Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2608.17376

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

Artus Krohn-Grimberghe

quant-phSubmitted Aug 18, 2026• Announced Aug 19, 2026• 0 citations• Revised Aug 25, 2026 (v2)
v1 · Aug 18, 2026v2 · Aug 25, 2026
arXivHTMLPDF

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 recent 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, $S_p^{\min}(\mathrm{N}_R\otimes\mathrm{N}_{\bar S}) < S_p^{\min}(\mathrm{N}_R)+S_p^{\min}(\mathrm{N}_{\bar S}),$ for every real order $0<p\le 1/22$. 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.

Comments
8 pages; v2: corrected proof of the envelope lemma, expanded literature discussion; verification artifact v1.2 archived at doi:10.5281/zenodo.22097554

Related papers

Closed-Form Arbitrary-Gap Rényi Nonadditivity Above Any Positive Order via Direct Products of Free Groups and High-Girth Graphs

Guocheng Zhen, Ranyiliu Chen, Xin Wang

We give a closed-form finite-dimensional construction of quantum channels with real algebraic entries exhibiting arbitrarily large violations of minimum-output Rényi entropy additivity uniformly above any prescribed positive Rényi order, using a different finite realization based on direct products of free groups and fixed-field high-girth graphs. More precisely, for every $p_0>0$ and $g>0$, we construct one channel and one Bell input for which the additivity gap is greater than $g$ simultaneously for all $p\in[p_0,\infty]$. With $r=\lceil2g\rceil$, the construction uses $15r$ output qubits and $O(r2^{30r}+r^2(1+p_0^{-1}))$ input qubits. Compared with the stated parameters of Shou and Gorshkov \cite{ShouGorshkov2026}, our construction uses fewer output qubits for every target gap and has a smaller asymptotic input-qubit bound as the gap grows with the positive cutoff fixed. At the von Neumann order, a separate specialization gives an explicit channel with gap greater than $1/1024$ using $37608898$ input qubits and $8$ output qubits.

quant-pharXiv:2610.06533•Oct 6, 2026•0 citations

Counterexamples to additivity of minimum output $p$-Rényi entropy of quantum channels for all $p\ge 0$

Debbie Leung, Benjamin Lovitz, Ion Nechita +1 more

The additivity of minimum output entropies is a central problem in quantum information theory. Nonadditivity is known for every Rényi order $p>1$, at the von Neumann point $p=1$, and for sufficiently small positive $p$, while much of the interval $0<p<1$ has remained open. In this work, we prove that for every $p>0$ there exist finite-dimensional quantum channels whose minimum output $p$-Rényi entropies violate additivity. Our proof combines two constructions: random projection-induced channels yield nonadditivity for $p>3/4$, and antisymmetric postprocessing extends the violation to all positive Rényi orders. Our estimates also improve the output-dimension bound obtained by Belinschi, Collins, and Nechita for additivity violation of minimum output von Neumann entropy.

quant-pharXiv:2607.15210•Jul 17, 2026•0 citations

Almost One Bit Violation of Minimum-Output Rényi Entropy Additivity Simultaneously at All Orders

Guocheng Zhen, Chengkai Zhu, Ranyiliu Chen +1 more

We prove that minimum-output Rényi-entropy additivity can fail by almost one bit simultaneously at every nonnegative order. For every $\varepsilon\in(0,\log2)$, there exists a finite-dimensional quantum channel with a real Stinespring isometry such that the same maximally entangled input witnesses a tensor-square entropy gap of at least $\log2-\varepsilon$ for all $p\in[0,\infty]$. The output dimension can be chosen to be $O(\varepsilon^{-3})$ as $\varepsilon\downarrow0$. The construction uses direct products of free groups: tensorized Haagerup estimates control the one-copy outputs, while commutation between distinct factors forces exact Bell-branch collisions at two copies. Strong convergence gives both an existential realization through finite-dimensional representations of right-angled Artin groups followed by realification, and a Haar-orthogonal model whose success probability tends to one as the matrix dimension grows. We also determine the exact Bell quotient, prove asymptotically sharp regular-radius bounds, and show that the cubic output-dimension scale is optimal within the present purity--rank certificate.

quant-pharXiv:2609.18747•Sep 17, 2026•0 citations

The Lovász number is not alone: a continuum of Rényi points in the entanglement-assisted spectrum of graphs

