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Private information of the qubit depolarizing channel beyond the single-use threshold: explicit finite-n witnesses up to p = 0.28467, and what one-way key distillation already implies

Harish Santhanalakshmi Ganesan

Published Oct 21, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.5281/zenodo.23516557

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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 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.
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.
When does a difference in information become a way to communicate? Imagine two quantum channels as competing routes through noise. One might preserve a distinction that the other loses. That difference is interesting on its own; the question here is whether it can be turned into something operational: a channel that can carry private or quantum information at a positive rate. This paper presents a construction intended to make that step. Starting from a failure of complete quantum-channel information order, it constructs a channel claimed to have strictly positive ordinary private and quantum capacity. “Complete” matters because the comparison allows an input to be correlated with a finite quantum reference: information may be visible in the joint system even when it is hidden from the channel output alone. The construction accounts for the resources it uses, including transmission of the reference and uses of an activation helper. Its conclusion concerns strict positivity: a rate greater than zero, however small. It does not claim a uniform positive gap or fixed-error capacity inapproximability. The paper also develops reductions connecting the capacity questions to one-way distillable entanglement and secret key. The state constructions are designed to retain any prescribed lower bound on two-way distillation. On this basis, the manuscript claims exact-zero undecidability classifications for finite rational-complex memoryless channels and corresponding classifications for the one-way quantities. The information-order equivalences and undecidability premises used in these conclusions are developed in the companion paper, Complete information orders of quantum channels: equivalence and undecidability. A further introduction and citation details are available on the manuscript page.
This thesis provides bounds on the performance of quantum error correcting codes when used for quantum communication and quantum key distribution. The first two chapters provide a bare-bones introduction to classical and quantum error correcting codes, respectively. The next four chapters present achievable rates for quantum codes in various scenarios. The final chapter is dedicated to an upper bound on the quantum channel capacity. Chapter 3 studies coding for adversarial noise using quantum list codes, showing there exist quantum codes with high rates and short lists. These can be used, together with a very short secret key, to communicate with high fidelity at noise levels for which perfect fidelity is, impossible. Chapter 4 explores the performance of a family of degenerate codes when used to communicate over Pauli channels, showing they can be used to communicate over almost any Pauli channel at rates that are impossible for a nondegenerate code and that exceed those of previously known degenerate codes. By studying the scaling of the optimal block length as a function of the channel's parameters, we develop a heuristic for designing even better codes. Chapter 5 describes an equivalence between a family of noisy preprocessing protocols for quantum key distribution and entanglement distillation protocols whose target state belongs to a class of private states called twisted states. In Chapter 6, the codes of Chapter 4 are combined with the protocols of Chapter 5 to provide higher key rates for one-way quantum key distribution than were previously thought possible. Finally, Chapter 7 presents a new upper bound on the quantum channel that is both additive and convex, and which can be interpreted as the of the channel for communication given access to side channels from a class of zero cloning channels. This clone assisted capacity is equal to the unassisted for channels that are degradable, which we use to find new upper bounds on the of a depolarizing channel.
Secret-key distillation from quantum states and channels is a central task of interest in quantum information theory, as it facilitates private communication over a quantum network. Here, we study the task of secret-key distillation from bipartite states and point-to-point quantum channels using local operations and one-way classical communication (one-way LOCC). We employ the resource theory of unextendible entanglement to study the transformation of a bipartite state under one-way LOCC, and we obtain several efficiently computable upper bounds on the number of secret bits that can be distilled from a bipartite state using one-way LOCC channels; these findings apply not only in the one-shot setting but also in some restricted asymptotic settings. We extend our formalism to private communication over a quantum channel assisted by forward classical communication. We obtain efficiently computable upper bounds on the one-shot forward-assisted private capacity of a channel, thus addressing a question in the theory of quantum-secured communication that has been open for some time now. Our formalism also provides upper bounds on the rate of private communication when using a large number of channels in such a way that the error in the transmitted private data decreases exponentially with the number of channel uses. Moreover, our bounds can be computed using semidefinite programs, thus providing a computationally feasible method to understand the limits of private communication over a quantum network.
The quantum capacity of a noisy quantum channel determines the maximal rate at which we can code reliably over asymptotically many uses of the channel, and it characterizes the channel's ultimate ability to transmit quantum information coherently. In this paper, we derive single-letter upper bounds on the quantum and private capacities of quantum channels. The quantum capacity of a quantum channel is always no larger than the quantum capacity of its extended channels, since the extensions of the channel can be considered as assistance from the environment. By optimizing the parametrized extended channels with specific structures such as the flag structure, we obtain new upper bounds on the quantum capacity of the original quantum channel. Furthermore, we extend our approach to estimating the fundamental limits of private communication and one-way entanglement distillation. As notable applications, we establish improved upper bounds to the quantum and private capacities for fundamental quantum channels of interest in quantum information, some of which are also the sources of noise in superconducting quantum computing. In particular, our upper bounds on the quantum capacities of the depolarizing channel and the generalized amplitude damping channel are strictly better than previously best-known bounds for certain regimes.