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arXiv:2607.15210

Counterexamples to additivity of minimum output $p$-Rényi entropy of quantum channels for $p>3/4$ and $0\leq p<1/4$

Debbie Leung, Benjamin Lovitz, Peixue Wu

quant-phmath-phmath.PRSubmitted Jul 16, 2026• Announced Jul 17, 2026• 0 citations
v1 · Jul 16, 2026v2 · Oct 5, 2026
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Abstract

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 near $p=0$, while most of the interval $0<p<1$ has remained open. We prove that for every Rényi order $p$ satisfying either $p>3/4$ or $0\leq p<1/4$, there exist finite-dimensional projection-induced quantum channels such that additivity of the minimum output $p$-Rényi entropy fails. The proof combines two correlated random-projection constructions: a product-conjugate Bell-state witness for $p>3/4$, and a transpose-complement rank-defect witness for $p<1/4$. Thus the unresolved part of $0<p<1$ is reduced to $[1/4,3/4]$. Our estimates also improve the output dimension threshold for additivity violation of minimum output von Neumann entropy, first established in Belinschi, Collins and Nechida.

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

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

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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.

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Counterexamples to additivity of minimum output ppp-Rényi entropy of quantum channels for p>3/4p>3/4p>3/4 and 0≤p<1/40\leq p<1/40≤p<1/4
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 near p=0p=0p=0, while most of the interval 0<p<10<p<10<p<1 has remained open. We prove that for every Rényi order ppp satisfying either p>3/4p>3/4p>3/4 or 0≤p<1/40\leq p<1/40≤p<1/4, there exist finite-dimensional projection-induced quantum channels such that additivity of the minimum output ppp-Rényi entropy fails. The proof combines two correlated random-projection constructions: a product-conjugate Bell-state witness for p>3/4p>3/4p>3/4, and a transpose-complement rank-defect witness for p<1/4p<1/4p<1/4. Thus the unresolved part of 0<p<10<p<10<p<1 is reduced to [1/4,3/4][1/4,3/4][1/4,3/4]. Our estimates also improve the output dimension threshold for additivity violation of minimum output von Neumann entropy, first established in Belinschi, Collins and Nechida.
We give an explicit non-random example of nonadditivity of minimum output von Neumann entropy. The proof uses a finite-dimensional construction which imitates free Haar unitary behavior. The same channel gives a violation of additivity for the minimum output Rényi-ppp entropy for any p∈[1,∞]p\in[1,\infty]p∈[1,∞]. Additionally, given any p0>0p_0>0p0​>0 and g>0\mathfrak g>0g>0, we extend the construction to obtain an explicit non-random channel which has entropy gap ≥g\ge\mathfrak g≥g uniformly over p∈[p0,∞]p\in[p_0,\infty]p∈[p0​,∞].
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.
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.
We investigate non-random quantum channels generated from the representation the- ory of orthogonal quantum groups to perform numerical computations for estimates of their minimum output entropies. In doing so, we address the violation of the additiv- ity conjecture for minimum output entropies, previously established through probabilistic proofs. This violation is relevant because given two quantum channels whose minimum output entropies sum to a value greater than the minimum output entropy of their tensor product channel it can be shown that more information can be sent with greater fidelity via the two channels in parallel than by using each separately. This is directly related to the relevance of quantum entanglement in quantum information theory. \nWe numerically compute minimum output entropies for single quantum channels and tensor products of quantum channels with respect to both Von Neumann and Renyi en- tropies. We then proceed to compute entropy estimates for nonplanar channels directly with Temperley-Lieb Category Theory. We conclude by discussing which of our choices for channels are optimal for entangled inputs.
Entanglement of purification is a fundamental measure of total correlations whose additivity remains unresolved. We study its additivity for classical states on two qubits at different Rényi orders. For every α∈[0,1)α\in[0,1)α∈[0,1), we prove nonadditivity within this family, witnessed by two copies of a single state. We first solve the one-copy optimization exactly for the entire family at every Rényi order. We then restrict the two-copy optimization to a natural finite set of purifications and exhibit one whose entropy is strictly below the product value. In contrast, for α∈[2,∞]α\in[2,\infty]α∈[2,∞] we prove additivity under tensor products within this family. The interval α∈[1,2)α\in[1,2)α∈[1,2), including the von Neumann case α=1α=1α=1, remains open, and we conjecture additivity there throughout the same family.
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.