Curved Channels, Communication, and Entanglement Harvesting

A curved spacetime does not by itself define a communication channel. A channel appears only after one fixes an encoding, localized interactions, causal propagation, an accessible receiver algebra, a decoding, and a resource-bounded task. This chapter turns those choices into operational maps, then separates genuine signal transmission from pre-existing field correlations, deterministic redshift from noise, and restricted access from destruction of information.

Helpful background. Completely Positive Maps and Causal Quantum Channels supplies the abstract channel language, while Localized Probe and Detector Models supplies local couplings. Green Operators, Causal Propagators, and State-Dependent Two-Point Functions distinguishes causal response from state covariance. Detector Response Along Curved and Accelerated Worldlines supplies the proper-time detector response used in accelerated protocols.

A channel is an operational construction

Let a sender system $A$ and receiver system $B$ couple to a real scalar field through compactly supported interaction densities. Schematically,

$$\mathcal N_{A\to B} =\mathcal D_B\circ\operatorname{Tr}_{\mathrm{inacc}} \circ\mathcal U_B\circ\mathcal U_\Phi\circ\mathcal U_A\circ\mathcal E_A .$$

Here $\mathcal E_A$ encodes a message, $\mathcal U_A$ and $\mathcal U_B$ are localized couplings, $\mathcal U_\Phi$ is field propagation on the stated geometry and state, $\operatorname{Tr}_{\mathrm{inacc}}$ denotes only a declared restriction to inaccessible observables or modes, and $\mathcal D_B$ is the receiver decoding. The task must additionally name a metric—error probability, fidelity, distinguishability, or an asymptotic rate—and a resource constraint.

For a free field, two kernels play different roles. With the site convention

$$E=G_{\mathrm{ret}}-G_{\mathrm{adv}}, \qquad [\Phi(f),\Phi(h)]=-iE(f,h),$$

the causal response is fixed by $E$ and the field equation. The symmetrized two-point function

$$H_\omega(f,h)=\frac12\,\omega\!\left(\{\Phi(f),\Phi(h)\}\right)$$

depends on the state and supplies fluctuations. A nonzero $H_\omega$ between two interaction regions can generate correlated detector outcomes even when $E$ vanishes there; it is therefore not a signaling kernel. This response–fluctuation separation is the organizing principle of the chapter and is consistent with the algebraic construction of relativistic communication channels on globally hyperbolic spacetimes Barcellos and Landulfo 2024, §§ II–IV.

Domain and failure conditions

This is the chapter’s canonical comparison table. Every row specifies a task rather than attributing an intrinsic “channel quality” to curvature. “Validation” names the decisive control, not a maturity rating.

Protocol class	Geometry, state, and worldlines	Coupling and encoding	Accessible output algebra	Causal exchange	Noise or restriction mechanism	Resource and task	Metric or capacity notion	Decisive validation or downgrade
Local probe channel	Globally hyperbolic background; declared Hadamard state; compact probe supports	Smeared monopole or oscillator coupling; finite code alphabet	Probe observables after field trace	Smeared retarded kernel $E$	Field covariance, detector preparation, switching	Coupling budget and sender energy; one-shot transmission	Output distinguishability or fidelity	Delete the output algebra or resource bound: the operational task becomes undefined
Stationary Gaussian channel	Stationary geometry and quasifree state	Linear oscillator couplings; quadrature encoding	Selected receiver quadratures plus declared leakage modes	Gain matrix from retarded response	Added covariance from field and receiver	Mean excitation bound; repeated-use assumptions	Gaussian fidelity, coherent information, or classical rate	Replacing $E$ by a Wightman function confuses correlations with signal
Tail-mediated channel	Curved spacetime violating strong Huygens propagation	Compact source and detector switching	Receiver detector algebra	Timelike support inside the null cone	State noise remains separate	Finite interaction time and energy	Signaling estimator or induced channel norm	Correlation outside direct null support is not evidence until the commutator is tested
Accelerated-probe channel	Declared inertial and accelerated trajectories; proper-time clocks	Local detector coupling and timed modulation	Accelerated receiver observables	Trajectory-pulled retarded kernel	Unruh response, switching transients, mismatch	Proper-time duration, acceleration, coupling, energy	Gain and added noise; not detector temperature alone	Background clicks can change while causal gain does not, and conversely
Redshifted wavepacket link	Static or stationary geometry; specified emitter and receiver frames	Normalized wavepacket code with redshift-aware decoder	Full received mode family or an explicit subsystem	Causal propagation between wavepackets	Mode mismatch, loss, tracing, receiver noise	Local sender and receiver energy conventions	Overlap, transmissivity, fidelity	On the complete mode algebra a unitary frequency relabeling is not decoherence
Spacelike harvesting	Hadamard state; spacelike-separated compact supports	Two initially uncorrelated probes	Joint probe algebra	Zero between the supports	State anticommutator and local excitation	Coupling order, separation, switching, smearing	Negativity or another declared witness	Curvature has no universal monotonic effect; use a matched flat control
Timelike or tail-connected extraction	Timelike or interior-null-cone support	Same detector protocol with causal connection retained	Joint probe algebra	Generally nonzero	State correlations plus exchange	Same perturbative and detector controls	Final entanglement with exchange decomposition	If the result disappears when the commutator term is removed, call it exchange-assisted rather than pure harvesting
Horizon-restricted operations	Wedge or exterior local algebras; state and split collar declared	Completely positive local operations	Named von Neumann algebras, not a geometric tensor factor	Limited by causal accessibility	Algebraic restriction and operational coarse graining	Allowed operations, communication, copies	Witness value or distillable task	A formal trace over a region is not licensed without a subsystem construction
State-transfer link	Static or accelerated laboratories with calibrated clocks	Bosonic or qubit code plus receiver mode matching	Full decoding algebra including leakage	Retarded transfer amplitude	Loss, mode mismatch, state noise	Energy, bandwidth, time, control pulses	Entanglement fidelity or worst-case fidelity	High overlap with one selected mode does not establish a high-fidelity channel
Energy-constrained coding	Declared single-use or memory channel and local Hamiltonians	Code family over $n$ uses	Decoder and any assistance stated	Already contained in the channel map	Loss, thermal noise, time dependence	Mean or maximum energy, bandwidth, coding limit	Classical, quantum, private, or assisted capacity	An unconstrained bosonic rate or untranslated redshifted energy is not comparable
Tomographic deployment	Fixed clocks, frames, wavepackets, calibration, and drift model	Informationally complete preparations and observables	Calibrated receiver instrument	Estimated causal gain with support check	Estimated covariance and systematic model	Sample size and preparation budget	Confidence region for channel parameters	Changing the clock or basis after calibration can mimic physical drift
Analogue or astronomical test	Laboratory effective metric or astrophysical propagation model	Platform-specific preparation	Measured platform observables	Platform response only unless a protocol is implemented	Instrumental and environmental systematics	Declared experimental resources	Observable-level likelihood or witness	Agreement with an analogue dispersion relation does not confirm gravitational information transfer

