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arXiv:2608.15870

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

Artus Krohn-Grimberghe

quant-phSubmitted Aug 16, 2026• Announced Aug 18, 2026• 0 citations
v1 · Aug 16, 2026v2 · Aug 18, 2026
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Abstract

The noise threshold below which the qubit depolarizing channel retains positive quantum capacity has been studied since 1996, and every improvement to date -- most recently the record of Agarwal et al. -- rests on floating-point evaluation of the coherent information. We give the first proven positivity results 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 the best previously reported numerical value. 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 two 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.

Comments
11 pages, 2 tables. Verification artifact: DOI 10.5281/zenodo.21962925

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The noise threshold below which the qubit depolarizing channel retains positive quantum capacity has been studied since 1996, and every improvement to date -- most recently the record of Agarwal et al. -- rests on floating-point evaluation of the coherent information. We give the first proven positivity results in this regime: an explicit 45-copy rank-two input state A, 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 the best previously reported numerical value. 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 A 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 two 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.
Short note with supporting code and data on Open Problem Zoo record op_d2813fe3fcdf09ad (private capacity P of the qubit depolarizing channel Lambda_p(rho) = (1-p) rho + p I/2). The record certifies P > 0 only for p ≤ 0.259824 (via the quantum capacity) and lists positivity on (0.259824, 1/3) as open. Two observations. (1) Published one-way key-distillation results already imply more: the one-way six-state rate with noisy preprocessing (Renes-Smith 2007, Kraus-Gisin-Renner 2005) is a single-use private information of this channel and is positive for p < 0.28223 (QBER = p/2), and the rates of Kern and Renes (2008) with repetition codes of length 250 are positive up to QBER 14.5741%, i.e. p = 0.291482; a short in-band lemma (send the publicly announced syndromes through extra channel uses with a classical code; I(S;B) = H(S) ≥ I(S;E) up to Fano terms) turns any such forward-announcement rate into a Devetak private information over finitely many uses, so P(Lambda_p) > 0 for all p ≤ 0.291482 with no public channel. (2) Explicit finite-n witnesses: ensembles of two classical mixed states on n ≤ 8 qubits whose private information I(X;B) - I(X;E) is positive at p = 0.28336, 0.28372, 0.28405, 0.28437, 0.28466 for n = 4, 5, 6, 7, 8 (thresholds 0.283375 ... 0.284671), above the single-use threshold 0.28223. They were found by an autonomous AI research agent (the depositor's EVO harness); the members are distributions over bit strings whose weights depend only on the Hamming weight, with the second member the bit complement of the first. Verification uses an exact closed-form block decomposition of the environment state's spectrum (no 4^n-dimensional objects); every value is recomputed in 30-digit arithmetic and agrees with double precision to about twelve digits. Negative findings of the same searches: the family is locally optimal within classical ensembles, gains ~0.0003 per added qubit, and general (quantum) members at n ≤ 3 give no gain; the finite-n frontier will not reach the key-distillation frontier at n ≤ 10. Claim boundary: lower bounds on the positivity threshold and on finite-n private informations; no capacity value, optimality or priority claim. Files: the note (PDF + LaTeX), the ensembles as JSON, the exact evaluator and verifier, the high-precision certifier, the agent's notes and search code, SHA-256 sums.
The quantum capacity captures the value of a quantum channel for transmitting quantum information, establishing the fundamental limits on quantum communication. In spite of its central role in quantum information theory, the quantum capacity of most channels is unknown, with wide gaps between the best upper and lower bounds. Even deciding whether a channel has nonzero capacity -- finding its capacity threshold -- is difficult. In this paper we report significant increases in the capacity thresholds of two prototypical noise models: the depolarizing channel and Pauli channels. In the case of the depolarizing channel, this is the first improvement in 18 years, giving a bigger increase beyond the hashing bound than all previous improvements combined. Our starting point is the representation theoretic framework recently proposed by Bhalerao and Leditzky (2025) to compute coherent information for special permutation invariant states. We generalize their framework to the full symmetric subspace, which allow us to optimize coherent information over rank two states in that space. A representation theoretic calculation shows that exponentially many Kraus operators of the channel annihilate the symmetric space, corresponding to a massive decrease in environment entropy for states on the symmetric space compared to the maximally mixed state. This explains the enhanced coherent information as a manifestation of degeneracy for the resulting codes.
We study the noise threshold for positive quantum capacity for the qubit depolarizing channel. We explore analytically the action of the qubit depolarizing channel on the symmetric subspaces of the input qubits, in the limit of asymptotically many uses of the channel. We observe the emergence of a bosonic Gaussian channel. Furthermore, the codes previously developed for the depolarizing channel can be translated to codes for the emergent Gaussian channel, and it is easier to further optimize these codes for the simpler emergent Gaussian channel. Translating these codes back to the depolarizing channel leads to extremely good input states for the coherent information of the depolarizing channel producing new lower bounds on the noise threshold for positive capacity. In addition to improved lower bounds on the threshold, this newly found link between depolarizing noise and Gaussian channels offers a novel perspective contributing to our understanding of these symmetric codes.
Short note with supporting code and data. For the qudit depolarizing channel N(rho) = (1-p) rho + p I/d, the only achievable rate quoted for d ≥ 3 is the hashing bound (maximally mixed input), which vanishes at p_h(3) = 0.2870867. An input maximally mixed on a two-dimensional subspace does better near the threshold: for one use of the qutrit channel its coherent information has the closed form I_c = 2η((1-p)/2 + p/3) + η(p/3) - η(1-p+p/6) - 5η(p/6) (bits, η(x) = -x log2 x) and is positive for all p < p_2(3), with p_2(3) in [0.29106723603359969, 0.29106723603359988] certified by interval arithmetic. Hence Q > 0 for all p < 0.2910672 (previous documented frontier 0.2870867, gain 3.98e-3), and the hashing expression is not the one-shot coherent information of the channel in a window below threshold for every 3 ≤ d ≤ 64 (certified enclosures for d = 3,4,5,6,8,16). Mechanism: the component leaking out of the code subspace heralds the error (qubit depolarizing channel with a flagged erasure). The input was found by an autonomous AI research agent (EVO harness, Claude Sonnet 5.5 proposers) as a two-qutrit rank-four state and identified by the author as a product of two single-qutrit rank-2 inputs. Multi-copy searches (n ≤ 7 qutrits, rank ≤ 4, including repetition codes inside the subspace) found nothing above p_2(3). Claim boundary: a lower bound on the threshold and on the one-shot coherent information; no capacity value, optimality or priority claim. Files: the note (PDF + LaTeX source), interval certificate script, exact multi-copy evaluator, the agent's state with verification scripts, baselines, the agent's notes and search scripts, SHA-256 sums.
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.
In this paper I revisit my 2006 conjecture on correlated errors in entangled physical qubits, originally proposed as a potential obstruction to quantum fault tolerance. The conjecture asserts that, in any physical implementation of a quantum computer, the effective noise channel acting on entangled physical qubits contains a joint fully depolarizing component, with a rate comparable to that of two-qubit gate errors. This hypothesized structural constraint goes beyond standard noise models and, if valid, would pose a significant challenge to scalable quantum fault tolerance. The conjecture remains open, but recent advances in experimental quantum computing bring it within reach of empirical testing on current devices. I also discuss two related directions in my critical study of quantum computation: the role of noise sensitivity and computational complexity in noisy intermediate-scale quantum systems, and the statistical analysis of experimental claims of quantum advantage. Finally, since this paper is written for a volume honoring Yuri Gurevich, I include some reflections on the ways in which my scientific and personal trajectory became intertwined with Yuri's.