Energy accounting of relativistic quantum communication: generality of the vanishing communication-specific energy cost and the particle-creation contribution in dynamical spacetimes

Barcellos and Landulfo (Phys. Rev. D 104, 105018, 2021) proved that when two observers communicate through a massless scalar quantum field using gapless two-level detectors in an asymptotically flat spacetime, the total energy variation splits into a particle-creation term $W_\phi$ , the switching works $W_A$ and $W_B$ , and a communication-specific term $W_{AB}$ that can be made to vanish while the channel capacities stay maximal: communicating through an already-built pair of qubits costs no extra energy. The result rests on three restrictions: Hadamard field states, a massless minimally coupled scalar carrier, and one exactly solvable detector model with a specific receiver ready state. The open problem is to complete this energy accounting, its core being to determine whether $W_{AB}=0$ is generic across physically admissible detector-field coupling models or whether reliable communication can carry an unavoidable energy cost. The result's scope restrictions further suggest open extension directions: to massive, non-minimally coupled, and gauge-field carriers and to state classes beyond Hadamard states, as well as an explicit computation of the renormalized particle-creation contribution $W_\phi$ in a dynamical spacetime such as black-hole formation and evaporation.

Background

Relativistic quantum information (RQI) studies how the relativistic structure of spacetime — causal horizons, particle creation, and observer dependence of the quantum vacuum — affects tasks such as communication and entanglement distribution. A standard tool is the particle detector: a localized quantum system, often a two-level system (a qubit), coupled for a finite time to a quantum field along an observer's worldline (the Unruh-DeWitt model and its relatives). Landulfo (2016) built a nonperturbative communication channel of this type: a sender Alice and a receiver Bob, each carrying a two-level system with degenerate (gapless) energy levels, couple linearly to a real scalar field through a $\sigma_z$ -type interaction, and the field mediates a linear completely positive trace-preserving map from Alice's qubit state to Bob's. In that model the classical capacity (the maximum rate at which classical bits can be sent with arbitrarily small error) is nonzero, while the unassisted quantum capacity vanishes because the channel is entanglement-breaking (the environment effectively measures the input); prior shared entanglement restores quantum transmission at half the classical rate.

A long-standing companion question asks what energy such transmission requires. Bekenstein (1981) showed that reliably transferring one bit with a signal of frequency $\omega$ and duration $\Delta t$ costs at least a universal multiple of $\hbar$ ; Jonsson, Martín-Martínez and Kempf (2015) showed that in $1+1$ -dimensional massless fields information can propagate in field commutators with strictly zero local energy flow; and a separate line quantified minimal energy costs of entanglement extraction from fields (Hackl and Jonsson, 2019) and of entangling Hawking radiation with the late-time vacuum (Wald, 2019).

Barcellos and Landulfo (2021) analyzed the energy cost of the full communication process in general asymptotically flat, globally hyperbolic spacetimes, possibly containing black holes. They use null-surface quantization — a quantization of the field restricted to three-dimensional null hypersurfaces such as future null infinity $\mathscr{I}^+$ and a future event horizon $h^+$ , where the field algebra is built from data on the null surface — together with the assumption (valid for massless minimally coupled fields, labeled (BB1),(BB2) in the paper) that every bulk solution extends uniquely to $h=\mathscr{I}^+\cup h^+$ with a conserved symplectic form, which identifies the bulk field algebra with the algebra on $h$ . Evolving the total two-qubit-plus-field system from the asymptotic past to the asymptotic future, they proved that the total energy variation decomposes as

\Delta E = W_\phi + W_A + W_B + W_{AB},

where $W_\phi$ is the energy change due to particle creation by the changing spacetime geometry (present even without Alice and Bob), $W_A$ and $W_B$ are the works needed to switch each qubit on and off, and $W_{AB}$ is the communication-specific contribution, proportional to $\langle\sigma_z^A\rangle\langle\sigma_z^B\rangle$ and to a null-surface integral of the two smeared causal solutions. Three restrictions define the scope of the result. First, $W_\phi$ involves divergent field-energy expectation values and is rendered finite only for Hadamard states — the physically admissible states whose short-distance singularity structure matches the universal Hadamard form, so that point-splitting renormalization (defining products of fields at coincident points by subtracting the universal singularity) applies. Second, the bulk-to-null-surface identification is established only for a massless, minimally coupled real scalar field; massive fields develop tails (backscattering off curvature) so that not all field information reaches $\mathscr{I}^+\cup h^+$ . Third, the proof that $W_{AB}$ can be made to vanish — achieved by preparing Bob's qubit in the ready state $|y_+\rangle$ with $\sigma_y|y_+\rangle=|y_+\rangle$ , which simultaneously maximizes the channel capacities — uses the specific gapless linear-coupling qubit model. For pointlike inertial detectors the switching cost was estimated as $W_A=\epsilon_A^2\alpha_A/4\pi$ (with coupling $\epsilon_A$ and switching timescale $\alpha_A^{-1}$ ), while the energy cost of creating and initializing the qubit devices themselves was not computed. The paper motivates the whole analysis by the black-hole information problem — whether information that falls into a black hole can return during evaporation — but performs no dynamical-spacetime computation.

Problem Statement

Complete the energy accounting of relativistic quantum communication as formulated by Barcellos and Landulfo (2021), by removing its scope restrictions: determine the energy budget of communicating classical and quantum information through quantum fields in general spacetimes, and in particular whether the communication-specific energy cost is generically zero.

The central open question is the generality of $W_{AB}=0$ : determine whether the vanishing of the communication-specific contribution $W_{AB}$ — proven for gapless two-level detectors with linear $\sigma_z$ coupling to a massless minimally coupled scalar field and for the receiver ready state $|y_+\rangle\langle y_+|$ — holds for arbitrary physically admissible detector models and coupling Hamiltonians, that is, models in which the detector-field interaction defines a well-defined communication channel (a completely positive trace-preserving map from sender to receiver) and the decomposition $\Delta E=W_\phi+W_A+W_B+W_{AB}$ is well defined with each contribution finite after renormalization (for example gapped Unruh-DeWitt detectors, delta-coupled detectors, or harmonic-oscillator detectors), or exhibit a physically motivated model in which reliable communication (nonzero classical capacity) carries an unavoidable nonzero communication-specific energy cost.

The remaining scope restrictions of the source result indicate further open directions that a fuller energy accounting would address; these describe the program's context rather than joint requirements that an answer must satisfy: extending the decomposition $\Delta E=W_\phi+W_A+W_B+W_{AB}$ beyond the massless minimally coupled real scalar field — to massive and/or non-minimally coupled fields and, if possible, gauge-field carriers — including a replacement for the bulk-to-null-surface algebra identification where assumptions (BB1),(BB2) fail due to tail effects, and characterizing for which state classes beyond Hadamard states a finite renormalized $W_\phi$ can still be defined; and computing the renormalized particle-creation contribution $W_\phi$ explicitly in a dynamical spacetime such as a black-hole formation and evaporation geometry. The source paper motivates the analysis by the black-hole information problem — whether information that falls into a black hole can return during evaporation — but performs no dynamical-spacetime computation; that motivation is programmatic rather than an auditable sub-question.

An answer may also supply the missing device-side accounting — the energy required to create and initialize the qubit devices themselves, which the source leaves uncomputed; this last component is the least sharply posed and does not by itself define completion.

References

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