What Are the Energy-Constrained Two-Way Capacities of the Pure-Loss Bosonic Channel?

Four different tasks share one capacity formula

The paper treats four operational tasks over the pure-loss channel L_eta with transmissivity 0 < eta < 1: quantum transmission (Q2), entanglement distribution (D2), private classical transmission (P2), and secret-key agreement (K2). Each capacity is the supremum of asymptotic rates log2(dn)/n under the mean-energy constraint, where dn is the target dimension or classical alphabet size and the error tends to zero. Rates are measured in qubits, ebits, or secret bits per channel use.

Theorem 1 states that all four coincide: Q2 = D2 = P2 = K2 = g(N) - g((1-eta)N), with g(x) = (x+1)log2(x+1) - x log2 x and g(0) = 0. The equality is not a coincidence of notation. Quantum transmission distributes entanglement, and entanglement in turn transmits quantum states by teleportation; private transmission of a random message generates key, and a key transmits a private message via a one-time pad with public ciphertext. These conversions add only local operations and public communication, so they preserve the optical energy budget.

Two boundary values anchor the formula. At N = 0 the capacities vanish, and as N goes to infinity they approach the PLOB value -log2(1-eta). The finite-energy result therefore interpolates between no communication and the unconstrained benchmark, with the gap set jointly by channel loss and input energy.

The constraint is on the mean, not the peak

For an n-use protocol, the constraint is (1/n) * sum over i of xi <= N, where xi = Tr(n_i rho_i) is the unconditional mean photon number of the transmitted mode at use i. Every random choice, public outcome, and unsuccessful branch is included, and all n channel slots count toward the rate. No peak or branch-wise energy constraint is imposed.

This matters because the paper shows the same capacity identities and the same strong-converse thresholds hold under the stronger per-use constraint xi <= N for every channel use, where xi is still averaged over all classical outcomes and random choices. The per-use condition implies the block-average constraint, so both converse bounds continue to apply, and the matching sector-hashing and vacuum-dilution constructions also satisfy the per-use constraint. Both energy conventions therefore give identical capacities and thresholds for all four tasks.

The protocol model is deliberately permissive: Alice and Bob start with local preparations and public randomness, without shared entanglement or secret key, and may interleave channel uses with adaptive local operations and unlimited authenticated two-way public classical communication. Local memories have no energy or entropy constraint, and Eve is assumed to receive the loss environments and the entire public transcript.

Sector teleportation moves the energy budget into entanglement

The central technical obstacle is that a converse argument must bound what any adaptive protocol can achieve while respecting the input photon budget. The paper's tool, sector teleportation simulation, addresses this by transferring the input-energy constraint to the shared entanglement resource.

A multimode state with a fixed total photon number occupies a finite-dimensional subspace and can be teleported perfectly using a maximally entangled pair whose two shares each contain that same photon number. Writing H_{s,m} for the subspace of s modes containing exactly m photons and Phi_{s,m} for a maximally entangled state between two copies of that subspace, the pure-loss resource is J_{s,m} = (id_R tensor L_eta^{tensor s})(Phi_{s,m}). It depends on the channel and the photon-number sector, but not on the particular input state. Passive-mode covariance then provides an LOCC map T_{s,m} such that the channel output equals T_{s,m} applied to the input together with J_{s,m}, whenever the transmitted block is supported on H_{s,m}, including arbitrary correlations with Alice's and Bob's retained memories.

Extending this sectorwise construction to general finite-energy inputs requires estimating the resource entanglement and controlling coherences between sectors. For adaptive protocols, these estimates are applied to the inputs at the same round across independent replicas of the entire protocol, preserving all feedback within each run. For an n-use protocol with unconditional input photon numbers xi satisfying sum xi <= nN, the resulting bound is E_R^infinity(zeta_n) <= sum over i of c_eta(xi) <= n c_eta(N), where zeta_n is the retained final quantum state and c_eta(x) = g(x) - g((1-eta)x). Target-state and privacy tests then yield the quantum and secret-key weak converses.

Strong-converse thresholds stay at the unconstrained PLOB value

For a capacity C, its strong-converse threshold C-dagger is the infimum of rates R such that every admissible sequence of n-use protocols with lim inf (log2 dn)/n > R has error tending to one. Theorem 2 states that for every N > 0, under the mean constraint, Q2-dagger = D2-dagger = P2-dagger = K2-dagger = -log2(1-eta).

The consequence is a sharp separation: the energy-constrained capacity depends on the available energy, but its strong-converse threshold does not, remaining equal to the unconstrained PLOB value for every nonzero mean-energy budget. The lower threshold is proved by vacuum dilution, and the upper threshold uses the established unconstrained adaptive strong converse.

This is the point where the paper's framing is easiest to misread. The threshold is not a second capacity and does not say that rates above g(N) - g((1-eta)N) are achievable. It says that under the unconditional mean constraint, the rate region where error can still be driven to zero is governed by the energy-dependent capacity, while the rate above which error necessarily tends to one is governed by the unconstrained value.