Nidhal Mghirbi, Seth Douglas

Li and Zuiddam's entanglement-assisted asymptotic spectrum of graphs consists of the normalized graph parameters that are additive under disjoint unions, multiplicative under strong products and monotone under entanglement-assisted cohomomorphisms; the entanglement-assisted Shannon capacity is their pointwise minimum. To our knowledge, the Lovász number $\vartheta$ was its only known element. We find a continuum of others. Three quantities of Duan and Winter, $\log_2\vartheta$, the minimal Holevo capacity $C_{\min}$ and the logarithm of the no-signalling simulation cost $Σ$, are the smallest information radii of classical-quantum channels compatible with a graph, measured with Petz-Rényi divergences of orders $0$, $1$ and $2$. At order zero this is due to Schrijver (as credited by Dalai) and to Duan and Winter; at order one it is the min-max formula for the Holevo capacity. Interpolating in the order gives a family $\vartheta_α$, $0\leα\le2$, and we prove that every member lies in the spectrum. The main step, multiplicativity under strong products for channels that need not factor, follows from an exact nesting identity for compressions. Sandwiched Rényi radii of order $β\in[1/2,\infty]$ give the same family, at $α=2-1/β$. On the five-cycle the family takes every value in $[\sqrt5,5/2]$. A Hilbert-Schmidt inequality for the pentagon gives $C_{\min}(C_5)=\log_2(5/2)$, proving a conjecture of Duan and Winter. We also settle the two open links of their inequality chain: the graph-level no-signalling simulation rate always equals its single-letter value $\log_2Σ(G)$, and a 32-vertex graph from the Mermin-Peres magic square has $C_{\min}<\log_2Σ$, certified in exact rational arithmetic. Finally, $Σ$ never exceeds the projective rank of the complement; a rank inequality for the same square makes this strict on a 1440-vertex graph.

quant-pharXiv:2610.20775•Oct 16, 2026•0 citations

Deterministic Minimum-Output-Entropy Nonadditivity via Haagerup's Inequality and Near-Free Permutation Representations

Guocheng Zhen, Chengkai Zhu, Ranyiliu Chen +1 more

We give a deterministic realization of the finite-dimensional quadratic certificate underlying Collins's mixed-unitary proof of minimum-output-entropy nonadditivity. For every fixed integer $K\ge 2$ and rational $η>0$ satisfying $\log K>2(3+η)^2$, a deterministic polynomial-time algorithm, for every sufficiently large target size $N$, outputs $K$ permutations on $N'=N+o_{K,η}(N)$ points. Restricting their permutation matrices to the nontrivial standard representation yields real orthogonal Stinespring blocks and a channel $Φ_{N'}:M_{N'-1}(\mathbb{C})\to M_K(\mathbb{C})$ such that \[ 2H_{\min}(Φ_{N'}) -H_{\min}(Φ_{N'}^{\otimes 2}) \ge \frac{\log K}{K} -2\log\left(1+\frac{(3+η)^2}{K}\right) >0. \] The construction combines Haagerup's length-two inequality with the simultaneous deterministic spectral approximation of O'Donnell and Wu. We further show that the constant $3$ is asymptotically sharp on the relevant Hermitian zero-diagonal coefficient class and that the finite spectral transfer is nearly saturated, thereby isolating the finer geometry of the full output body as the natural next level of refinement beyond the scalar-radius method. Finally, a standard covariant extension converts the same deterministic entropy gap exactly into self-tensor superadditivity of the one-shot Holevo quantity.

quant-pharXiv:2608.31081•Sep 1, 2026•0 citations

A certified lower bound on the quantum-capacity threshold of the depolarizing channel

Artus Krohn-Grimberghe

The noise threshold below which the qubit depolarizing channel retains positive quantum capacity has been studied since 1996. The classic constructions come with exact finite formulae, but in every reported threshold to date -- most recently the record of Agarwal et al. -- the final step, the sign of the coherent information at the reported point, is a floating-point evaluation. We give the first exact-arithmetic, independently machine-checkable positivity proofs in this regime: an explicit 45-copy rank-two input state $A$, published as a 1,472-byte witness, with certified $I_c > 0$ at the exact rational $p = 16239/250000 = 0.064956$ (per-Pauli convention; total error rate $3p = 0.194868$) -- beyond both the best value printed in the Agarwal et al. paper ($0.064657$) and the strongest state in the same authors' public repository ($\approx 0.064911$), both numerical. Because the depolarizing family is a semigroup under composition, fixed-input coherent information is nonincreasing in $p$ on $[0,1/4]$, so a single certified point extends to the entire interval below it. The same machinery confines the positivity boundaries of $A$ and of the strongest public state of Agarwal et al. to disjoint rational intervals separated by more than $1/25000$ -- to our knowledge the first proven ordering of the positivity boundaries of two explicit competing code states. Every computational claim in the paper reduces to a finite list of big-integer comparisons, checkable by a dependency-free few-hundred-line verifier whose soundness rests on three self-contained half-page lemmas. Payloads, certificates, and verifier accompany the paper as a supplementary artifact.

