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Monotonicity of the R\'enyi channel capacity under non-signaling assisted channel simulation

Tina Shariat, Matt Hoogsteder-Riera, Marco Tomamichel

arXiv:2609.34213v2quant-phcs.ITmath.IT

TL;DR

Whether non-signaling simulation can improve a classical channel’s reliability cannot be settled by comparing Shannon capacity alone, since coding exponents depend on the full Rényi-capacity family. The paper gives a variational information-radius proof of monotonicity for that family and derives consequences for coding exponents.

  • Problem

    Shannon capacity alone does not determine how error or success probabilities behave with block length, so channel reliability across rates depends on the full Rényi-capacity family.

    Source: §1 · p. 1

  • Method

    A Legendre-transform variational representation of the information radius makes channel dependence affine, enabling comparison with the linear non-signaling constraints.

    Source: §1 · p. 2

  • Results

    The Rényi capacity of a classical channel is monotone under non-signaling simulation at every order.

    Source: §1 · p. 2

  • Takeaways & Limitations

    Non-signaling simulation cannot increase random-coding or sphere-packing exponents below capacity or decrease the strong-converse exponent above capacity.

    Source: §1 · p. 3

    The variational approach is presented for classical channels; extending it to classical-quantum channels remains open and requires treating noncommuting outputs.

    Source: §5.2 · p. 23

Abstract

from arXiv · show

We give an alternative variational proof that non-signaling correlations shared between the sender and receiver cannot increase the R\'enyi capacity of a classical channel. This monotonicity follows from the data-processing inequality for channel divergences established by Oufkir, Yao and Berta. Our approach works directly with the information-radius representation of the capacity. The constraints defining non-signaling boxes are linear in the box, while the R\'enyi capacity depends on the channel nonlinearly. We use a Legendre transform to express the capacity as an extremum over a family of affine functionals in which the channel appears linearly. Combining this method with separate arguments for the Shannon capacity and endpoint orders, we recover monotonicity at every order from zero to infinity. We then derive consequences for standard coding exponent functions: non-signaling simulation cannot increase the random-coding or sphere-packing exponent below capacity, or decrease the strong-converse exponent above capacity.

1 Introduction

The paper proves that non-signaling simulation cannot increase any order of a classical channel’s Rényi capacity. Its variational information-radius approach also connects this monotonicity to channel coding exponents.

1 Introduction

  • The full Rényi-capacity family is needed to describe reliability at all rates: orders below one govern error exponents, while orders above one govern the strong-converse exponent.At α = 1, the family gives Shannon capacity.Source: §1 · p. 1
  • Non-signaling boxes allow the sender-side and receiver-side processing to share correlations, including shared randomness or entanglement.They form a convex set strictly larger than the set of quantum correlations.Source: §1 · p. 2
  • Non-signaling assistance has made some operational quantities tractable, including optimal coding as a linear program and assisted zero-error capacity and simulation formulas.Matthews connected the coding value to the finite-blocklength converse; Cubitt, Leung, Matthews and Winter studied zero-error capacities and simulation [CLMW11].Source: §1 · p. 2
  • The entire Rényi capacity family is monotone under non-signaling simulation, so the Shannon-capacity result extends to every order.The paper notes that Shannon-capacity monotonicity alone would not rule out improved error exponents below capacity or a weakened strong converse above it.Source: §1 · p. 2
  • The monotonicity follows from channel-divergence data processing by comparing a channel with a constant-output channel; the paper supplies an alternative variational derivation [OYB26, Lemma 24].The endpoint orders follow by limits.Source: §1 · p. 2
  • A Legendre transform rewrites the capacity as an extremum of affine functionals, making the channel appear linearly alongside the linear non-signaling constraints.Young’s inequality underpins the representation, and the sign of the Hölder conjugate separates proof regimes.Source: §1 · p. 2
  • At α = ∞, the proof uses only no-forward-signaling, so monotonicity extends to a larger class of processing boxes when the simulation formula still defines a valid channel.The no-backward-signaling condition is not invoked in that proof.Source: §1 · pp. 2–3
  • Non-signaling simulation cannot increase random-coding or sphere-packing exponents below capacity, or decrease the strong-converse exponent above capacity.The guarantee holds at every rate; above the critical rate it also applies to the reliability function where that function is known.Source: §1 · p. 3

2 Preliminaries

The preliminaries establish the information measures, channel-capacity definitions, and non-signaling simulation framework used in the proof. They identify Rényi capacity with the channel’s information radius.

