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arxiv: 2604.26874 · v1 · submitted 2026-04-29 · 🪐 quant-ph

Convex combinations of bosonic pure-loss channels

Giuseppe Catalano , Marco Fanizza , Francesco Anna Mele , Giacomo De Palma , Vittorio Giovannetti This is my paper

Pith reviewed 2026-05-07 11:11 UTC · model grok-4.3

classification 🪐 quant-ph
keywords fading channelspure-loss channelsbosonic quantum channelsquantum capacityentanglement distributionquantum key distributionnon-Gaussian statescoherent information
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The pith

Any bosonic fading channel that is not completely noisy supports positive-rate entanglement distribution and quantum key distribution.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that bosonic fading channels, formed as convex combinations of pure-loss channels with varying transmissivity, always allow entanglement distribution and quantum key distribution at strictly positive rates whenever the transmissivity distribution includes any positive value. This holds regardless of how severe the average noise or fluctuations are. The result is relevant for free-space quantum links where loss varies unpredictably. The work further shows that Gaussian thermal inputs are suboptimal for the entanglement-assisted classical capacity, and that non-Gaussian Fock-diagonal states can produce positive coherent information in regimes where thermal states yield zero, thereby activating the channel for quantum tasks. Exact closed-form solutions are given for binary fading, and a variational algorithm handles general distributions.

Core claim

We prove that entanglement distribution and quantum key distribution can always be achieved at a strictly positive rate over any fading channel, no matter how noisy it is or how strongly the transmissivity fluctuates, provided the channel is not completely noisy. Thermal states fail to achieve the entanglement-assisted classical capacity of fading channels, with non-Gaussian Fock-diagonal states strictly outperforming all Gaussian encodings. In some regimes the coherent information of thermal inputs vanishes while optimized non-Gaussian states achieve strictly positive values, activating the channel for quantum communication. For a binary fading model the exact capacity-achieving state is 1.

What carries the argument

The fading channel, defined as the convex combination of pure-loss bosonic channels weighted by a known probability distribution over transmissivity values.

If this is right

  • Non-Gaussian Fock-diagonal states strictly outperform Gaussian encodings for the entanglement-assisted classical capacity.
  • Optimized non-Gaussian states achieve positive coherent information in regimes where thermal inputs give zero, activating the channel for quantum communication.
  • For binary fading the exact capacity-achieving input state is obtained in closed form.
  • An iterative variational algorithm computes the coherent and mutual information for arbitrary fading distributions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Quantum-network design can assume positive rates without needing real-time compensation for every transmissivity fluctuation, provided some light transmission remains possible.
  • The same convex-combination robustness may apply to other bosonic noise models that admit similar decompositions.
  • The variational method could be used to pre-compute optimal input states for experimentally realized fading channels.

Load-bearing premise

The overall channel must be exactly a convex combination of pure-loss channels whose transmissivity distribution has support that includes at least one strictly positive value.

What would settle it

An explicit computation or laboratory simulation of a fading channel with positive transmissivity support in which the coherent information is zero for every possible input state would falsify the positive-rate claim.

Figures

Figures reproduced from arXiv: 2604.26874 by Francesco Anna Mele, Giacomo De Palma, Giuseppe Catalano, Marco Fanizza, Vittorio Giovannetti.

Figure 1
Figure 1. Figure 1: FIG. 1. Quantum circuit representation of a fading channel. view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Regions of Antidegradability and Non-Antidegradability. Left: Parameter space ( view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Absolute quantum mutual information gain view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Convergence of the iterative optimization algorithm. As the order view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Activation of the quantum capacity for the binary fading family Φ view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Regions of non-antidegradability (green) and degradability (red) for specific binary fading configurations defined in view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Absolute quantum mutual information gain view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Absolute capacity gain view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Single-shot quantum capacity analysis for the erasure-lossy channel Φ view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Evolution of the optimal photon number distribution view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Non-Gaussian advantage for the coherent information of the LNW fading channel at low energy ( view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Non-Gaussian advantage for the coherent information of the LNW fading channel at high energy ( view at source ↗
read the original abstract

