Em03-F Source-Dependent Information in Physical Scalar Radiation Compact source–field dynamics, outgoing packet identification, bounded witnesses, and quantum-transfer limitations

We construct an exactly solvable effective exterior quantum source coupled to the massless scalar field on Schwarzschild. The benchmark is restricted to the s-wave sector and uses a spatially extended source with full angular-shell support. A degenerate qubit controls a smooth compact linear field interaction in normalized Killing time. We construct its unitary propagator, including the finite time-ordering phase, in a regular representation of the inherited Unruh state.

Tracing the source gives a measure-and-prepare field channel. Its restriction to finitely many identified outgoing scalar packets is a normal channel whose two conditional outputs are oppositely displaced versions of the semiclassical Gaussian baseline. All source matrix units are evaluated. The channel transmits a classical source observable but is entanglement breaking. For a finite packet output, its optimal deterministic recovery error is expressed through the trace distance of the two conditional radiation states and is strictly greater than one half at every finite coupling.

We prove a nonzero outgoing signal for a compact separable exterior profile. A bounded sine-Weyl effect distinguishes the source signs, whereas all selected joint number probabilities are identical for those signs. The phase-sensitive gap is linear at weak coupling, while the occupation-mean departure is quadratic.

A compact local scalar probe realizes the sine-effect comparison. We verify a common second-moment bound for the source-displaced states and give separate packet, transfer, probe-noise, profile, readout, and statistical allowances. The local construction uses growing full-shell access, a scaled processing test, and a specified squeezed probe preparation. Causal support, retained-source dephasing, finite excess radiative energy, and switching work are calculated separately.

The source law is a declared exterior model, not a derivation from microscopic quantum gravity. No information escape from a black-hole interior, coherent local field transfer, or evaporation theorem is established.

The construction follows

Em03-E Em03E uses an independently specified information-bearing payload beside occupation registers. Here there is no such payload. Every radiation output used for the information claim is a restriction of the massless scalar field already appearing in Em03-A–D.

The interaction is a deliberately solvable benchmark. Its simple-generated structure and exact qubit–field calculations have established precedents Tjoa; neither that interaction class nor controlled Weyl displacement is claimed as new. The present construction identifies the output with Schwarzschild scattering packets, proves a compact-profile signal, evaluates its witnesses and recovery limitations, and composes it with a quantitative local probe scheme.

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Em03-B and Em03-C Em03B,Em03C provide operational context, not a local source or an automatically available coherent scalar receiver. Em03-D Em03D supplies the relevant full-shell local-observable architecture. Its cosine readout is not used as though it distinguished opposite displacements: the imaginary part is treated explicitly here.

boundary. The manuscript supplies analytical arguments. Compilation, formal verification, numerical execution, calibration, and physical implementation are not asserted. “Certified” refers to the printed mathematical estimates under their stated hypotheses.

conventions. For normal states and channels,

The channel metric retains arbitrary external references. An expectation comparison is not silently promoted to a diamond-distance comparison.

In physical units,

The time generator is normalized at infinity.

For the estimates, measure length in units of $r_{\mathrm{h}}$. Thus, after rescaling,

A physical frequency is ${r_{\mathrm{h}}}^{-1}\omega$, and a physical energy is ${r_{\mathrm{h}}}^{-1}$ times the displayed dimensionless energy. Packet normalization is preserved by

A dimensionless bulk smearing used below corresponds to a physical smearing with factor ${r_{\mathrm{h}}}^{-3}$.

The right exterior is globally hyperbolic. We use

The free scalar action is

Let $Y_0=(4\pi {)}^{-1/2}$. In the s-wave sector,

For a real reduced test $f\in {C_c}^{\infty }({\mathbb{R}}^2)$, put

Then

The factor follows from $\mathrm{d}{\mathrm{vol}}_g=r^2\mathsf{f}(r)\mathrm{d}t\mathrm{d}x\mathrm{d}\Omega$.

The incoming horizon and infinity modes give complete stationary scattering for the massless Schwarzschild scalar. In the s-wave sector there is a unitary spectral transform

intertwining $\mathsf{L}$ with multiplication by ${\omega }^2$. The flux-normalized two-channel scattering matrix is unitary. The free Cauchy problem and advanced/retarded Green operators have their usual causal support.

The classical scattering input is the Schwarzschild wave-scattering construction Dimock; the Green-operator input is standard normally hyperbolic theory BGP. The specific weighted estimates needed here are proved below.

