Energy-Constrained Capacities and Coding Tasks

A channel capacity is an asymptotic coding rate, not a generic measure of signal size. In a field theory, its definition must name the task, physical input Hamiltonian, energy constraint, allowed frequency and spatial modes, channel-use clock, and error criterion. Without those data an infinite-dimensional optimization may include states or modes that the localized sender cannot prepare and the receiver cannot collect.

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

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

The chapter’s task map, protocol comparison, and failure controls connect the coding problem to its causal implementation without repeating the overview figures.

Capacity begins with a code definition

Let $\mathcal N_{A\to B}$ be a channel, let $H_A\geq0$ be its physical input-energy observable, and let $E$ be the allowed mean energy per use. For an ensemble $\{p_x,\rho_x\}$ with average state $\bar\rho=\sum_xp_x\rho_x$, the one-use Holevo information is

$$\chi(\{p_x,\rho_x\},\mathcal N) =S(\mathcal N(\bar\rho)) -\sum_xp_xS(\mathcal N(\rho_x)),$$

subject, for example, to $\operatorname{tr}(H_A\bar\rho)\leq E$. Its optimization is not automatically the classical capacity. In general one regularizes over blocklength,

$$C(E)=\limsup_{n\to\infty}\frac1n \sup_{\operatorname{tr}(H_A^{(n)}\bar\rho^{(n)})\leq nE} \chi(\{p_x,\rho_x^{(n)}\},\mathcal N^{\otimes n}),$$

where $H_A^{(n)}=\sum_{j=1}^nH_{A_j}$. Additivity can turn this expression into a one-use formula, but it must be proved for the channel class.

Quantum capacity uses coherent information. For a complementary channel $\mathcal N^c$,

$$I_c(\rho,\mathcal N) =S(\mathcal N(\rho))-S(\mathcal N^c(\rho)).$$

Its energy-constrained regularization gives the quantum rate under the corresponding code definition. For a degradable channel, the complementary output can be simulated from the receiver output, coherent information is additive and concave, and the optimization becomes single-letter under the standard finiteness hypotheses.

Other names denote other resources:

• The entanglement-assisted classical capacity optimizes $I(R{:}B)$ for a purification $\psi_{RA}$ and permits unlimited pre-shared entanglement.
• The private classical capacity requires the receiver to decode while the complementary output learns asymptotically nothing; its private-information expression is regularized in general.
• An entanglement-transmission capacity asks for high fidelity on a growing entangled code subspace. Its equivalence to other quantum coding formulations requires matching energy and error conditions.

In infinite dimensions, these equivalences and formulas need domain assumptions. A useful sufficient setting is that $H_A$ is a Gibbs observable, $\operatorname{tr}e^{-\beta H_A}\lt\infty$ for every $\beta\gt0$, and the channel has finite output entropy on the energy-bounded set. The distinctions between uniform and average code constraints are made explicit in Wilde and Qi 2018, Definitions 1–4 and Theorems 3 and 7, pp. 7805–7812.

Mean, uniform, and peak energy are different

The phrase “energy at most $E$” can describe inequivalent admissible code families.

An ensemble-average constraint bounds

$$\sum_m p_m\operatorname{tr}(H_A^{(n)}\rho_m)\leq nE.$$

It allows an infrequent message to carry much more than $nE$ if other messages carry less. A uniform per-codeword constraint instead requires $\operatorname{tr}(H_A^{(n)}\rho_m)\leq nE$ for every message. A hard peak or occupation constraint restricts each codeword to a bounded-energy or bounded-total-photon-number subspace, which is stronger than bounding only its expectation. Average and maximum decoding error criteria can further change the conversion between code formulations.

For example, send vacuum with probability $1-\epsilon$ and a pulse with mean photon number $N/\epsilon$ with probability $\epsilon$. The ensemble mean remains $N$ as $\epsilon\to0$, while the pulse energy diverges. A source with a peak bound rejects that family. This adversarial change does not prove that one particular capacity formula increases; it demonstrates that the two optimizations do not range over the same codes.