The pure-loss channel is a fundamental model for describing noise in bosonic quantum platforms. It is characterised by a single parameter, the transmissivity, which quantifies the fraction of the input energy that reaches the output of the channel. In realistic scenarios, however, such as free-space quantum communication, the transmissivity is not fixed but fluctuates from one channel use to another. In this setting, the overall channel is effectively described as a convex combination of pure-loss channels, known as a fading channel. Despite its practical relevance, the quantum Shannon theory of the fading channel has remained largely unexplored. Here, we address this gap, specifically investigating degradability, anti-degradability, entanglement breakingness, and capacities of the fading channel. Of particular relevance to practical quantum-internet applications, we prove that entanglement distribution and quantum key distribution can always be achieved at a strictly positive rate over any fading channel, no matter how noisy it is or how strongly the transmissivity fluctuates, provided the channel is not completely noisy. Moreover, we prove that thermal states, which are optimal for a broad class of static bosonic Gaussian channels, fail to achieve the entanglement-assisted classical capacity of fading channels: non-Gaussian Fock-diagonal states strictly outperform all Gaussian encodings. Most strikingly, we identify regimes where the coherent information of thermal inputs vanishes, while optimized non-Gaussian states achieve strictly positive values, thereby activating the channel for quantum communication. For a paradigmatic binary fading model we establish this result analytically, deriving the exact capacity-achieving state in closed form. For general fading distributions, we design an iterative variational algorithm to optimize the coherent and mutual information.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript models bosonic fading channels as convex combinations of pure-loss channels with a distribution over transmissivity. It analyzes their degradability, anti-degradability, and entanglement-breaking properties, proves that the single-letter coherent information is strictly positive (hence quantum and private capacities are positive) whenever the distribution has support on at least one η > 0, shows that non-Gaussian Fock-diagonal states strictly outperform thermal states for the entanglement-assisted classical capacity, provides a closed-form optimizer for the binary-fading case, and introduces an iterative variational algorithm for general distributions.

Significance. If the central claims hold, the work is significant for free-space quantum communication: it guarantees strictly positive rates for entanglement distribution and QKD over any non-trivial fading channel and demonstrates that Gaussian encodings are suboptimal, with concrete analytical and algorithmic tools for capacity evaluation.

minor comments (3)
  1. The abstract and introduction would benefit from an explicit statement of the mean-photon-number constraint used when optimizing the coherent and mutual information, as this is standard for bosonic capacities and affects the comparison with thermal states.
  2. The description of the iterative variational algorithm (mentioned in the abstract) lacks details on convergence criteria, initialization, or stopping conditions; adding these would improve reproducibility without altering the central claims.
  3. A brief comparison table or remark contrasting the fading-channel results with the known optimality of thermal states for static pure-loss and thermal-loss channels would clarify the novelty of the non-Gaussian activation result.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our manuscript, recognition of its significance for free-space quantum communication, and recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The central result—that coherent information (hence quantum and private capacities) remains strictly positive for any fading channel whose transmissivity distribution has support on at least one η > 0—follows directly from the standard fact that the pure-loss channel with η > 0 has positive coherent information for a suitable input, combined with the convexity of coherent information under convex combinations of channels. The paper supplies an explicit optimizer for the binary-fading case and a variational procedure for the general case, but these are computational tools to exhibit the positive value, not definitions that presuppose the result. No self-definitional steps, fitted-input predictions, or load-bearing self-citations appear in the derivation chain; the argument relies on external, independently established properties of bosonic Gaussian channels and coherent information.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on the standard definition of bosonic pure-loss channels and the mathematical definition of convex combinations; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Bosonic pure-loss channels are completely characterized by their transmissivity parameter and obey standard quantum channel properties.
    Invoked throughout the analysis of degradability, capacities, and coherent information.
  • domain assumption The fading channel is defined as the convex combination of pure-loss channels according to a probability distribution over transmissivity.
    This is the central modeling assumption stated in the abstract.

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discussion (0)

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Reference graph

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    Protocol Description Alice and Bob useMuses of the channel Φ to share a target entangled state|Ψ⟩ AB. •Encoding:Alice prepares a maximally entangled state|Ψ⟩ AB = 1√ D PD k=1 |k⟩A |k⟩B, where the systemB(to be sent through the channel) is encoded in the subspaceH K,M ofKphotons distributed acrossMbosonic modes. The dimension of this subspace isD=D K,M = M...