The initial field is the centered, regular, gauge-invariant quasifree Unruh prescription. The incoming horizon sector has occupation $n_{\mathrm{H}}(\omega )=(e^{\beta \omega }-1{)}^{-1}$; the incoming infinity sector is vacuum; the sectors are independent. This prescription has the bulk Hadamard identification of Dappiaggi–Moretti–Pinamonti DMP.

The global Hadamard proof is not reproduced. Hadamard regularity alone would not select this state.

Let $f_{+}(x,\omega )\sim e^{\mathrm{i}\omega x}$ at $+\infty$ and $f_{-}(x,\omega )\sim e^{-\mathrm{i}\omega x}$ at $-\infty$. Define

Let ${\rho }_{\mathrm{I}}$ be the outgoing-infinity reflection amplitude of the infinity-incident mode. The spectral convention is

The canonical field expansion is

Annihilation tests are conjugate-linear:

Write

For a compact reduced test,

The common covariance norm is

The ordinary two-channel norm is denoted $\Vert k{\Vert }_0$. It is dominated by $\Vert k{\Vert }_{\mathrm{U}}$. In the baseline,

The potential is smooth and nonnegative,

The zero-energy Jost solutions are

Choose ${\omega }_c>0$ with $|\mathcal{W}(\omega )|\ge 1/2$ on $[0,{\omega }_c]$. For every $X>0$, a finite constant $C_X$ exists with

One explicit choice is

Proof. Use $\mathrm{d}x={\mathsf{f}}^{-1}\mathrm{d}r$ to obtain $\int V\mathrm{d}x={{\int }_1}^{\infty }{r}^{-3}\mathrm{d}r=1/2$. The first moment follows from logarithmic horizon behavior and $x=r+O(\log r)$ at infinity.

Direct substitution gives the zero-energy solutions and their Wronskian. The first-moment condition makes the Jost Volterra series continuous at zero, so its nonzero Wronskian supplies ${\omega }_c$.

In the low-frequency Volterra series use $|\sin (\omega z)/\omega |\le |z|$. This gives $|f_{\pm }|\le e^{m_X}$ on the compact region; $|\tau |\le 4\omega$ gives the threshold bound. For $\omega \ge {\omega }_c$, the series is bounded by $e^{\Vert V{\Vert }_1/{\omega }_c}$, and current conservation gives $|\tau |\le 1$.

Define

For a real compact smooth test supported in $|x|\le X$,

Furthermore,

Proof. At low frequency,

At high frequency, one time integration by parts gives

Use $1+2n_{\mathrm{H}}\le 1+2/(\beta \omega )$, sum both channels, and integrate the two intervals. This gives (eq:Bbound). Integrating the same estimates with weight $\omega$ gives (eq:energybound).

These estimates retain the thermal horizon sector. They do not replace unselected radiation by vacuum.

The message and source are qubits. The encoding is the fixed unitary identification

where

The symbols $|+⟩$, $|-⟩$ here denote $Z_S$ eigenstates, not $X$ eigenstates.

The initial state is

There is no message dependence in the initial field and no additional information-bearing radiation factor.

Choose a real compact reduced profile $f$, and set $F=\mathfrak{F}[f]$. Its support is strictly exterior for every fixed compact spatial support. The same qubit controls the full shell. This is an effective extended source in a specified slicing, not a covariant microscopic matter field.

The lapse and spatial volume on a time slice are

Consequently

and

The interaction-picture Hamiltonian is

The corresponding action term is

Define the instantaneous coefficient

Then $k_F=\int k_t\mathrm{d}t=k(f)$.

The map $t\mapsto k_t$ is smooth in the weighted norm and has compact temporal support. Each ${\Phi }_F(t)=X(k_t)$ is a well-defined essentially self-adjoint field operator on the standard finite-particle core of a quasifree Fock realization.

Proof. The compact-region threshold bound controls the weighted low-frequency integral uniformly on compact time intervals. At high frequency,

Time derivatives introduce finite powers of $\omega$ and derivatives of $f$. Spectral unitarity and the compact spatial smoothness of $f$ give the required bounds by applying sufficiently many powers of $\mathsf{L}$ to those derivatives. Thus all time derivatives exist in the weighted norm.

For clarity, a doubled Fock realization is available. On its one-particle space use the real-linear map

It obeys

The vacuum Weyl characteristic function is the required one. The Segal field $a(Tk)+a(Tk{)}^{†}$ is essentially self-adjoint on finite-particle vectors: the usual creation/annihilation estimates make those vectors analytic for each such field. This realizes $X(k_t)$ and its closure. It does not posit a global stationary density in the physical, undoubled radiation Fock space.

