Exact certification of a positive-order Rényi additivity violation for an explicit channel pair

Abstract: Cubitt, Harrow, Leung, Montanaro, and Winter (CHLMW) exhibited an explicit pair of quantum channels whose minimum output Rényi entropy is nonadditive at order zero, and reported numerical violations at positive orders close to zero. Their paper states that a semidefinite-programming argument yields a rigorous positive-order interval for the same example, without printing the endpoint or a verifiable certificate; a paper by Leung, Lovitz, and Wu records that for this pair "no rigorous endpoint was obtained". We close that gap for the trace-preserving normalization of the printed pair fixed in Section 2. Small rational witness matrices prove that every output of either channel has all eigenvalues between $301/100000$ and $2/3$; a single explicit entangled input has an exact rational joint output spectrum of rank eight; and two independent elementary interval arguments turn these three facts into a proof of strict additivity violation, $S_p^{{\min}(\mathrm{N}<em>R\otimes\mathrm{N}</em>{\bar} S}) &lt; S_p^{{\min}(\mathrm{N}<em>R)+S_p^{{\min}(\mathrm{N}}</em>{\bar} S}),$ for every real order $0&lt;p\le 1/22$. Every step of the verification reduces to comparisons of integers, and the complete certificate is a few small rational matrices that a reader can check with a short program -- or, for any single order, by hand. To our knowledge, consistent with the assessment of Leung, Lovitz, and Wu, this is the first printed, computer-verifiable certified positive-order endpoint for this explicit pair. We claim no novelty for the phenomenon or for the eigenvalue-floor mechanism, both due to CHLMW, and no optimality of the endpoint.

• The paper proves that the explicitly constructed CHLMW channel pair violates minimum-output Rényi entropy additivity for every real order 0 < p ≤ 1/22, converting a prior numerical observation into a rigorous result.
• The paper uses exact rational certificates, partial-transpose bounds, and independent verification to show that every single-channel output eigenvalue lies between 301/100000 and 2/3, while an entangled input yields a rank-eight joint output with an exactly rational spectrum.
• The paper reduces the violation to the certified inequality A(p)^2 > B(p) through both staircase and derivative interval arguments, while noting that the endpoint is not optimal and the intermediate regime 1/4 ≤ p ≤ 3/4 remains unresolved.

Background and problem

The minimum output Rényi entropy of a quantum channel $\mathcal{N}$ at order $p$ is $(\mathcal{N})=\min_\rho S_p(\mathcal{N}(\rho))$, and its additivity under tensor products was a central open question in quantum information theory until counterexamples were found across most orders. Cubitt, Harrow, Leung, Montanaro, and Winter (CHLMW) exhibited an explicit pair of subspaces of $\mathbb{C}^4\otimes\mathbb{C}^3$ whose associated channels violate additivity at $p=0$, and reported numerical violations for positive $p$ up to roughly $0.11$ without proof (0712.3628). Subsequent work closed the parameter range in existence: Leung, Lovitz, and Wu prove violations for all $p>3/4$ and all $0\le p<1/4$ via random constructions that yield no explicit pair (Leung et al., 16 Jul 2026), and Derksen and Lovitz give explicit constructions for all $p>1$ (Derksen et al., 8 Oct 2025). For the explicit CHLMW pair at positive order, however, reference (Leung et al., 16 Jul 2026) records plainly: "no rigorous endpoint was obtained."

The paper under review closes this gap. Its main theorem states that, for the CHLMW channels $p$0 and $p$1 under a fixed trace-preserving normalization,

$p$2

for every real order $p$3. The proof is certificate-based and deliberately elementary: every computer-verified inequality reduces to a comparison of integers, and the complete certificate consists of four rational witness matrices, one rational spectrum, and a short list of rational grid points.

The channel pair and normalization

CHLMW print two six-dimensional orthogonal subspaces $p$4 with entries in $p$5. At $p$6 only ranks matter, but positive-order output eigenvalues are not invariant under changes of channel normalization, so the paper fixes the canonical inverse-input-marginal trace-preserving choice: with input marginals $p$7 (both exactly diagonal and invertible), the Choi matrix is

$p$8

and $p$9. A short lemma establishes that product expectations $(\mathcal{N})=\min_\rho S_p(\mathcal{N}(\rho))$0 equal output matrix elements $(\mathcal{N})=\min_\rho S_p(\mathcal{N}(\rho))$1, so controlling all product expectations of $(\mathcal{N})=\min_\rho S_p(\mathcal{N}(\rho))$2 controls all pure-input outputs.