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    We have⟨λ n⟩ ≥ ⟨λ⟩ n

    Universal Lower Bound To obtain a universal bound that depends only on the mean transmissivity⟨λ⟩, we apply Jensen’s inequality for the convex functionf(x) =x n (forx≥0, n≥1). We have⟨λ n⟩ ≥ ⟨λ⟩ n. Substituting this into the expression forP succ: Psucc ≥ 1 DK,M X ⃗ n∈HK,M MY i=1 ⟨λ⟩ni (B5) = 1 DK,M X ⃗ n∈HK,M ⟨λ⟩ P ni (B6) = 1 DK,M X ⃗ n∈HK,M ⟨λ⟩K (B7) = ...

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    Substituting the decomposition forρ, we obtain: S(ρ) =H 2(p0) + (1−p 0)S(σ),(C3) whereH 2(x) =−xlog 2 x−(1−x) log 2(1−x) is the binary entropy function

    Input Entropy For a Fock-diagonal stateρ= P n ρnn |n⟩ ⟨n|, the von Neumann entropy is the Shannon entropy of its diagonal elements. Substituting the decomposition forρ, we obtain: S(ρ) =H 2(p0) + (1−p 0)S(σ),(C3) whereH 2(x) =−xlog 2 x−(1−x) log 2(1−x) is the binary entropy function

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    The vacuum component is shifted due to the erasure term, while the excited components are simply scaled byp

    Output Entropy The channel maps the inputρto the output state Φ p(ρ) =pρ+ (1−p)|0⟩ ⟨0|. The vacuum component is shifted due to the erasure term, while the excited components are simply scaled byp. The spectrum of the output state is: spec(Φp(ρ)) ={pρ 00 + (1−p), pρ 11, pρ22, . . .}.(C4) Using the decompositionρ 00 =p 0 and the expression forS(ρ), the outp...

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    Exchange Entropy To compute the exchange entropyS( ˜Φp(ρ)), we construct the Stinespring dilation and, from that, we can compute the Kraus operators of Φ. The channel admits a Kraus representation with operators{N k}∞ k=0 defined as: N0 = √p1, N k = p 1−p|0⟩ ⟨k−1|fork≥1.(C6) Using the general formula for the complementary channel ˜Φ(ρ) = P i,j |i⟩E ⟨j|Tr[...

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    To maximize the capacity, we must therefore maximize the entropyS(σ) subject to the constraints imposed onσ

    Closed-F orm Mutual Information and Optimization Combining the entropy terms derived above, the mutual informationI(Φ p, ρ) simplifies to: I(Φ p, p0, σ) = 2p(1−p 0)S(σ) + 2pH2(p0)−plog 2 p+ (1−p) log 2(1−p) +pp 0 log2(pp0)−(pp 0 + 1−p) log 2(pp0 + 1−p) −(1−p)p 0 log2((1−p)p 0) + X α∈{+,−} λα(p0) log2 λα(p0).(C10) Notice that for a fixedp 0, the mutual inf...

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    Notation and preliminary: thermal state and binomial kernel We begin by recalling the definition of a thermal state with mean photon numberE≥0: ρth,E = 1 E+ 1 ∞X n=0 E E+ 1 n |n⟩ ⟨n|.(D2) and its von Neumann entropy is the well-known function g(E) :=S(ρ th,E) = (E+ 1) log 2(E+ 1)−Elog 2 E, with the conventiong(0) = 0. For reference we also define the bino...

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    Input State and Input Entropy At thek-th step of the optimization algorithm, the ansatz stateρ (k) E is constructed as a mixture of the firstkFock states and a “shifted” thermal tail. Explicitly: ρ(k) E = k−1X n=0 wn |n⟩ ⟨n|+w ≥k σ(k) N ,(D5) where we defined the weightsw n =p n Qn−1 i=0 (1−p i) for the discrete part, and the total weight of the tailw ≥k ...

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    We compute the diagonal elements [Eλ(ρ(k) E )]mm for a generic transmissivityλ

    Output State and Output Entropy The action of the fading channel on the input state yields the output state: Φ{pn,λn}(ρ(k) E ) = dX j=1 pjEλj(ρ(k) E ).(D11) Since the input is Fock-diagonal and the channel is phase-insensitive, the output is also Fock-diagonal. We compute the diagonal elements [Eλ(ρ(k) E )]mm for a generic transmissivityλ. Due to the stru...