2.1 Information measures

  • Conditional mutual information is defined by I(X; Y | Z) = H(X | Z) − H(X | Y, Z), and the mutual-information chain rule decomposes I(X; Y, Z) into I(X; Y) + I(X; Z | Y).Conditional mutual information is nonnegative, with equality exactly when X and Y are conditionally independent given Z.Source: §2.1 · p. 3

2.2 The Shannon capacity

  • Shannon capacity is the maximum mutual information between channel input and output over input distributions.The maximum is attained because mutual information is concave and continuous on the compact input-distribution simplex.Source: §2.2 · p. 4

2.3 Super-channels and non-signaling conditions

  • A non-signaling simulation wraps a processing box around the base channel, with constraints preventing the box from bypassing the channel to transfer information.The processing box jointly generates the base-channel input and simulated output from the outer input and channel output; the constraints ensure the simulated mapping is a valid channel.Source: §2.3 · p. 4, Figure 1 · p. 4

2.4 Rényi divergence and the information radius

  • The α-information radius minimizes over output distributions the maximum Rényi divergence from the channel outputs, and equals the Rényi channel capacity.At α = 1, it reduces to Shannon capacity; orders 0, 1, and ∞ are defined by limits.Source: §2.4 · p. 5

3 Monotonicity of the R´enyi capacity

This section establishes that non-signaling simulation cannot increase Shannon or Rényi capacity, covering orders from zero to infinity. It combines a variational argument with separate treatments of the Shannon capacity and endpoint orders.

3 Monotonicity of the R´enyi capacity

  • The proof treats the open intervals 0 < α < 1 and α > 1 variationally, while α = 1 uses classical information inequalities; α = 0 and α = ∞ are identified as separate regimes.The α = 0 case is approached as a limit, and the α = ∞ information-radius expression reduces the claim to a combinatorial inequality.Source: §3 · p. 5

3.1 Monotonicity for α = 1

  • At α = 1, non-signaling simulation cannot increase Shannon capacity, as shown using the chain rule and the non-signaling constraints.The proof derives the comparison from a joint distribution for the simulated channel and bounds a logarithmic term.Source: §3 · p. 5, §3.1 · p. 8

3.2 Monotonicity for α = 1

  • For 0 < α < 1, non-signaling simulation cannot increase Rényi capacity; the proof uses an exponentiated objective, a Rényi center, and a Legendre representation.The representation is derived from Young’s inequality and helps compare the base and simulator channels.Source: §3.2 · pp. 8–9

3.3 Monotonicity for α > 1

  • For α > 1, non-signaling simulation cannot increase Rényi capacity; the variational proof compares the channels using a feasible output distribution and a Legendre representation.The proof introduces an auxiliary quantity to bridge the capacity comparison.Source: §3.3 · pp. 11–13
  • For α > 1, the variational proof uses a locally averaged test function, Jensen’s inequality, and Legendre optimization to complete the monotonicity argument.The local test function averages against a conditional distribution supplied by the no-backward-signaling condition; convexity for β > 1 bounds the nonlinear term.Source: §3.3 · pp. 13–15

3.4 Monotonicity for α = ∞

  • At α = ∞, Rényi capacity is monotone under non-signaling simulation, and the proof requires only the no-forward-signaling condition if the simulation identity defines a valid channel.The proof linearizes the pointwise maximum and uses the no-forward-signaling condition to bound the resulting channel expression.Source: §3.4 · p. 17

3.5 The endpoint α →0 and zero-error capacity

  • At order zero, the information radius is monotone under non-signaling simulation and equals the non-signaling-assisted zero-error capacity, so codes achieve zero error below R0(W).The capacity is log α*(W), the fractional packing number of the channel’s bipartite graph [CLMW11].Source: §3.5 · p. 17
  • Unlike R0, the unassisted zero-error capacity can increase under entanglement assistance, so monotonicity does not hold for all capacity-like quantities.The cited result states that entanglement-assisted codes can strictly increase unassisted zero-error capacity [CLMW11].Source: §3.5 · p. 17

3.6 The main Theorem

  • Theorem 1 shows that non-signaling simulation cannot increase Rényi capacity at any order α ∈ [0, ∞], as a consequence of [OYB26, Lemma 24].The paper’s alternative proof combines variational arguments for 0 < α < 1 and α > 1 with separate arguments for the Shannon capacity and endpoint orders.Source: §3.6 · p. 18

4 Applications

The applications translate R´enyi-capacity monotonicity into bounds on error and success probabilities for simulated channels and non-signaling assisted codes. They compare standard exponent functions at every rate and establish strong-converse bounds for assisted codes.