A hard photon cutoff restores an energy-dependent strong converse

The picture changes in a more restricted setting. Suppose all n optical inputs are prepared before transmission, with arbitrary two-way LOCC before and after the parallel channel uses but no feedback into later inputs. Impose a hard cutoff: the joint input is supported on states with at most floor(nN) photons in total, allowing coherence between different photon-number sectors. Alice and Bob still start without shared entanglement or secret key.

Theorem 3 states that in this parallel cutoff model, for 0 < N < infinity, all four capacities and their strong-converse thresholds equal c_eta(N) = g(N) - g((1-eta)N). At any fixed rate above this value, the quantum target fidelity vanishes exponentially, and the operational key or private-message error approaches one exponentially. The same threshold holds when the probability of exceeding the cutoff vanishes with n, with exponential convergence guaranteed when this tail also decays exponentially.

The support constraint excludes the vacuum-dilution construction responsible for Theorem 2, which is why an energy-dependent strong converse becomes available here. The paper is explicit about the boundary: this result concerns parallel inputs and does not assert an energy-dependent strong converse for interleaved adaptive protocols. Establishing that for interleaved adaptive protocols under a suitable photon-number constraint remains an open problem.

The amplifier extension reuses the same sector calculation

The paper extends the analysis to the quantum-limited amplifier A_kappa with gain kappa > 1 and vacuum environment, whose Heisenberg-picture output annihilation operator is b = sqrt(kappa) a + sqrt(kappa - 1) e-dagger. The energy constraint still counts only photons in the transmitted input: an input of mean x produces an output of mean kappa x + kappa - 1, and the photons supplied by the amplifier do not enter the input budget.

Theorem 5 gives Q2 = D2 = P2 = K2 = r_kappa(N), with r_kappa(x) = g(kappa x + kappa - 1) - g((kappa - 1)(x + 1)). For every N > 0 the strong-converse thresholds are all log2(kappa/(kappa - 1)). Both identities remain valid under the per-use unconditional constraint, and in the parallel photon-cutoff model all four capacities and thresholds equal r_kappa(N) with exponential convergence above this rate.

The mechanism is a correspondence between conditional resources. Conditioning the amplifier resource on the number of added photons yields a pure-loss resource from a shifted sector, conditioned on receiving a certain photon number, with the parties exchanged. That correspondence transfers the conditional resource entanglement from pure loss to amplification, and degradability then bounds its average in terms of the input energy. The paper notes that the capacity in the amplifier theorem is already achievable by unassisted quantum and private codes, so the extension's content is that adaptive two-way classical assistance cannot increase it.

A corollary pins down the entanglement of finite-energy outputs

As a consequence of the sector-resource calculation, the paper determines the regularized relative entropy of entanglement of a finite-energy channel output. For a bipartite state rho, the relative entropy of entanglement is E_R(rho) = inf over separable sigma of D(rho || sigma), and the regularized version is E_R^infinity(rho) = lim over r of E_R(rho^{tensor r})/r.

Corollary 4 states that if J_{eta,N} results from sending one half of a two-mode squeezed vacuum of local mean photon number N through L_eta, then E_R^infinity(J_{eta,N}) = c_eta(N). The amplifier analogue states that if J^amp_{kappa,N} results from sending half of a two-mode squeezed vacuum of local mean N through A_kappa, then E_R^infinity(J^amp_{kappa,N}) = r_kappa(N).

These identities matter because they connect the operational capacities to an entanglement measure evaluated on a specific Gaussian output state, rather than leaving the capacity formula as an isolated coding statement. They also show that the same function c_eta(N) that fixes the pure-loss capacities fixes the regularized entanglement of the corresponding finite-energy output.

Where the finite-energy framework changes the comparison

The most decision-relevant split in this paper is between the constraint type, not between the four tasks. Under an unconditional mean transmitted-photon-number constraint, the four capacities collapse to g(N) - g((1-eta)N) while the strong-converse thresholds stay at -log2(1-eta) for every N > 0. Under a parallel hard total-photon-number cutoff, the same capacity formula becomes its own strong-converse threshold, with exponential decay of fidelity or exponential approach of error to one above it.

For a reader comparing bosonic communication benchmarks, that means the finite-energy capacity alone does not determine the strong-converse behavior. The relevant question is whether the protocol model permits the vacuum-dilution construction, which the mean constraint allows and the hard cutoff excludes. The paper also flags that the cutoff results use one simulator for a parallel optical block, whereas the adaptive weak converse groups the same round across independent protocol replicas, so the cutoff proofs do not extend to adaptive inputs.

The paper's own stated next step is to apply sector teleportation simulation to the thermal-loss channel, aiming to tighten finite-energy upper bounds and clarify the remaining gap to achievable rates. Further directions listed include finite-blocklength and second-order refinements, experimentally accessible implementations of sector-resolved measurements, and extensions to other Gaussian and non-Gaussian bosonic channels.

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