Energy-constrained two-way capacities of the pure-loss channel

The gist: The two-way quantum and secret-key capacities of the pure-loss channel under an average input photon-number constraint are determined to be equal to a specific function involving the entropy of a thermal state, providing a complete formula for bosonic quantum communication under pure loss

Theorem 1.1: Energy-constrained two-way capacities

The theorem establishes that for every η ∈ [0, 1] and every finite N ≥ 0, the two-way quantum and secret-key capacities are equal to g(N) − g((1 − η)N), where g is the entropy of a thermal state

Key Results and Formulas

The main result states that both capacities equal g(N) − g((1 − η)N), where g is the entropy of a thermal state This coincides with the rate achieved by the known reverse coherent information of a two-mode squeezed vacuum The formula gives 0 at N = 0 or η = 0, and g(N) at η = 1 For 0 ≤ η < 1, it also gives limN→∞ fη(N) = − log2 (1 − η), recovering the unconstrained capacity [PLOB17]

Relative Entropy Tools

The paper utilizes quantum relative entropy D(ρ∥σ) and its regularized version ER(A: B)ρ, defined as the infimum of D(ρ∥σ) over separable states. The properties used include convexity, subadditivity on tensor products, and monotonicity on average under local instruments and LOCC Lemma 4.1 provides a block bound for entanglement increase under an energy constraint, stating that ER(X: BmY)ω ≤ ER(XSm: Y)ρ + mfη(ν) + g(qmν)

Privacy and Converse Bounds

The paper relates the regularized relative entropy to an approximate secret key using a privacy test that survives regularization Lemma 5.1 provides a regularized privacy bound, stating that if F(ω, γ) ≥ 1 − ε for a private state of key dimension d ≥ 2 and ER(ω) < ∞, then E∞ R (ω) ≥ (1 − ε) log2 d − h2(ε) Proposition 5.2 proves the finite-block converse by showing that for an n-use adaptive protocol, (1 − ε) log2 d ≤ nfη(N) + h2(ε)

Achievability

The achievability proof involves preparing independent copies of the two-mode squeezed vacuum and sending them through Lη. The output purification RBE is pure, ωR = τN, and ωE = τqN This leads to the rate fη(N) ≤ Q2(Lη, N) ≤ K(Lη, N) ≤ fη(N), completing the proof of Theorem 1.1

The paper provides a complete picture of bosonic quantum communication under pure loss by determining the two-way quantum and secret-key capacities under an average input photon-number constraint. The results show that these capacities are given by g(N) − g((1 − η)N), which is the last missing capacity formula for the celebrated pure-loss channel.

How it works

The single-mode pure-loss channel Lη mixes the input with a vacuum environment on a beam splitter [HW01, WPG+12] The complementary channel is L1−η, up to the choice of an environment phase

Relative Entropy Tools

The quantum relative entropy is defined as D(ρ∥σ) = Tr ρ(log2 ρ − log2 σ) The regularized relative entropy of entanglement and its regularization are [VPRK97, VP98]

The cost of local dephasing

Lemma 2.1 provides a finite-dimensional observation for non-lockability, stating that ER(ρ) ≤ ER(∆ρ) + D(ρ∥∆ρ), D(ρ∥∆ρ) ≤ H(p)

Simulation from an irreducible symmetry

A finite-dimensional channel T is simulable by LOCC using its normalized Choi state omegaT = (id ⊗T)(Φd) whenever it is covariant under a compact group whose input representation is irreducible This implies ER(output) ≤ ER(input) + ER(omegaT)

A fixed number of photons

The distribution of the total number J lost has distribution bk,j:= Pr(J = j k) = k j! q j η(k−j), 0 ≤ j ≤ k The resulting identity implies that the environment marginal of Vk,jπkV†k,j is πj

Entanglement increase under an energy constraint

Lemma 4.1 states that ER(X: BmY)ω ≤ ER(XSm: Y)ρ + mfη(ν) + g(qmν) Step 4 removes all cutoffs by choosing increasing finite-rank projections P XL → IX and P YL → IY strongly, and defining the product projection TL = P X L ⊗ Π S m m,≤L ⊗ P Y L, pL = Tr(TLρ)

Private states and the capacity theorem

The privacy test is defined by the projector Πγ = U(Φd,AB ⊗ IA′B′)U† The regularized privacy bound shows that for operational key error e, Proposition 5.2 applies with ε = 2e − e squared whenever this is at most 1/2 The final achievability step proves fη(N) ≤ Q2(Lη, N) ≤ K(Lη, N) ≤ fη(N)

The main result of this work was obtained on 17 September 2026 with a single prompt of ChatGPT Astra (OpenAI). While refining the manuscript, we became aware of [Pir26a], whose author independently obtained the same result.

References

[BBC+93] Charles H. Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William Wootters. Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels. Physical Review Letters 70(13), 1895–1899 (1993)

[BSST02] Charles H. Bennett, Peter W. Shor, John A. Smolin, and Ashish V. Thapliyal. Entanglementassisted capacity of a quantum channel and the reverse Shannon theorem. IEEE Transactions on Information Theory 48(10), 2637–2655 (2002). arXiv:quant-ph/0106052

[CLMOW14] Eric Chitambar, Debbie Leung, Laura Mančinska, Maris Ozols, and Andreas Winter.

1. Bold header: Direct Capacity Calculation for Energy-Constrained Communication

The improved AI system can determine that Q2(Lη, N) = K(Lη, N) = g(N) − g((1 − η)N) provides the exact two-way quantum and secret-key capacities under an average input photon-number constraint. This allows the system to calculate achievable rates for quantum entanglement generation and secret key extraction in pure-loss channels with high precision, completing the picture of bosonic quantum communication under pure loss.

2. Bold header: Optimized Adaptive LOCC Protocol Design

The system can optimize protocols by considering all adaptive LOCC protocols satisfying (2) when maximizing capacities. This enables the AI to design communication strategies that utilize local operations and classical communication optimally for a given energy budget, as the optimization includes all adaptive LOCC protocols satisfying (2).

3. Bold header: Error-Aware Secret Key Generation Bounds

The AI can establish tighter bounds on secret key length by using the privacy bound derived from Lemma 5.1, specifically relating error to key dimension: (1 − ε) log2 d ≤ nfη(N) + h2(ε). This allows the system to predict the minimum required number of channel uses for a desired security level, as for operational key error e, Proposition 5.2 applies with ε = 2e − e squared whenever this is at most 1/2.