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From Symmetry to Secrecy: Covariant Classical--Quantum Wiretap Channels

Masahito Hayashi, Farzin Salek

arXiv:2610.03580v1quant-phcs.ITmath.IT

TL;DR

The paper asks how symmetry can make correlated randomized encoding tractable for cq wiretap channels, where multi-use correlations can raise rates. It uses covariant group-orbit structure to construct finite-block preprocessing schemes and analyzes their achievable rates and leakage. Numerical comparisons improve on the tested noisy-repetition benchmark at three and four uses, and a separate two-use construction proves nonadditivity at an explicit parameter.

  • Problem

    Optimizing randomized encoders across multiple uses is difficult, although correlations across uses can increase the rate per use for cq wiretap channels.

    Source: §I · pp. 1–2

  • Method

    The paper uses channel symmetry to express finite-block randomized encoders through two group distributions and their convolution, then applies coding and hashing to achieve rates with leakage bounds.

    Source: §I · p. 2

  • Results

    At three and four uses, broader noise distributions give larger rates than the noisy-repetition benchmark estimate wherever it is positive on the tested grid, and a two-use encoder proves nonadditivity at p = 621/5000.

    Source: §I · p. 2

  • Takeaways & Limitations

    The construction provides a symmetry-based way to study correlated finite-block preprocessing, with an explicit leakage analysis and a uniform-input bound that can equal the single-use private information.

    Source: §I · p. 2, §X · p. 26

Abstract

from arXiv · show

For classical--quantum wiretap channels with a transitive covariant input action, both output ensembles admit a description as unitary group orbits of fixed seed states. We use this structure to construct finite-block preprocessing schemes whose private information is an entropy difference evaluated at two distributions related by group convolution. Applying the construction to product groups allows correlations across any fixed number of channel uses. We give a sufficient condition for the associated uniform-input bound to equal the single-use private information. Constant-composition channel codes and universal$_2$ hashing achieve the resulting rates, with trace-distance leakage controlled by sandwiched R\'enyi information. For uniform outer randomization, the representation structure reduces the comparison-state optimization to Eve's invariant states. For the hybrid binary channel considered by Tikku, Berta and Renes, numerical optimization at three and four uses gives rates above our numerical estimate of the optimized noisy-repetition benchmark wherever that estimate is positive on the tested grid. The improvement comes from noise distributions beyond the family of independent physical flips and a symmetric logical flip. We also prove nonadditivity at an explicit channel parameter using a two-use encoder, an analytic single-use converse, and certified entropy bounds. Finally, we relate the hybrid model to binary phase-shift keying with a fixed receiver measurement.

I. INTRODUCTION

The introduction uses input symmetry to make randomized encoding more explicit for cq wiretap channels, where correlations across uses can increase rates. It develops a finite-block preprocessing construction and previews its operational realization, numerical comparisons, and nonadditivity result.

I. INTRODUCTION

  • The paper addresses the difficulty of optimizing correlated encoders across uses, using channel symmetry to make auxiliary distributions and stochastic encoders more explicit.For cq wiretap channels, correlations across several uses can increase the rate per use; the construction extends to correlated preprocessing on Gn.Source: §I · pp. 1–2
  • The construction represents randomized encoders using two distributions on a finite group, with the marginal input distribution given by their group convolution.For uniform outer randomization, the bound simplifies to Δ(Pu) − min_P Δ(P) and equals the single-use private information whenever Pu maximizes Δ.Source: §I · p. 2
  • Constant-composition coding and balanced universal2 hashing operationalize the construction, while sandwiched Rényi mutual information controls leakage for arbitrary P3.For uniform P3, the representation reduces the comparison-state optimization to invariant states; a cyclic example permits explicit evaluation.Source: §I · p. 2
  • At three and four uses, optimizing beyond independent physical flips and a symmetric logical flip gives larger rates than the noisy-repetition benchmark estimate wherever it is positive on the tested grid.The three-use distribution makes the physical flip probability depend on whether the logical bit was flipped; the four-use example requires a broader distribution.Source: §I · p. 2
  • At p = 621/5000, an analytic single-use converse and certified entropy bounds establish zero single-use private information but a positive two-use rate.This is an explicit nonadditivity example.Source: §I · p. 2

A. Notation

  • The notation section sets density operators and von Neumann entropy on finite-dimensional Hilbert spaces as defaults, with natural logarithms and information measured in nats.It defines h(x) as binary entropy and Pu as the uniform distribution on a finite group.Source: A. Notation · p. 3