For $t\ge t_0$, define

The interaction (eq:HI) has the unitary propagator

It satisfies the composition law and the Schrödinger equation on the common field-number form domain in the doubled Fock realization.

After the source has switched off,

Its adjoint action depends only on the free test class of $F$; the time-ordering phase is common to both source branches.

Proof. All integrals in (eq:phase) are finite by Lemma lem:instant and $|\sigma (k,l)|\le 2\Vert k{\Vert }_{\mathrm{U}}\Vert l{\Vert }_{\mathrm{U}}$. Weyl operators are unitary, so the displayed formula is unitary.

A chronological product of short interactions has factors $W(-sg{k}_{t_j}\Delta t_j)$, with later times on the left. The exact Weyl product gives the phase

Smooth weighted continuity makes its limit (eq:phase); the summed coefficients converge to $K(t,t_0)$. Strong continuity of the Weyl representation gives (eq:propagator). Splitting the same integrals proves the propagator composition law.

The number form domain is invariant under Weyl operators. Indeed the displacement identity for annihilators and the creation/annihilation estimates bound its graph norm after a finite displacement. On this domain, a differentiable one-particle path $r(t)$ obeys

This follows from the Weyl relation and the Stone derivative of $W(\delta r)$; approximation by coherent finite-particle vectors extends it in the form-domain graph topology.

For $r=-sgK$, the scalar derivative term is $\mathrm{i}{g}^2\sigma (k_t,K)/2$. It cancels $\mathrm{i}\dot{\gamma }$, leaving ${\partial }_{t}U=-\mathrm{i}{H}_I(t)U$. This also establishes uniqueness on that domain by comparing two solutions and differentiating their unitary relative propagator.

Finally $\varphi (Pu)=0$, so replacing $F$ by $F+Pu$ does not change $k_F$. A different path phase is an overall phase and leaves the adjoint action unchanged.

No operator-norm estimate for an unbounded field Hamiltonian has been inferred from the finite-generator estimate of Em03-E. The construction uses exact Weyl products and a stated domain.

For the rest of the manuscript abbreviate

Define the conditional field states

The field-output map is the channel into field state functionals

Its Heisenberg adjoint is the unital CP map

In particular,

For every reference-source input, its referenced output is

Proof. Expand the controlled unitary (eq:UF). Tracing the source keeps exactly its diagonal blocks. The global phase cancels. This proves the channel formula and all matrix-unit identities. Positivity of each state functional proves complete positivity of the displayed adjoint; their normalization proves unitality. The same block calculation retains $R$.

The retained-source channel preserves diagonal matrix units and multiplies each off-diagonal one by

Equivalently,

Its half-diamond distance from the identity is $(1-{\chi }_F)/2$, and its Bell-input fidelity is $(1+{\chi }_F)/2$.

Proof. The coefficient of $|s⟩⟨t|$ after tracing the field is

The branch directions are collinear, so there is no Weyl phase. For $s\ne t$, $(s-t{)}^2=4$, giving (eq:sourcecoherence).

The resulting qubit channel is

On a Bell input its two Pauli outputs are orthogonal; the triangle inequality supplies the matching diamond upper bound. Their identity weight is the Bell fidelity.

For superposed source inputs, the joint evolution creates source–field correlations. Removing the retained source removes the off-diagonal source blocks from the radiation channel; it is not a claim that the joint unitary itself destroys coherence.

-channel boundary. The retained-source channel is not the complete complement of a selected packet channel. That complement would also contain unselected scalar modes, the future-horizon output, and a specified purification of the mixed background. No complementary distance is asserted from ${\mathcal{B}}_F$ alone.

For a normalized compact-band packet $h$, stationary scattering gives

Thus

Define the outgoing source coefficient

If $\mathsf{S}(\omega )$ is the complete scattering matrix, the other component of $\mathsf{S}{k}_F$ is denoted $d_{\mathrm{hor}}$.

The conditional outgoing packet amplitude is

Consequently,

These are calculated from the bulk profile and scattering map.

Proof. The Weyl displacement identity gives

Insert $r=-sg{k}_F$. The initial state is centered, so its conditional mean is (eq:alpha). Taking the quadrature combinations proves the result.