The vacuum pure-loss packet channel

For one bosonic input mode $a$, one accessible output mode $b$, and an environmental vacuum mode $e$, the pure-loss channel is

$$b=\sqrt\eta\,a+\sqrt{1-\eta}\,e, \qquad 0\leq\eta\leq1.$$

Let $N=\langle a^\dagger a\rangle$ be the allowed mean input photon number and define, continuously at $x=0$,

$$g(x)=(x+1)\log_2(x+1)-x\log_2x.$$

The exact unassisted classical capacity is

$$C(\eta,N)=g(\eta N) \quad\text{bits per use}.$$

A circular Gaussian ensemble of coherent states achieves it. For multiple independent modes, one maximizes

$$\sum_k g(\eta_kN_k) \quad\text{subject to}\quad \sum_k\hbar\omega_kN_k\leq E.$$

The matching upper bound, coherent-state achievability, and narrowband specialization are proved in Giovannetti et al. 2004, Eqs. (4)–(14), pp. 1–3.

The energy-constrained quantum capacity of the same vacuum pure-loss channel is

$$Q(\eta,N)= \max\{0,\,g(\eta N)-g((1-\eta)N)\} \quad\text{qubits per use}.$$

For $\eta\gt1/2$ the channel is degradable and the displayed coherent information is optimal; for $\eta\lt1/2$ it is antidegradable, so no-cloning forces $Q=0$. At $\eta=1/2$ the expression vanishes. The unconstrained limit for $\eta\gt1/2$ is

$$\lim_{N\to\infty}Q(\eta,N) =\log_2\frac{\eta}{1-\eta},$$

but that limit is not the capacity at finite transmitter energy. Degradability and the unconstrained attenuation result appear in Wolf, Pérez-García, and Giedke 2007, Eqs. (8)–(12), pp. 2–4; the finite-energy equality is stated and derived in Wilde and Qi 2018, Eq. (1) and § VII.D, pp. 7803, 7821–7823.

Neither formula above applies unchanged to a thermal-loss channel. If $e$ has nonzero occupation, both receiver and complementary entropies change, and the quantum capacity is not generally given by the simple difference above. Amplification, additive noise, memory between uses, and frequency-dependent mixing likewise define different channels.

A finite-energy, finite-bandwidth benchmark

Choose a normalized packet $h(\omega)$ supported in the declared band

$$\omega_0-\frac{B}{2} \leq\omega\leq \omega_0+\frac{B}{2}, \qquad \int d\omega\,|h(\omega)|^2=1,$$

and let $a_h=\int d\omega\,h(\omega)a_\omega$. For states diagonal in the packet photon number, the mean field energy is

$$E=\hbar\bar\omega_hN, \qquad \bar\omega_h=\int d\omega\,\omega|h(\omega)|^2.$$

Suppose propagation and matched-mode extraction give a vacuum pure-loss channel with $\eta=0.75$, and impose $N=2$ photons per packet use. Then

$$\begin{aligned} C&=g(1.5)=2.4273765\ \text{bits per use},\\ Q&=g(1.5)-g(0.5) =1.0499327\ \text{qubits per use}. \end{aligned}$$

The unconstrained quantum value would be $\log_2 3=1.5849625$ qubits per use, about $0.5350$ larger; substituting it at $N=2$ would overstate the rate. The benchmark is reproducible from the four declared inputs $h$, $B$, $\eta$, and $N$. If $\eta(\omega)$ varies appreciably across the band, one must diagonalize the frequency-dependent input–output map or use a multimode allocation rather than insert an averaged transmissivity into a theorem for one pure-loss mode.

These numbers are per orthogonal packet use. If packets can be prepared and decoded every $\tau$ seconds with negligible memory, the corresponding rates are $C/\tau$ and $Q/\tau$. Bandwidth and localization constrain $\tau$; because no clock has been specified here, quoting bits or qubits per second would invent information absent from the model.