B. Symmetries of a classical input alphabet

  • For a classical input model, a symmetry must preserve the diagonal algebra and therefore permutes basis labels; wiretap covariance requires the same input action for Bob and Eve.The phases of the input unitaries have no effect on classical states, and covariance of a quantum extension alone does not impose this input-model requirement.Source: B. Symmetries of a classical input… · p. 3

C. Transitive symmetries, seed states and the regular action

  • Under a transitive group action, each receiver’s outputs form a unitary orbit of a seed state, and stabilizer-invariant seeds conversely define covariant channels.Redundant group labels leave finite-use private-information quantities unchanged; transitivity matters because nontransitive actions may have unrelated orbits.Source: C. Transitive symmetries, seed states… · p. 4

D. Translates, convolutions and the invariance of entropies

For group-orbit cq wiretap channels, translating a noise distribution leaves Bob’s and Eve’s entropies unchanged, while mixing translates changes the input marginal through convolution. This yields a finite-block achievable private-information formula and allows correlated preprocessing across product-group channel uses.

D. Translates, convolutions and the invariance of entropies

  • Left translation preserves Δ, while convolution mixes translates; thus conditional entropies stay constant and the marginal input distribution determines the entropy difference.The uniform distribution is absorbing under convolution, and a point mass at the identity is its unit.Source: D. Translates, convolutions and the… · p. 5

III. CQ WIRETAP CHANNELS AND THE FINITE-USE PRIVATE INFORMATION

  • For n uses, correlations in classical preprocessing can yield nonadditivity when an achievable n-use rate per use exceeds a converse bound on C(1).Conditioned on the classical input block, the channel outputs remain product states.Source: §III · pp. 5–6
  • For binary inputs, single-use private information is the maximum gap between Δ and its lower convex envelope, and C(1) = 0 exactly when Δ is convex.For a fixed input mean, the lower convex envelope gives the minimum averaged Δ over finitely supported distributions.Source: §III · p. 6

IV. THE COVARIANT PREPROCESSING BOUND

  • For independent V and Z with X = VZ, private information equals Δ(P3 ∗ P1) − Δ(P1), giving an achievable bound for each pair of distributions.The conditional input is a translate of P1, so its output entropies are unchanged across values of V.Source: §IV · pp. 6–7
  • Uniform P3 gives the uniform-input bound, which equals single-use private information whenever the uniform distribution maximizes Δ.On the binary parameter line discussed in the paper, the uniform input does not maximize Δ.Source: §IV · p. 7
  • On product groups G^n, both preprocessing distributions may be correlated across uses, though the family need not be optimal among all auxiliary-variable constructions.A fixed preprocessing block can be used as a super-symbol in a longer channel code.Source: §IV · pp. 7–8

V. OPERATIONAL REALIZATION OF THE COVARIANT BOUND

The section turns covariant preprocessing into an operational coding scheme: a fixed preprocessing block is treated as a super-symbol, then constant-composition coding and balanced hashing realize the resulting private-information difference with vanishing error and leakage.

A. Induced channel and information difference

B. Codes, hashes and conventions

  • A constant-composition code carries the randomized index, while a balanced universal2 hash partitions indices into equal fibers with collision probability at most 1/MN.Source: B. Codes, hashes and conventions · p. 8

C. The protocol

  • The public code and hash seed coexist with private fiber and P1 draws; balanced fibers preserve a uniform index for seed selection without changing decoding error.Source: C. The protocol · p. 9

D. Reliability

  • Constant-composition coding makes Bob's average error vanish when the index rate is below the induced-channel mutual information.Source: D. Reliability · p. 9

E. Secrecy for an arbitrary outer distribution

F. Uniform outer randomization and an explicit exponent

For uniform outer randomization, the group orbit yields a single-copy Rényi leakage bound without a code-composition restriction. The representation structure also narrows the comparison-state optimization to invariant states.

F. Uniform outer randomization and an explicit exponent

A. Cyclic diagonal representation

The examples use a cyclic diagonal representation and relate quantum BPSK to a hybrid channel with classical output for Bob and pure states for Eve. In the hybrid example, the single-use entropy difference is minimized at nonzero input probability.

A. Cyclic diagonal representation

  • The cyclic diagonal example has inequivalent one-dimensional representation sectors, so invariant comparison states are mixtures over those sectors.The exponent is evaluated for the point-mass specialization of P1.Source: A. Cyclic diagonal representation · p. 13

B. Quantum BPSK

  • When |α′|>|α|, BPSK has zero private capacity; when |α′|<|α|, it is degraded and the private capacity is additive.Source: B. Quantum BPSK · p. 14

C. Hybrid binary example: a symmetric channel to Bob and pure states to Eve

D. The two-copy functional

The two-copy functional compares Bob's output entropy with Eve's output-state entropy. The paper also specifies how the noisy-repetition family sits inside the preprocessing choices considered for two and larger blocks.