Choose orthonormal $h_1,\dots ,h_m$ in a common compact positive-frequency band. Set $b_i=b_{h_i}$ and

This convention retains the covariance transposition: ${\omega }_{\mathrm{U}}({b_i}^{†}{b}_j)=A_{ji}$.

Let

The selected baseline has a unique normal density ${\rho }_0$ on ${\mathcal{F}}_{\mathrm{s}}({\mathbb{C}}^m)$ with

If $A=W\mathrm{diag}({\lambda }_1,\dots ,{\lambda }_m)W^{†}$, it is the passive rotation of the product densities

It commutes with total number.

Proof. The restricted covariance is finite, positive, centered, and has zero anomalous part. Construct independent circular complex Gaussian amplitudes of variances ${\lambda }_j$, rotate them by $W$, and form their coherent-state mixture. It is positive trace class of trace one and has exactly the displayed characteristic function.

For one variance $\lambda >0$, angular integration kills off-diagonal number terms and radial integration gives ${\lambda }^n/(1+\lambda {)}^{n+1}$. Zero variance means vacuum. Equality of finite-dimensional Weyl characteristic functions identifies the normal density and proves uniqueness. Passive rotation preserves total number.

Define

The conditional finite packet states are

The source-to-packet channel is the normal CPTP map

This includes all source matrix units and all references.

Proof. Conjugating a selected Weyl operator by $W(r_s)$ multiplies its baseline characteristic function by the displacement phase determined by (eq:alpha). This is the characteristic function of (eq:packetstates). The latter is a normal density by Lemma lem:gaussian, so characteristic-function uniqueness proves normality of the restricted output. The channel formula follows from Theorem thm:fieldchannel and is CPTP by its measure-and-prepare form.

The selected packets do not exhaust the source signal. The components of $d_{\infty }$ orthogonal to their span and the coefficient $d_{\mathrm{hor}}$ remain unselected. The retained source also remains a physical factor of the model.

The measure-and-prepare characterization of entanglement-breaking channels is standard EntanglementBreaking. The following bounds are proved directly for the present normal packet channel.

For every reference-source density,

is separable across reference and radiation. Every radiation-only deterministic CPTP decoder to a qubit has Bell-input fidelity at most $1/2$ and recovery error at least $1/2$:

Proof. The displayed positive product decomposition proves separability. Local decoding preserves it. A normalized product vector has overlap at most $1/2$ with

by Cauchy–Schwarz. Convexity extends this bound to separable states. Testing the Bell projector gives trace-distance error at least $1-F_e\ge 1/2$, hence the diamond lower bound.

For the declared finite packet output,

These are information-theoretic decoder optima, not local decoder-implementation claims.

At every finite $g$ and compact smooth source profile,

Thus the optimal recovery error is strictly greater than $1/2$.

Proof. For any decoder, let $E={\mathcal{D}}^{\ast }(|+⟩⟨+|)$. Its Bell overlap is

using the positive-part characterization of trace distance.

Let $\Pi =e^{\mathrm{i}\pi N_{\mathrm{tot}}}$. Lemma lem:gaussian gives $\Pi {\rho }_{+}{\Pi }^{†}={\rho }_{-}$. For $T={\rho }_{+}-{\rho }_{-}$, choose the Helstrom effect $E_{+}=(I+\mathrm{sign}T)/2$, with value $1/2$ on the kernel. Then $\Pi E_{+}{\Pi }^{†}=I-E_{+}$. Both conditional correct-decision probabilities equal $q=(1+d_{\mathbf{h}})/2$.

Measure this effect and prepare the corresponding source basis state. The decoded qubit channel has Pauli probabilities

For a Pauli channel its half-diamond distance from identity is $1-{\lambda }_I$: the Bell outputs are orthogonal and the triangle inequality gives the matching upper bound. This decoder attains both values in (eq:recovery). Every decoder has error at least $1-F_e$, proving optimality.

For the strict upper bound on $d_{\mathbf{h}}$, compare the two vector states $W(-a){\Omega }_{\mathrm{U}}$ and $W(a){\Omega }_{\mathrm{U}}$ in the regular quasifree realization. Their overlap modulus is $e^{-2\Vert a{\Vert }_{\mathrm{U}}^2}$. Their pure-state distance is $\sqrt{1-e^{-4\Vert a{\Vert }_{\mathrm{U}}^2}}$. Restriction to packets is contractive. Lemma lem:compact makes the exponent finite.

A nonzero classical signal is therefore compatible with a quantified impossibility of faithful unknown-qubit transfer.

For a weighted coefficient $l$, define

where

Both are effects because the real and imaginary parts of a unitary are self-adjoint contractions.