quant-pharXiv:2608.15870•Aug 18, 2026•0 citations
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 recent 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 301/100000301/100000301/100000 and 2/32/32/3; 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, Spmin⁡(NR⊗NSˉ)<Spmin⁡(NR)+Spmin⁡(NSˉ),S_p^{\min}(\mathrm{N}_R\otimes\mathrm{N}_{\bar S}) < S_p^{\min}(\mathrm{N}_R)+S_p^{\min}(\mathrm{N}_{\bar S}),Spmin​(NR​⊗NSˉ​)<Spmin​(NR​)+Spmin​(NSˉ​), for every real order 0<p≤1/220<p\le 1/220<p≤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.
We give a closed-form finite-dimensional construction of quantum channels with real algebraic entries exhibiting arbitrarily large violations of minimum-output Rényi entropy additivity uniformly above any prescribed positive Rényi order, using a different finite realization based on direct products of free groups and fixed-field high-girth graphs. More precisely, for every p0>0p_0>0p0​>0 and g>0g>0g>0, we construct one channel and one Bell input for which the additivity gap is greater than ggg simultaneously for all p∈[p0,∞]p\in[p_0,\infty]p∈[p0​,∞]. With r=⌈2g⌉r=\lceil2g\rceilr=⌈2g⌉, the construction uses 15r15r15r output qubits and O(r230r+r2(1+p0−1))O(r2^{30r}+r^2(1+p_0^{-1}))O(r230r+r2(1+p0−1​)) input qubits. Compared with the stated parameters of Shou and Gorshkov \cite{ShouGorshkov2026}, our construction uses fewer output qubits for every target gap and has a smaller asymptotic input-qubit bound as the gap grows with the positive cutoff fixed. At the von Neumann order, a separate specialization gives an explicit channel with gap greater than 1/10241/10241/1024 using 376088983760889837608898 input qubits and 888 output qubits.
Counterexamples to additivity of minimum output ppp-Rényi entropy of quantum channels for all p≥0p\ge 0p≥0
The additivity of minimum output entropies is a central problem in quantum information theory. Nonadditivity is known for every Rényi order p>1p>1p>1, at the von Neumann point p=1p=1p=1, and for sufficiently small positive ppp, while much of the interval 0<p<10<p<10<p<1 has remained open. In this work, we prove that for every p>0p>0p>0 there exist finite-dimensional quantum channels whose minimum output ppp-Rényi entropies violate additivity. Our proof combines two constructions: random projection-induced channels yield nonadditivity for p>3/4p>3/4p>3/4, and antisymmetric postprocessing extends the violation to all positive Rényi orders. Our estimates also improve the output-dimension bound obtained by Belinschi, Collins, and Nechita for additivity violation of minimum output von Neumann entropy.
We prove that minimum-output Rényi-entropy additivity can fail by almost one bit simultaneously at every nonnegative order. For every ε∈(0,log⁡2)\varepsilon\in(0,\log2)ε∈(0,log2), there exists a finite-dimensional quantum channel with a real Stinespring isometry such that the same maximally entangled input witnesses a tensor-square entropy gap of at least log⁡2−ε\log2-\varepsilonlog2−ε for all p∈[0,∞]p\in[0,\infty]p∈[0,∞]. The output dimension can be chosen to be O(ε−3)O(\varepsilon^{-3})O(ε−3) as ε↓0\varepsilon\downarrow0ε↓0. The construction uses direct products of free groups: tensorized Haagerup estimates control the one-copy outputs, while commutation between distinct factors forces exact Bell-branch collisions at two copies. Strong convergence gives both an existential realization through finite-dimensional representations of right-angled Artin groups followed by realification, and a Haar-orthogonal model whose success probability tends to one as the matrix dimension grows. We also determine the exact Bell quotient, prove asymptotically sharp regular-radius bounds, and show that the cubic output-dimension scale is optimal within the present purity--rank certificate.
Li and Zuiddam's entanglement-assisted asymptotic spectrum of graphs consists of the normalized graph parameters that are additive under disjoint unions, multiplicative under strong products and monotone under entanglement-assisted cohomomorphisms; the entanglement-assisted Shannon capacity is their pointwise minimum. To our knowledge, the Lovász number ϑ\varthetaϑ was its only known element. We find a continuum of others. Three quantities of Duan and Winter, log⁡2ϑ\log_2\varthetalog2​ϑ, the minimal Holevo capacity Cmin⁡C_{\min}Cmin​ and the logarithm of the no-signalling simulation cost ΣΣΣ, are the smallest information radii of classical-quantum channels compatible with a graph, measured with Petz-Rényi divergences of orders 000, 111 and 