D. The two-copy functional

E. The pair family at two uses and the repetition subfamily

VII. REPETITION-CODE BENCHMARKS AND NUMERICAL COMPARISON

The section defines three nested noisy-repetition benchmarks and compares them on a grid near the single-use threshold. Optimizing a logical flip can materially raise the repetition rate, so the full benchmark is the relevant comparator.

A. The Tikku–Berta–Renes line and three repetition benchmarks

  • The full repetition benchmark optimizes the logical prior for the repetition-preprocessed channel, while the simpler rates restrict the logical input or its preprocessing.The repetition code maps a logical bit to a repeated block and independently flips physical bits with probability λ; the next benchmark also allows a logical flip with probability µ.Source: A. The Tikku–Berta–Renes line and… · pp. 17–18
  • The three benchmarks are nested: no logical flip is a special case of logical-flip preprocessing, which is itself a particular decomposition in the full single-letter repetition benchmark.The full benchmark need not generally belong to the covariant pair family, but it does when its optimal decomposition is symmetric and two-point.Source: A. The Tikku–Berta–Renes line and… · p. 18
  • At p = 6081/49000, Rrep,0_3 ≃ 1.40 × 10^-7 while Rrep,log_3 ≃ 2.57 × 10^-5, a factor of about 180.The comparison motivates using the benchmark that includes logical preprocessing.Source: A. The Tikku–Berta–Renes line and… · p. 18
  • On the tested grid, the full benchmark's maximizing decomposition was symmetric and two-point wherever its value was positive, agreeing with the optimized logical-flip construction within 7.1 × 10^-9 nats per use.The authors evaluated the full benchmark by sampling logical priors and computing the lower convex hull; the tolerance is consistent with prior discretization.Source: A. The Tikku–Berta–Renes line and… · p. 18

B. The single-use reference curve

The numerical comparison finds that three- and four-use covariant preprocessing can outperform the optimized noisy-repetition benchmark across the tested positive-rate grid. Two uses agree with logically prefixed repetition, while the single-use reference curve is numerically identified with the single-use capacity on this grid.

B. The single-use reference curve

  • On the tested grid, the single-use reference R1 numerically matches C(1): the maximizing input is uniform where capacity is positive, and both vanish at the remaining points.The match is numerical except at the parameter treated in Theorem VIII.1, where C(1)=0 is proved.Source: B. The single-use reference curve · p. 19

C. Two uses: agreement with logically prefixed repetition

  • The achieved two-use curve lies on the logically prefixed repetition benchmark; the search found no improvement beyond its numerical tie tolerance.The whole gap between uniform-logical-input repetition and the two-use value is supplied by the logical flip.Source: C. Two uses: agreement with logically… · p. 19

D. Three and four uses: noise beyond physical and logical flips

  • For three and four uses, the retained pairs use a uniform outer distribution on the all-zero/all-one blocks and exchangeable noise optimized over its Hamming-weight profile.These profiles achieve rates above the complementary-binomial family, moving beyond independent physical flips plus a symmetric logical flip.Source: D. Three and four uses: noise beyond… · p. 19
  • The four-use noise profile cannot be a mixture of two binomial distributions, as shown by the moment-matrix rank test for the exact decimal distribution.The distribution is operationally realized with different physical flip probabilities depending on whether the logical bit was flipped.Source: D. Three and four uses: noise beyond… · pp. 19–20
  • Across the tested grid, the best pair at blocklength at most four exceeds the numerical envelope of repetition benchmarks wherever that envelope is positive.Lengthening the block and optimizing its internal structure are complementary routes to higher rates.Source: D. Three and four uses: noise beyond… · pp. 20–21
  • At n=3, the achieved rate exceeds the numerical repetition-benchmark estimate at all 44 positive grid points; at n=4 it exceeds the estimate at all 46.The absolute gains reach 7.76 × 10^-6 and 2.06 × 10^-5 nats per channel use, respectively; the corresponding gain plots are in Fig. 7.Source: D. Three and four uses: noise beyond… · p. 20, Table III · p. 20, Figure 7 · p. 22

VIII. AN EXPLICIT NONADDITIVITY WITNESS

At p = 621/5000, the paper proves zero single-use capacity but exhibits a positive two-use rate. The proof combines a convexity-based single-use converse with an explicit encoder and certified entropy bounds.

