rotation generator in plane $r$ is

These generators are self-adjoint on their total-number-sector direct sums. Purification, the spectral inequality $|e^{\mathrm{i}x}-1|\le |x|$, and the vector triangle inequality give the bound

For a centered gauge-invariant Gaussian two-mode marginal, put $m=⟨a(g_r{)}^{†}a({h^{\prime }}_r)⟩$. Wick contraction gives

using $|m{|}^2\le E_{{h^{\prime }}_r}{E}_{g_r}$. This depends only on the participating marginal, so arbitrary reference extensions obey the same estimate.

The principal target occupations are bounded by $n_{\max }$. The residual vectors ${f^{\prime }}_r-\cos {\theta }_r{h^{\prime }}_r$ have weighted occupation tending uniformly to zero by Lemma lem:windows. Their occupations are ${\sin }^2{\theta }_r{E}_{g_r}$. Therefore ${{\theta }_r}^2{E}_{g_r}\to 0$, proving the last assertion.

For a non-Gaussian input the same purification proof gives

provided these moments are finite. Otherwise the trivial bound one remains valid. A proposed correction must justify its own participating-mode moments; the baseline's Wick estimate cannot be transferred to it.

Figure fig:plane is the comparison between the desired packet plane and the plane spanned by the compact tests. A small weighted residual can still be a normalized mode of large occupation. The bound uses the residual occupations, not the norm of the residual alone.

This section constructs a combined apparatus. It does not add two one-packet capture budgets.

Fix the orthonormal temporal tests $f_1,f_2$ just constructed, supported in $(a,b)=(-2L,2L)$, and $0<\tau <1$. Set

Orthonormality gives $D(a)=(1-\tau )I$, $D(b)=I$.

Two memory amplitudes $z(t)\in {\mathbb{C}}^2$ and one incoming port amplitude $u(t)$ evolve by

The quantum evolution is the second quantization of this one-particle map.

The initial memories have a named joint state ${\sigma }_c$, independent of radiation, spectators, and references. Write $p_c=d_{\mathrm{tr}}({\sigma }_c,|0,0⟩⟨0,0|)$. An explicit allowed preparation is

It may have internal correlations.

A subsequent common storage interval has vacuum transmissivity $s=e^{-{\kappa }_s{T}_s}$ on each memory, with independent vacuum loss ports and no additional unpriced interaction. Define ${\tau }_{\ast }=\tau s$. The clock is the predecessor's normalized asymptotic time.

This is a specified linear input–output model, in the setting of passive memory models GardinerCollett,YamamotoJames. The following synthesis and comparison are proved directly.

Equation (eq:vectorODE) is unitary from ${\mathbb{C}}^2\oplus L^2([a,b])$ to itself. Its final memory annihilators obey

The interaction is smooth and finitely switched, and

Its resources satisfy

Proof. Differentiating $D=G^2$ gives $\dot{D}=\dot{G}G+G\dot{G}=\tau q{q}^{†}$. It follows that

Consequently

Integration proves isometry. Substituting $u=u_{\mathrm{out}}+{\lambda }^{†}z$ gives a bounded backward linear equation with a unique solution from the final memory and outgoing amplitude. The isometry is therefore onto.

The final-time fundamental solution is $G(t)$, since $\dot{G}=-GC$ and $G(b)=I$. Thus

Second quantization proves (eq:capturepair).

The integrated coupling follows from

The peak bound uses $D⪰(1-\tau )I$. The Sylvester equation for $\dot{G}$ has the integral solution

Hence $\Vert \dot{G}\Vert \le \Vert \dot{D}\Vert /(2\sqrt{1-\tau })$, and $\Vert C\Vert \le \tau \Vert q{\Vert }^2/(2(1-\tau ))$. The same bounds $\Vert \Omega \Vert$. Smoothness and finite switching follow from the supports of $f_i$ and the positive lower bound on $D$.

The off-diagonal memory Hamiltonian $\Omega (t)$ is part of the construction, not an omitted cross-coupling error. The underlying bosonic evolution is not treated as a bounded full-oscillator Hamiltonian.

Figure fig:port is the shared-port synthesis. One scalar input $u(t)$ drives both memories because $\lambda (t)$ is a vector. The generator $C(t)$ contains the memory mixing required by passivity. Two separate window approximations are not this joint port.

Vacuum pure loss of common transmissivity $t$ on a finite mode family satisfies

when $E=\mathrm{tr}(\rho N_{\mathrm{tot}})<\infty$.

Proof. Purify the input and append vacuum loss modes. The overlap of the initial and beam-splitter-evolved purifications is