4.1 Error exponent curves of the simulated channel

  • For every rate R ≥ 0, non-signaling simulation cannot increase the random-coding or sphere-packing exponent, or decrease the strong-converse exponent.Above the base channel’s critical rate and below its capacity, the reliability function itself is monotone; below the critical rate, only the two bounds are shown to be monotone.Source: §4.1 · pp. 19–20
  • The finite-blocklength comparisons order bounds, not the exact optimal error probabilities.Below the critical rate, the reliability function is not known in general, so monotonicity of the bounds does not establish monotonicity of that function.Source: §4.1 · p. 20

4.2 A strong converse for non-signaling assisted codes

  • The strong-converse exponent bound is recovered from R´enyi-capacity monotonicity, alongside the earlier channel-interconversion result.The passage identifies the equality of assisted and unassisted strong-converse exponents as already covered by Oufkir, Yao and Berta [OYB26, Theorem 3 and Remark 1].Source: §4.2 · pp. 20–21
  • For rates R > C(W), the assisted-code success probability is forced to decay exponentially, and non-signaling assistance cannot decrease the strong-converse exponent.The bounds hold at every block length n, while tightness of the optimized exponent is asymptotic; the same exponent conclusion applies to shared randomness and entanglement assistance.Source: §4.2 · p. 21
  • At α = ∞, Corollary 3 gives a closed-form bound on the average success probability of non-signaling assisted codes.The bound is stated in terms of quantities associated with the original channel.Source: §4.2 · p. 21

5 Discussion

Non-signaling assisted coding reaches the sphere-packing exponent at every positive rate below capacity, though this does not mean channel simulation can enhance reliability. The discussion also identifies rate regions where reliability monotonicity is known, contradicted, or still open, and leaves a quantum-channel extension for future work.

5.1 What non-signaling assistance can still do

  • Channel simulation cannot increase the sphere-packing exponent, even though non-signaling assisted coding can attain that exponent where unassisted codes may not.The classical-quantum comparison is qualified: the cited sphere-packing characterization for activated assistance includes an additional noiseless bit, while unactivated results require extra assumptions or rate restrictions.Source: §5.1 · p. 22
  • For every 0 < R < C(W), the non-signaling assisted reliability function equals the sphere-packing exponent; it has no critical rate and gives zero error below R0(W), unlike unassisted codes at low rates.The paper also notes that assistance doubles the binary symmetric channel’s zero-rate exponent while leaving capacity unchanged.Source: §5.1 · p. 22

5.2 Open questions

  • Below C0(W), reliability monotonicity is trivial, and above Rcrit(W) it follows from Corollary 1; below the critical rate, the random-coding lower and sphere-packing upper bounds can separate.The middle inequality in the rate ordering holds because the sphere-packing exponent is infinite below R0(W), while the random-coding bound is finite.Source: §5.2 · pp. 22–23
  • For rates in (C0(W), R0(W)), zero-error assisted coding yields counterexamples to reliability-function monotonicity under simulation.For a nonempty such interval, the simulated noiseless channel can have infinite exponent where the base channel’s exponent is finite.Source: §5.2 · p. 23
  • For rates in (R0(W), Rcrit(W)), both reliability bounds are finite and ordered but separated, so monotonicity remains open to the authors’ knowledge.Their separation prevents Corollary 1 from establishing reliability-function monotonicity in this interval.Source: §5.2 · p. 23
  • Extending the proof to classical-quantum channels remains future work, requiring a treatment of noncommuting outputs and quantum non-signaling constraints together.The extension would also need to distinguish capacities defined by Petz and sandwiched Rényi divergences and specify the order range.Source: §5.2 · p. 23

LLM use disclaimer

The authors state that Claude assisted with proofs of Lemmas 2–5 and manuscript writing.

LLM use disclaimer

  • Claude assisted in deriving the proofs of Lemmas 2 to 5 and in writing the manuscript.The authors claim full responsibility for the manuscript’s statements, including any mistakes.Source: LLM use disclaimer · p. 23
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