With the evolution sign fixed in (eq:UF),

Hence

and the sign gap is

The cosine expectations are identical for both signs. If $l$ belongs to the selected outgoing packet algebra, $d_{\mathbf{h}}\ge {\Delta }_{\sin }$.

Proof. The exact Weyl identity is

Insert $r=-sg{k}_F$, then the centered Gaussian characteristic function. Real and imaginary parts give the two effects. An effect probability difference is bounded by state trace distance, proving the last assertion.

For the packet family in Theorem thm:packetnormal, every joint number-diagonal probability is the same in ${\rho }_{+}$ and ${\rho }_{-}$. Each marginal occupation mean is

For one packet, its generating function is

Proof. The total parity $\Pi$ maps ${\rho }_{+}$ to ${\rho }_{-}$ and acts on each joint number vector by a scalar sign. Its diagonal probabilities are therefore unchanged, even if the baseline packet covariance has off-diagonal correlations.

Conjugation by displacement replaces $b_i$ by $b_i+{\alpha }_{s,i}$. The centered baseline gives ${\nu }_i+|{\alpha }_{s,i}{|}^2$.

For one packet the baseline is a circular coherent-state Gaussian mixture of variance ${\nu }_h$. After displacement its coherent amplitude has mean ${\alpha }_{s,h}$. Conditional number counts are Poisson. The circular Gaussian integral of $e^{-(1-t)|z{|}^2}$ gives (eq:displacedPGF); the vacuum case follows by its zero-variance limit.

The two labels have identical count laws, but those laws need not equal the undisplaced Hawking baseline.

If $B=\sigma (k_F,l)\ne 0$, then for fixed profiles and witness,

For $|gB|\le \pi /2$,

Thus a phase-sensitive bounded witness can be first order while the occupation-mean departure is second order.

This section supplies an analytical benchmark, not an evaluated experiment.

Choose any ${\omega }_0>0$. The two incoming generalized eigenfunctions are linearly independent, and $\tau ({\omega }_0)\ne 0$. Therefore

is not the zero solution. Choose a finite $x_0$ with $M_0=M(x_0,{\omega }_0)\ne 0$.

Let

Use normalized rescalings to choose real nonnegative bumps

with both integrals one. Choose $\delta$ small enough that

These finite choices exist by continuity.

Set

This has fixed compact exterior shell support.

For this profile,

where ${\hat{\chi }}_{\mathrm{plain}}(\omega )=\int {\chi }_{\delta }(t)e^{\mathrm{i}\omega t}\mathrm{d}t$. In particular $d_0=d_{\infty }({\omega }_0)\ne 0$.

Choose a compact positive-frequency band $B_0$ around ${\omega }_0$ on which

Let $H$ be a nonnegative normalized smooth bump inside $B_0$, and define

Then

The explicitly specified compact profile and packet satisfy $c_h\ne 0$. For $\vartheta >0$, choose

Then

Consequently

For $0<|g|\sqrt{2}\vartheta |c_h|\le \pi /2$, the gap is strictly positive.

Proof. The preceding bump estimates prove $d_0\ne 0$, and the band estimate proves $c_h\ne 0$. Since

its imaginary part is $\vartheta |c_h|/\sqrt{2}$. Scattering unitarity and the baseline covariance give the stated norm. Apply Theorems thm:witness and thm:counts.

No independently chosen packet displacement is used in this construction. The phase of the packet and of the witness is a declared experimental reference.

The fixed source shell is sufficient for nonzero coupling. Growing shells will enter the local approximation to its noncompact outgoing packet observable, not the source-existence proof.

For $C\ge 1$, let ${\mathfrak{S}}_C$ be the normal states in the regular Unruh representation satisfying

on the real weighted s-wave coefficient domain. The second moment includes any nonzero mean.

Both source-displaced states, and every mixture of them, belong to ${\mathfrak{S}}_{C_{\mathrm{src}}}$, where

Their two-point functions differ from the baseline by smooth terms, so the conditional states and their mixtures retain the Hadamard two-point singularity.

Proof. Weyl conjugation shifts $X(u)$ by $-sg\sigma (k_F,u)$. The baseline is centered, so

Convexity gives the mixture statement.

The displacement mean is the pairing with the smooth solution $EF$. The two-point change is its smooth product. For mixtures the analogous two-point change is still smooth. No change of the Hadamard singularity occurs.

For $\omega \in {\mathfrak{S}}_C$,