VIII. AN EXPLICIT NONADDITIVITY WITNESS

  • At p = 621/5000 = 0.1242, C(1)=0 while the explicit two-use rate lies between 10^-6 and 1.1 × 10^-6 nats per channel use.The resulting strict separation establishes nonadditivity at this parameter.Source: §VIII · p. 22
  • The two-use witness uses a logically prefixed repetition-family encoder, with a uniform outer distribution on a translate of the repetition subgroup.The noise parameters are approximately λ=0.31834 and µ=0.27706.Source: §VIII · pp. 22–23

A. The single-use converse: convexity of the entropy difference

  • The single-use converse establishes convexity of the entropy difference using the sufficient conditions k atanh a ≥ a^3 and k^2 ≥ u^3.At the stated parameter, finite rational computations verify these conditions, so the lower convex envelope is the entropy difference itself and C(1)=0.Source: A. The single-use converse: convexity… · pp. 23–24
  • The proof's single-use bound applies to every auxiliary-variable construction: Jensen's inequality makes its private information nonpositive, and a constant auxiliary variable attains zero.The explicit positive two-use rate is certified separately by exact rational entropy bounds.Source: §VIII · p. 22, A. The single-use converse: convexity… · p. 24

B. Certified positivity of the two-use rate

The section certifies a positive two-use rate bound, relates the hybrid binary channel to quantum BPSK under a fixed receiver measurement, and states the scope and open questions of the construction.

B. Certified positivity of the two-use rate

IX. FROM BPSK TO THE HYBRID BINARY CHANNEL

  • The hybrid binary channel arises from quantum BPSK when Bob uses a fixed binary measurement while Eve retains her quantum output.The section identifies the induced channels and then gives a beam-splitter realization.Source: §IX · p. 24

A. Eve’s channel

  • For nonzero α′, Eve’s cat signals span an even/odd two-dimensional subspace, identifying her channel with Section VI C at q′ < 1/2; at zero amplitude it is constant-output.The parity acts as diag(1, −1) on this subspace.Source: A. Eve’s channel · p. 25

B. Bob’s channel

  • Bob’s equal-prior Helstrom measurement induces the Section VI C binary symmetric channel, but excludes unrestricted collective measurements on his quantum outputs.The model is the cq channel seen by a receiver with a fixed symbol-by-symbol detector.Source: B. Bob’s channel · p. 25

C. A physical realization

  • A lossless three-output beam-splitter realizes the channel pair when the environment is inaccessible to Eve or its output amplitude is zero.If Eve collects the environment mode, her effective energy includes both Eve’s and the environment’s output amplitudes.Source: C. A physical realization · pp. 25–26

X. CONCLUSION

  • The conclusion leaves exactness for additional channel classes, larger-block correlated noise, and finite-blocklength secrecy refinements as open questions.It also reports numerical benchmark improvements at three and four uses and a rigorous two-use separation from single-use private information.Source: §X · p. 26

Appendix A: Choi-operator form of covariance

The appendix expresses covariance as a commutation condition on the Choi operator and records the conventions, numerical checks, and optimization procedures used for the computations.

Appendix A: Choi-operator form of covariance

  • Covariance is equivalently expressed by commutation of the Choi operator with U*_g ⊗ V_g for every group element g.For full-unitary covariance with V_g = U_g, the commutant yields the isotropic Choi family of depolarizing channels.Source: Appendix A · p. 26
  • For cq channels, the input action in the Choi covariance condition is the permutation action defined in Section II B.Changing basis conjugates the condition, while exchanging Choi factors exchanges representation factors.Source: Appendix A · p. 26

1. Grid, conventions and verification

  • Section VII computations use natural logarithms, a 50-point exact grid, binary-integer ordering, and normalized distributions before rate evaluation.The entropy calculation sets 0 log 0 = 0 and rejects larger negative-eigenvalue violations.Source: §1 · p. 27
  • Across all 150 achieved rates, direct tensor-product output calculations differed by at most 1.2 × 10^-15 nats per use from the exchangeable reduction.This check is separate from the exact rational bounds in Section VIII and the moment determinant in (75).Source: §1 · p. 27

2. Search and benchmark evaluation

  • At each grid point and n ∈ {2, 3, 4}, exchangeable-noise searches refine feasible encoder pairs but make no global-optimality claim.For n = 2, the search also uses differential evolution before full-pair local refinement.Source: §2 · p. 27
  • The repetition benchmark is evaluated by scanning physical-flip probabilities and binary priors, then refining the best values.The full benchmark is formed from the lower convex hull of the binary-prior function.Source: §2 · p. 27
  • At every positive benchmark point, the maximizing prior is x = 1/2, and the full benchmark agrees with Rrep,log_n within 7.1 × 10^-9 nats per use.The figures and tables therefore use Rrep,log_n as the numerical estimate of Rrep_n.Source: §2 · p. 27

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