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    Exchange Entropy The exchange entropy is defined asS( ˜Φ(ρ(k) E )), where ˜Φ is the complementary channel of Φ. For a fading channel defined by the ensemble{p n, λn}, the output of the complementary channel can be written in a block-diagonal form 24 with respect to the environment basis{|i⟩ E}: ˜Φ{pn,λn}(ρ) = ∞X i=0 |i⟩E ⟨i| ⊗ ˜Φi(ρ),(D15) where each ˜Φi(...

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    We now derive a rigorous upper bound on the error introduced by this truncation, ensuring the reliability of our numerical results

    T runcation Error Bound Since the Fock space is infinite-dimensional, any numerical evaluation of entropic functionals requires truncating the Hilbert space at a finite photon number ¯n. We now derive a rigorous upper bound on the error introduced by this truncation, ensuring the reliability of our numerical results. Letρ= P∞ n=0 γn |ψn⟩ ⟨ψn|be the exact ...

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    Summary: mutual information and numerical recipe Putting the pieces together, for the chosen inputρ=ρ (k) E and channel Φ = Φ {pn,λn} =Pd−1 n=0 pnEλn we compute: 1.S(ρ (k) E ) from (D8)

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    For each channel componentλ n compute the output populations (D12),(D13), then form the mixture populations Λm = Pd j=1 pj[Eλj(ρ(k) E )]mm by averaging on the distribution and evaluateS(Φ(ρ (k) E )) =− P m Λm log2 Λm by (D14), summing up to the truncation cutoff

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    For the complementary channel compute each block ˜Φi(ρ(k) E ) by (D18), diagonalize thed×dmatrix to obtain S( ˜Φi(ρ(k) E )) and sum overiup to the truncation cutoff to obtainS( ˜Φ(ρ(k) E )) up to a desired precision

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    26 Appendix E: Optimality of F ock-diagonal inputs Lemma 2.LetΦbe a phase-insensitive quantum channel

    The mutual information follows as I(Φ, ρ(k) E )≃S(ρ (k) E )− imaxX m=0 Λm log2 Λm − ˜imaxX i=0 S( ˜Φi(ρ(k) E )) (D29) wherei max,˜imax are chosen so that the truncation error bound (D25) is negligible for the desired precision. 26 Appendix E: Optimality of F ock-diagonal inputs Lemma 2.LetΦbe a phase-insensitive quantum channel. That is, for any input sta...

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    The numerical evaluation of this channel allows us to observe the core mechanism of non-Gaussian advantage

    Erasure-like channel We begin by analyzing the identity-erasure channel, defined by the map: Φp =pI+ (1−p)E 0.(F1) This channel represents a fundamental communication link suffering from complete signal dropouts with probability 1−p. The numerical evaluation of this channel allows us to observe the core mechanism of non-Gaussian advantage. As shown in Fig...

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    Erasure-lossy channel We now generalize the previous model to the erasure-lossy channel, where successful transmission events occur with a finite transmissivityλ: Φ(0) p,λ =pE λ + (1−p)E 0.(F2) This model captures the essence of imperfect transmission interspersed with total signal loss. In Fig. 8, we report the absolute capacity gain provided by our Fock...

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    clear-air

    Identity-lossy channel As a complementary scenario to the erasure models, we consider the identity-lossy channel: Φ(1) p,λ =pI+ (1−p)E λ.(F3) This family describes intermittent “clear-air” turbulence, where the signal is occasionally attenuated but never com- pletely erased. Figure 10 illustrates the capacity gain for this channel. The results demonstrate...

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    Figure 11 displays the optimized coherent information alongside the absolute gain over the thermal benchmark

    Coherent information and channel activation To further investigate the ultimate quantum limits of the channel, we focus on the single-shot coherent information Ic, which provides a rigorous lower bound for the quantum capacity. Figure 11 displays the optimized coherent information alongside the absolute gain over the thermal benchmark. Crucially, the righ...

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    In free-space quantum communications affected by beam wandering, the transmissivity statistics are accurately described by the Log-Negative Weibull (LNW) distribution

    General fading (Log-Negative W eibull distribution) Finally, we apply our numerical framework to a physically realistic continuous fading model. In free-space quantum communications affected by beam wandering, the transmissivity statistics are accurately described by the Log-Negative Weibull (LNW) distribution. We map the non-Gaussian advantage for the si...

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