Additivity of minimum output Rényi entropy can fail first at three copies

Yuxuan Zhang

Full manuscript (PDF, 5 pages) · Manuscript source, exact checker, reviews and provenance (ZIP)

A complete proof candidate. An internal AI review of the complete proof and a final internal audit of the manuscript and program found no mathematical gap. External specialist confirmation and literature priority remain unverified; arXiv was searched through 27 September 2026. Two steps are computer-certified — a two-copy positivity certificate and the numerical inequalities at \(p=1/1000\) — and both reproduce exactly with a program that needs only the Python standard library.

The question

For a channel \(\Phi\) and \(p>0\), the minimum output Rényi entropy is \(S_{p,\min}(\Phi)=\min_\rho S_p(\Phi(\rho))\). Product inputs give \(S_{p,\min}(\Phi^{\otimes n})\le n\,S_{p,\min}(\Phi)\), and additivity is known to fail: Hastings found violations at \(p=1\) [2], Cubitt, Harrow, Leung, Montanaro and Winter at \(p\) close to \(0\) [3], and Derksen and Lovitz gave explicit violations for every \(p>1\) [4]. The constructions we are aware of exhibit the violation already for two channel uses.

Ruskai asked whether additivity can hold for all tensor powers below some order \(m\) and fail first at the \(m\)th [1]. The question [6] asks for \(p>0\), \(m\ge3\) and a channel with \(S_{p,\min}(\Phi^{\otimes n})=n\,S_{p,\min}(\Phi)\) for \(n<m\) and strict inequality at \(n=m\).

It can, with \(m=3\).

The channel

Let \(\Psi\) map \(\mathbb C^4\) to \(\mathbb C^3\) with six Kraus operators, each supported on two matrix entries: \(K_j=(|r_1\rangle\langle k_1|+\theta_j|r_2\rangle\langle k_2|)/\sqrt3\) on the cell pairs \(((0,0),(2,2))\), \(((0,1),(1,2))\), \(((0,2),(2,1))\), \(((0,3),(1,0))\), \(((1,1),(2,3))\), \(((1,3),(2,0))\) of the \(3\times4\) grid, with \(\theta=(1,1,1,1,1,\omega)\) and \(\omega=e^{2\pi i/3}\). Let \(\Phi\) be the flagged direct sum of \(\Psi\) and the constant channel whose output is \(\tau_\star=\mathrm{diag}(1-2^{-20},2^{-21},2^{-21})\). Then at \(p=1/1000\)

At every sufficiently small \(p>0\), the same holds with a \(p\)-dependent constant block.

Why it works

At small \(p\) the Rényi entropy is governed by rank. Every output of \(\Psi\) has rank \(3\), and every output of \(\Psi\otimes\Psi\) has full rank \(9=3^2\), so the minimum output rank is multiplicative up to two copies. An input with a three-party GHZ structure makes an output of \(\Psi^{\otimes3}\) lose a dimension: its rank is at most \(26<27\). The flagged constant block turns these rank facts into exact additivity at two copies and strict subadditivity at three.

The two-copy statement is the heart of the proof. It is certified by an exact decomposable block-positivity certificate over \(\mathbb Z[\omega]\): a \(144\times144\) matrix \(W\) built from the Kraus operators satisfies \(W-2^{-14}I=P+Q^\Gamma\) with \(P\) positive definite and \(Q\) positive semidefinite. This bounds the smallest eigenvalue of every two-copy output below by \(2^{-14}/9\). The sixth phase matters: with all \(\theta_j=1\), the two-copy outputs lose rank.

Verification

The argument and the program were produced by an AI research agent. The program uses only the Python standard library and exact arithmetic, and prints 21 lines in under a second:

python3 check_delayed_onset.py | diff - expected_stdout.txt

It proves the positive definiteness of \(P\) twice, by independent routes: fraction-free elimination over \(\mathbb Z[\omega]\), and an embedded Gram factor with an exact Frobenius-norm bound. It checks the 216 three-copy identities and a negative control. It certifies the inequalities at \(p=1/1000\) as exact big-integer statements, including \(X^{500}>26^{333}\) and the admissible window for \(\tau_\star\). Thirty single-point mutations each make it fail, and isolated runs reproduce its output byte for byte.

The review rebuilt \(W\) from the Kraus operators and proved \(P\succ0\) with its own exact certificate. It also tested the partial-transpose convention against alternatives that would fail silently. The final audit recomputed every number in the manuscript; at its request the constant block was made explicit.

Reproduction establishes only that the program computes what it prints. The reductions it relies on are established by the written argument.

What this establishes, and what it does not

A first violation of additivity at \(m=3\) exists, at \(p=1/1000\), for an explicit finite-dimensional channel from \(\mathbb C^5\) to \(\mathbb C^6\). That answers the question as posed.

Nothing is claimed for \(p\ge1\), including the von Neumann entropy, or for a first violation at \(m\ge4\). Nor is anything claimed for a single channel without the flagged constant block. The mechanism is related to the non-multiplicative minimum output rank of Cubitt, Harrow, Leung, Montanaro and Winter [3], who used a pair of different channels. What is new here is that a single channel’s rank stays multiplicative at two copies and first fails at three.

References

1. M. B. Ruskai, Some open problems in quantum information theory, arXiv:0708.1902, Problem 19.
2. M. B. Hastings, Superadditivity of communication capacity using entangled inputs, Nature Physics 5, 255 (2009), arXiv:0809.3972.
3. T. Cubitt, A. W. Harrow, D. Leung, A. Montanaro and A. Winter, Counterexamples to additivity of minimum output p-Rényi entropy for p close to 0, Commun. Math. Phys. 284, 281–290 (2008), arXiv:0712.3628.
4. H. Derksen and B. Lovitz, Constructive counterexamples to the additivity of minimum output Rényi entropy of quantum channels for all p>1, arXiv:2510.07547.
5. L. Shou and A. V. Gorshkov, A constructive violation of additivity of minimum output von Neumann entropy, arXiv:2609.23946. Recent related work on two-use violations.
6. Quantum Information and Quantum Computation Open Problem Zoo, Delayed-onset additivity violation for minimum output Rényi entropy.
7. T. Kassis, V. Agarwal, Y. He, D. Patel and A. M. Brueckner, Scientific Agent Skills: A Library of Procedural Knowledge for Research Agents, arXiv:2609.00065v2. Writing guidance consulted at commit 330c8e764435.