222. At order zero this is due to Schrijver (as credited by Dalai) and to Duan and Winter; at order one it is the min-max formula for the Holevo capacity. Interpolating in the order gives a family ϑα\vartheta_αϑα​, 0≤α≤20\leα\le20≤α≤2, and we prove that every member lies in the spectrum. The main step, multiplicativity under strong products for channels that need not factor, follows from an exact nesting identity for compressions. Sandwiched Rényi radii of order β∈[1/2,∞]β\in[1/2,\infty]β∈[1/2,∞] give the same family, at α=2−1/βα=2-1/βα=2−1/β. On the five-cycle the family takes every value in [5,5/2][\sqrt5,5/2][5​,5/2]. A Hilbert-Schmidt inequality for the pentagon gives Cmin⁡(C5)=log⁡2(5/2)C_{\min}(C_5)=\log_2(5/2)Cmin​(C5​)=log2​(5/2), proving a conjecture of Duan and Winter. We also settle the two open links of their inequality chain: the graph-level no-signalling simulation rate always equals its single-letter value log⁡2Σ(G)\log_2Σ(G)log2​Σ(G), and a 32-vertex graph from the Mermin-Peres magic square has Cmin⁡<log⁡2ΣC_{\min}<\log_2ΣCmin​<log2​Σ, certified in exact rational arithmetic. Finally, ΣΣΣ never exceeds the projective rank of the complement; a rank inequality for the same square makes this strict on a 1440-vertex graph.
We give a deterministic realization of the finite-dimensional quadratic certificate underlying Collins's mixed-unitary proof of minimum-output-entropy nonadditivity. For every fixed integer K≥2K\ge 2K≥2 and rational η>0η>0η>0 satisfying log⁡K>2(3+η)2\log K>2(3+η)^2logK>2(3+η)2, a deterministic polynomial-time algorithm, for every sufficiently large target size NNN, outputs KKK permutations on N′=N+oK,η(N)N'=N+o_{K,η}(N)N′=N+oK,η​(N) points. Restricting their permutation matrices to the nontrivial standard representation yields real orthogonal Stinespring blocks and a channel ΦN′:MN′−1(C)→MK(C)Φ_{N'}:M_{N'-1}(\mathbb{C})\to M_K(\mathbb{C})ΦN′​:MN′−1​(C)→MK​(C) such that 2Hmin⁡(ΦN′)−Hmin⁡(ΦN′⊗2)≥log⁡KK−2log⁡(1+(3+η)2K)>0. 2H_{\min}(Φ_{N'}) -H_{\min}(Φ_{N'}^{\otimes 2}) \ge \frac{\log K}{K} -2\log\left(1+\frac{(3+η)^2}{K}\right) >0. 2Hmin​(ΦN′​)−Hmin​(ΦN′⊗2​)≥KlogK​−2log(1+K(3+η)2​)>0. The construction combines Haagerup's length-two inequality with the simultaneous deterministic spectral approximation of O'Donnell and Wu. We further show that the constant 333 is asymptotically sharp on the relevant Hermitian zero-diagonal coefficient class and that the finite spectral transfer is nearly saturated, thereby isolating the finer geometry of the full output body as the natural next level of refinement beyond the scalar-radius method. Finally, a standard covariant extension converts the same deterministic entropy gap exactly into self-tensor superadditivity of the one-shot Holevo quantity.
The noise threshold below which the qubit depolarizing channel retains positive quantum capacity has been studied since 1996. The classic constructions come with exact finite formulae, but in every reported threshold to date -- most recently the record of Agarwal et al. -- the final step, the sign of the coherent information at the reported point, is a floating-point evaluation. We give the first exact-arithmetic, independently machine-checkable positivity proofs in this regime: an explicit 45-copy rank-two input state AAA, published as a 1,472-byte witness, with certified Ic>0I_c > 0Ic​>0 at the exact rational p=16239/250000=0.064956p = 16239/250000 = 0.064956p=16239/250000=0.064956 (per-Pauli convention; total error rate 3p=0.1948683p = 0.1948683p=0.194868) -- beyond both the best value printed in the Agarwal et al. paper (0.0646570.0646570.064657) and the strongest state in the same authors' public repository (≈0.064911\approx 0.064911≈0.064911), both numerical. Because the depolarizing family is a semigroup under composition, fixed-input coherent information is nonincreasing in ppp on [0,1/4][0,1/4][0,1/4], so a single certified point extends to the entire interval below it. The same machinery confines the positivity boundaries of AAA and of the strongest public state of Agarwal et al. to disjoint rational intervals separated by more than 1/250001/250001/25000 -- to our knowledge the first proven ordering of the positivity boundaries of two explicit competing code states. Every computational claim in the paper reduces to a finite list of big-integer comparisons, checkable by a dependency-free few-hundred-line verifier whose soundness rests on three self-contained half-page lemmas. Payloads, certificates, and verifier accompany the paper as a supplementary artifact.