In a QFT implementation, derive $h$ and $\eta$ from the localized coupling, causal propagation, and receiver acceptance mode. Include energy spent on switching and detector preparation separately from the packet’s communication energy. Explicit relativistic detector protocols show that these contributions need not coincide Barcellos and Landulfo 2021, §§ III–VI, pp. 4–12.

Constraint adversaries and limiting checks

Replace mean by peak energy. The rare-pulse family above remains mean-admissible but violates any fixed peak constraint. Recompute the code optimization; do not keep a mean-energy capacity label on a peak-limited transmitter.

Remove the bandwidth or mode set. This changes both the input Hamiltonian’s admissible spectrum and what counts as one use. It can enlarge the optimization and alter the rate normalization. A result for one selected packet does not become a broadband field capacity merely by allowing the packet shape to vary without limit.

Warm the environment. The vacuum output of a coherent input is pure, which is crucial to $C=g(\eta N)$. A nonvacuum environment removes that step in the proof and invalidates the quoted $Q$ expression. Re-estimate the actual thermal or non-Gaussian channel.

Check the endpoints. At $N=0$, both capacities vanish. At $\eta=0$, the receiver always obtains the environment and both vanish. At $\eta=1$, $C=g(N)$ and $Q=g(N)$ for the identity bosonic mode under the mean-photon constraint. These controls catch sign swaps between $\eta$ and $1-\eta$.

Exercises

1. Reproduce the packet rates. Evaluate $g(1.5)$ and $g(0.5)$ for the benchmark.

Solution

Using base-two logarithms,

$$g(1.5)=2.5\log_2(2.5)-1.5\log_2(1.5) =2.4273765,$$ $$g(0.5)=1.5\log_2(1.5)-0.5\log_2(0.5) =1.3774438.$$

Their difference is $1.0499327$ qubits per use. The classical formula contains only the first value; subtracting the environment entropy belongs to coherent information, not to unassisted classical capacity.

2. Mean versus peak. With target mean photon number $N=2$ and $\epsilon=0.01$, find the rare pulse in the vacuum-plus-pulse ensemble. Is it allowed by a peak bound of ten photons?

Solution

The pulse has mean photon number

$$N_{\mathrm{pulse}}=\frac{N}{\epsilon}=200.$$

The ensemble average is $0.99\times0+0.01\times200=2$, so it obeys the mean constraint. It violates a ten-photon peak bound. The example distinguishes admissible sets; it does not by itself compute either capacity.

3. Loss threshold and high-energy limit. Show that the pure-loss quantum-capacity expression is zero for $\eta\leq1/2$ and tends to $\log_2[\eta/(1-\eta)]$ for $\eta\gt1/2$ as $N\to\infty$.

Solution

$g(x)$ is increasing for $x\geq0$. If $\eta\leq1/2$, then $\eta N\leq(1-\eta)N$, so the entropy difference is nonpositive and the maximum with zero vanishes. For large $x$,

$$g(x)=\log_2(ex)+O(x^{-1}).$$

Therefore

$$g(\eta N)-g((1-\eta)N) =\log_2\frac{\eta}{1-\eta}+O(N^{-1}).$$

The limit is positive precisely when $\eta\gt1/2$.

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.
• Giovannetti, V., Guha, S., Lloyd, S., Maccone, L., Shapiro, J. H., and Yuen, H. P. (2004). “Classical Capacity of the Lossy Bosonic Channel: The Exact Solution.” Physical Review Letters 92, 027902. DOI. Open PDF.
• Wilde, M. M., and Qi, H. (2018). “Energy-Constrained Private and Quantum Capacities of Quantum Channels.” IEEE Transactions on Information Theory 64(12), 7802–7827. DOI. Open PDF.
• Wolf, M. M., Pérez-García, D., and Giedke, G. (2007). “Quantum Capacities of Bosonic Channels.” Physical Review Letters 98, 130501. DOI. Open PDF.