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

arXiv:2610.12365 · quant-ph · Submitted 2026-10-08 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "Energy-constrained two-way capacities of the pure-loss channel".

Mira: The gist:

Kai: First, who's behind it and why it matters.

Paper summary: Kai: So, to recap where we are, this paper is tackling how to calculate the two-way quantum and secret key capacities for a pure-loss channel when you have an average input photon budget constraint. The main thesis they put forward is that these two capacities are equal to a specific function involving the entropy of a thermal state.

Mira: They claim that for transmissivity eta and photon budget N, both the two-way quantum capacity Q2(L eta, N) and the secret key capacity K(L eta, N) equal g(N) - g((one - eta)N), where g is defined as (x + one) two(x + one) - x two x <ref:2610.12365#pg1,and photon budget N, both>.

Lev: That function g is crucial because it's what they use to describe the entropy of a thermal state, and they establish this identity as the main result of Theorem one point one <ref:2610.12365#pg1,the entropy of a thermal state>. They’ve also noted that this result aligns with the known reverse coherent information for a two-mode squeezed vacuum state.

Kai: It matters because it provides what they call the last missing capacity formula for the pure-loss channel, completing the picture of bosonic quantum communication under pure loss, which is pretty big for this area.

Mira: They also show how this function behaves at the boundaries; it gives zero when N is zero or eta is zero, and it equals g(N) when eta equals one. For cases between those points, like zero eta < one it approaches minus log2 (one minus eta) as N goes to infinity <ref:2610.12365#pg1>.

Lev: That asymptotic behavior recovering the unconstrained capacity PLOB17 is a key piece of the puzzle, showing how the constrained result connects back to the theoretical ideal when you have plenty of photons.

Kai: So, for someone listening just for what this paper means in broad terms, it’s about getting an exact mathematical description of communication limits when you have physical energy constraints on a lossy channel.

Mira: It shifts the focus from just having upper bounds to having a precise formula that describes the actual achievable quantum and key rates under these specific constraints.

Lev: For someone looking at error correction, this gives us a concrete quantity to work with when designing protocols, knowing exactly what the capacity is in terms of N and eta.

Kai: It sets up the groundwork for how we can analyze performance when we’re dealing with realistic setups that involve loss.

Conclusion: Kai: So, wrapping up this discussion on "Energy-constrained two-way capacities of the pure-loss channel," we see that the authors have successfully determined that Q2 and K are equal to g(N) - g((one - eta)N) for every finite N and eta <ref:2610.12365#pg1,Energy-constrained two-way capacities of the pure-loss channel>.

Mira: This means they’ve found a way to precisely quantify both how much quantum information you can send and how much secret key you can extract, given the average photon budget N you put in. It’s a complete formula for this problem under pure loss.

Lev: The authors also noted that the achievable function f eta comes from reverse reconciliation and reverse coherent information, which is important because it connects their result to existing techniques in entanglement-assisted capacity calculations.

Kai: Ultimately, what this paper does is provide the final piece of the puzzle for bosonic quantum communication when dealing with pure loss channels. It’s a complete picture that was previously missing in our understanding of these limits.

Mira: In simple terms, they’ve given us the exact mathematical tool to calculate those two-way rates when you have limitations on your input energy for a lossy channel.

Lev: For the future, this formula is what we can start using to design protocols that operate right at the edge of what's possible with finite resources.

Kai: It moves us closer to knowing exactly how much secure communication we can expect from these systems in a real-world setting where loss is unavoidable.

Scuola Normale Superiore · Institute for Quantum Information and Matter, California Institute of Technology · NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR

quant-ph

Submitted: 2026-10-08

Updated: 2026-10-08

Comments: 13 pages, 1 figure

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 85/100

The gist: 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

Key concepts

Two-way capacities: These are two different measures of communication capability: one for sending quantum information (quantum capacity) and another for securely transmitting a secret key (secret-key capacity). The paper shows these two values are identical under the given constraints.
Pure-loss channel ($L_ η$): This is a specific type of communication channel where input photons are lost to an environment, modeled by mixing the input with a vacuum state on a beam splitter. The parameter $\eta$ controls the loss rate.
Regularized relative entropy ($ER(A: B)\rho$: This is a mathematical tool used to quantify the distinguishability between two quantum states, $A$ and $B$, especially when dealing with mixed states. It helps bound how much entanglement or information can be gained or lost under constraints.
Entropy of a thermal state ($g(N)$): This function represents the entropy associated with a thermal state, which is crucial for defining the capacity limits. The final capacity formula is expressed directly in terms of this entropy function.

Terminology

Entanglement — 25× in this paper · explained in 4 episodes
Trace — 7× in this paper · explained in 2 episodes
Fidelity — 5× in this paper · explained in 2 episodes
Regularization — 5× in this paper · explained in 3 episodes
Hilbert Space — 4× in this paper · explained in 2 episodes
Adaptivity — 1× in this paper · explained in 1 episode
Distillation — 1× in this paper · explained in 2 episodes
Geometric Distribution — 1× in this paper · explained in 2 episodes

Summary

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.

Improvements for AI systems

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.

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).

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.