Eigenvalue floor and cap

The core mechanism is a two-line partial-transpose argument: if $(\mathcal{N})=\min_\rho S_p(\mathcal{N}(\rho))$3 with $(\mathcal{N})=\min_\rho S_p(\mathcal{N}(\rho))$4 and $(\mathcal{N})=\min_\rho S_p(\mathcal{N}(\rho))$5, then every product expectation of $(\mathcal{N})=\min_\rho S_p(\mathcal{N}(\rho))$6 is at least $(\mathcal{N})=\min_\rho S_p(\mathcal{N}(\rho))$7, since $(\mathcal{N})=\min_\rho S_p(\mathcal{N}(\rho))$8. Applying this to exact rational decompositions

$(\mathcal{N})=\min_\rho S_p(\mathcal{N}(\rho))$9

with the stated semidefinite lower bounds, yields a uniform eigenvalue floor and cap: every eigenvalue of every output of either channel lies in $\mathbb{C}^4\otimes\mathbb{C}^3$0. These constants carry visible slack and are verified in exact arithmetic, not as numerical tolerances.

From floor and cap to the violation

Since $\mathbb{C}^4\otimes\mathbb{C}^3$1 is concave with decreasing derivative on $\mathbb{C}^4\otimes\mathbb{C}^3$2, a tangent-line majorization argument (a Karamata-style envelope over partial sums) shows that among all spectra consistent with the floor, cap, and unit trace, the extreme spectrum $\mathbb{C}^4\otimes\mathbb{C}^3$3 minimizes $\mathbb{C}^4\otimes\mathbb{C}^3$4. Hence

$\mathbb{C}^4\otimes\mathbb{C}^3$5

On the other side, an explicitly constructed entangled input $\mathbb{C}^4\otimes\mathbb{C}^3$6 — weighted by the square roots of the marginal diagonals, not the unweighted maximally entangled vector — produces a joint output whose spectrum is exactly rational and rank eight: $\mathbb{C}^4\otimes\mathbb{C}^3$7 each with multiplicity two, plus one zero. This gives

$\mathbb{C}^4\otimes\mathbb{C}^3$8

The theorem therefore reduces to the master inequality $\mathbb{C}^4\otimes\mathbb{C}^3$9 on $p=0$0. At $p=0$1 this degenerates exactly to CHLMW's rank count ($p=0$2 versus $p=0$3); the content here is that the witnesses keep it strict on a whole interval.

Two independent interval arguments

Two separately implemented certificates establish $p=0$4 on the full interval:

• Staircase argument: partitioning $p=0$5 at ten rational points, monotonicity of $p=0$6 in $p=0$7 reduces each cell to the single claim $p=0$8, which is decided by enclosing each rational power between nearby rationals — e.g., $p=0$9 iff $p$0 — via binary search on a dyadic grid with outward rounding.
• Derivative argument: $p$1 is enclosed with its derivative (logarithms handled by exact $p$2 series with controlled geometric tails); $p$3 throughout and $p$4 imply positivity everywhere.

The two arguments share only the certified primitives, not the interval-reasoning layer.

Verification and provenance

Every claim was checked by two independently written implementations: one performing exact Hermitian $p$5 factorization over the number field $p$6, the other reconstructing everything from printed vectors alone using rational-interval enclosures converted to operator-norm bounds. Both confirm the eight positivity statements, the rational joint matrix and characteristic polynomial, and the endpoint inequalities. Both also reject deliberately corrupted inputs, confirming the tests can fail. The witness matrices were located by an AI-assisted numerical search, which plays no role in the proof; the artifact (certificates, verifiers, SHA-256 manifest) is archived with the paper.

Limitations and open questions

The paper is explicit about what it does not establish. The endpoint $p$7 is not claimed optimal, and indeed the envelope method itself fails shortly after $p$8: extending toward CHLMW's numerically observed $p$9 would require a larger certified floor or a genuinely sharper single-channel bound, not a finer staircase. The true minimizing inputs and minimum output entropies are not identified, and no optimality is claimed for the floor, cap, or envelope. The middle-order regime $0.11$0 remains open even in existence, per (Leung et al., 16 Jul 2026). Finally, the result applies to one specific normalization; positive-order eigenvalues are not normalization-invariant, so the certificate does not automatically transfer to other Choi normalizations of the same support.

Conclusion

This note converts a decade-old numerical observation into a fully rigorous, machine-checkable statement: the explicit CHLMW channel pair violates Rényi minimum-output entropy additivity for all $0.11$1, certified entirely by small rational matrices and integer comparisons, with dual independent verification. It claims novelty neither for the phenomenon nor for the eigenvalue-floor mechanism — both due to CHLMW — but supplies the first printed, verifiable positive-order endpoint for this pair, while leaving the sharp endpoint and the intermediate-order regime open.