Energy-Constrained Capacities and Coding Tasks

A capacity is defined only after choosing the communication task, admissible codes, energy or cost operator, bandwidth or localization constraint, blocklength limit, and error criterion. In an infinite-dimensional field channel, omitting the constraint can make rates physically meaningless and continuity arguments fail. A single-use mutual information is generally an achievable expression or bound, not automatically an asymptotic capacity.

Required background. Infinite-dimensional channel–state methods supplies the energy domain. Field communication supplies the physical channel.

Helpful background. Quantum communication and entanglement distribution identifies the coherence-sensitive task.

Name the capacity

For a channel $\mathcal N$ and constraint $\operatorname{tr}(H\rho)\le E$, the one-shot Holevo information of an ensemble $\{p_x,\rho_x\}$ is

$$\chi= S\!\left(\sum_xp_x\mathcal N(\rho_x)\right) -\sum_xp_xS\!\left(\mathcal N(\rho_x)\right).$$

The classical capacity is obtained from a regularized block-code optimization unless additivity is established. Quantum capacity is governed by regularized coherent information in general. Entanglement-assisted classical capacity has a single-letter mutual-information formula under the stated constraint, but it consumes pre-shared entanglement. Private capacity is another task again.

The Hamiltonian $H$ must refer to the physical input modes or localized encoding. A mean-energy constraint, a peak-energy constraint, and a per-codeword constraint define different admissible sets.

Capacity is the coding branch of the physical channel. Its label must include the task and constraints; it is not a generic measure of field correlation or signal visibility. The diagram is schematic.

Lossy bosonic wavepacket

For a pure-loss mode channel of transmissivity $\eta$ with mean photon number $N$, the energy-constrained classical capacity in bits per use is

$$C(N)=g(\eta N), \qquad g(x)=(x+1)\log_2(x+1)-x\log_2x,$$

for the ideal single-mode model. The quantum capacity under the corresponding assumptions is

$$Q(N)=\max\!\left\{0,\, g(\eta N)-g((1-\eta)N) \right\}$$

in the degradable pure-loss regime, with appropriate limiting optimization. A localized field protocol must derive $\eta$ and the usable mode from sender–receiver overlap, and it must account for bandwidth, channel uses, switching time, and environmental noise. These formulas are not capacities of arbitrary QFT communication setups.

The underlying bosonic rate optimization and its physical constraints are reviewed by Caves and Drummond 1994, §§ IV–VI, pp. 500–528.

Constraint adversaries

Change mean energy to peak energy. Codes with rare high-energy pulses that were admissible on average can disappear, so a prior rate need not remain achievable. Alternatively remove a bandwidth bound: increasingly high-frequency modes may change the rate per unit time or invalidate the localized coupling model. A fair comparison holds the physical task, time normalization, and receiver access fixed.

Infinite-dimensional capacity theory and continuous ensembles are treated carefully by Holevo and Shirokov 2005, §§ 2–4, pp. 88–96. Detector-mediated field channels with explicit energy accounting are analyzed by Barcellos and Landulfo 2021, §§ III–VI. Their results illustrate why switching energy and communication energy should not be merged.

An omitted energy or bandwidth constraint can inflate a capacity without changing the causal channel. Localization tails and receiver mismatch are separate physical failures that a mode-only formula may conceal. The map is schematic.

Common pitfalls

Calling $I(X{:}Y)$ a capacity. Mutual information for one input distribution is an achieved one-use value. Capacity requires optimization and, generally, a blocklength limit.

Comparing rates with different clocks. Bits per channel use, bits per proper time, and bits per unit energy are inequivalent. State the normalization.

References

• Barcellos, I. B., and Landulfo, A. G. S. (2021). “Relativistic Quantum Communication: Energy Cost and Channel Capacities.” Physical Review D 104, 105018. DOI. Open PDF.
• Caves, C. M., and Drummond, P. D. (1994). “Quantum Limits on Bosonic Communication Rates.” Reviews of Modern Physics 66, 481–537. DOI.
• Holevo, A. S., and Shirokov, M. E. (2005). “Continuous Ensembles and the Capacity of Infinite-Dimensional Quantum Channels.” Theory of Probability and Its Applications 50, 86–98. DOI. Open preprint.