Read the chapter in order

1. Curved-Spacetime Channel Deployment Contract fixes the geometry, state, couplings, clocks, accessible observables, task, and resources.
2. From Propagators and Response Functions to Channel Maps derives causal gain and state-dependent noise from smeared kernels.
3. Causal Support, Signaling, and Global Geometry treats light-cone support, tails, horizons, and topology.
4. Accelerated Detector Communication Channels separates trajectory-dependent response, proper-time timing, signal gain, and added noise.
5. Redshift, Restricted Access, and Effective Channel Noise distinguishes deterministic frequency conversion from mismatch, tracing, and receiver noise.
6. Entanglement Harvesting in Curved Spacetime computes detector negativity and uses matched flat fixtures.
7. Correlation Extraction versus Causal Exchange on Curved Backgrounds decomposes anticommutator and commutator contributions.
8. Horizon-Restricted Local Operations and Distillability replaces informal region traces by accessible local algebras and split inclusions.
9. Entanglement Distribution and State Transfer Through Curved Fields evaluates a complete decoded channel rather than one mode overlap.
10. Energy-Constrained Capacity Under Redshift and Acceleration states coding tasks with local Hamiltonians and finite resources.
11. Relativistic Protocols, Clocks, Encoding, and Channel Tomography makes calibration and coordinate changes explicit.
12. Evidence, Analogue Systems, and Astrophysical Claim Limits classifies what present experiments do and do not establish.

Construction and failure maps

The construction map keeps the physical propagation stage between the sender and receiver couplings. Inspect its lower checkpoint: commutator support, resource accounting, and reference-frame transformation must be verified before the reduced receiver map is interpreted.

A reproducible channel separates causal propagation from receiver restriction, noise, and the task-specific performance measure; curvature alone is not a noise mechanism. Schematic; not to scale.

The failure map identifies four common overclaims. Each lower witness removes a required element of the operational statement, so the conclusion must stop or be narrowed at that boundary.

Channel claims are licensed only with specified observables, state, geometry, encoding, access, and resources; the four displayed witnesses force a precise downgrade. Schematic; not to scale.

Handoffs

Causal Channels and Relativistic Communication develops the abstract channel, coding, harvesting, and protocol theory used here. Particles, Detectors, and Nonadiabatic Production owns response functions and particle interpretation. A reproducible calculation can provide the computational checks; mutable experimental claims require a dated research record rather than silent promotion into a durable theorem.

References

• Barcellos, Ian Bernardes, and André G. S. Landulfo. “Relativistic Quantum Broadcast Channel.” Physical Review D 109 (2024): 065020. DOI. Open PDF.
• Fewster, Christopher J., and Rainer Verch. “Algebraic Quantum Field Theory in Curved Spacetimes.” In Advances in Algebraic Quantum Field Theory, edited by Romeo Brunetti, Claudio Dappiaggi, Klaus Fredenhagen, and Jakob Yngvason, 125–189. Cham: Springer, 2015. DOI. Open PDF.