A two-dimensional-subspace input beats the hashing point of the qudit depolarizing channel: positive single-letter coherent information at p = 0.2910672 for d = 3
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.
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.
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 A, published as a 1,472-byte witness, with certified Ic>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 (≈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.
Short note with supporting code and data on Open Problem Zoo record op_37f7b003ec78a028 (stabilizer rank of tensor powers of the magic state |T> = (|0> + e^{i pi/4}|1>)/sqrt2). The exact stabilizer rank chi(|T>^{(x)m}) is known for m ≤ 4 (2, 2, 3, 4); at m = 5, 6, 7 only the upper bounds 6, 6, 12 were known, the last being the product of the six-copy and one-copy decompositions (no direct search beyond six copies had been reported). This deposit gives an explicit decomposition of |T>^{(x)7} into ten stabilizer states, hence chi(|T>^{(x)7}) ≤ 10, and by sub-multiplicativity chi(|T>^{(x)13}) ≤ 60 (previously 72). The ten states are symmetric under permutations within the qubit blocks {1,2} and {3,...,7}; they are listed in Dehaene-De Moor normal form with exact coefficients in (1/4) Z[e^{i pi/4}]. The decomposition was found by an autonomous AI research agent (the depositor's EVO harness) that closed three seed states under block-symmetric Clifford gates to obtain a dictionary of a few hundred stabilizer states spanning the relevant symmetric subspace and then ran a subset search over it. Verification: exact Gaussian elimination over Q(e^{i pi/4}) (a proof of the identity), a 40-digit least-squares residual of 1.6e-41, and a brute-force recount of the stabilizer group of every state. Companion searches give evidence, not proof, that chi(|T>^{(x)5}) = 6. Claim boundary: an upper bound with an explicit certificate; no optimality, no asymptotic statement, no priority claim. Files: the note (PDF + LaTeX), the decomposition as JSON and as a readable listing, the exact verifier and the independent check with its log, the agent's search code and notes, the task text, 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.
Correction d’erreurs quantiques avec des qubits de chats Au cours des deux dernières décennies, les circuits supraconducteurs se sont imposés comme une plateforme prometteuse pour la construction d’un ordinateur quantique. Cependant, ils demeurent limités par leur temps de cohérence qui reste insuffisant pour démontrer un avantage quantique pratique. La correction d'erreur quantique offre une approche pour contrer la décohérence, le principe fondamental étant d'introduire de la redondance afin de définir un qubit dit logique. Ainsi, si un qubit physique isolé subit une erreur, celle-ci peut être détectée et corrigée sans affecter l'information contenue dans le qubit logique. L'une des approches les plus intuitives, se rapprochant le plus de la correction d'erreur classique, consiste à utiliser une multitude de qubits physiques pour réaliser la redondance recherchée.Cette thèse explore une approche alternative, basée sur l'encodage de l'information quantique dans des cavités supraconductrices, la redondance étant alors simplement fournie par la dimension infinie de l'espace de Hilbert. Nous utilisons des qubits de chat pour lesquels les états logiques 0 et 1 sont des états cohérents d'un oscillateur harmonique. Ces états sont stabilisés en utilisant la dissipation à notre avantage, de sorte que les échanges de photons entre le mode harmonique et son environnement se fassent principalement par paires. Ainsi, les erreurs de type « bit-flip » sont supprimées exponentiellement avec le nombre de photons contenus dans le mode, au prix d'une augmentation linéaire des erreurs de type « phase-flip ». Ces erreurs pourraient alors être corrigées par une couche supplémentaire de correction, tel qu'un code de répétition de qubits de chats. Le coeur de ce travail de thèse consiste à introduire un circuit supraconducteur autoparamétrique qui couple de manière non-linéaire un mode contenant le qubit de chat à un mode dissipatif dont la fréquence est réglée au double de celle du mode du chat. Contrairement à de précédentes réalisations, ce couplage passif ne nécessite pas de pompe paramétrique et atteint un fort taux de dissipation d'environ 2 MHz. Les erreurs de bit-flip sont alors évitées pendant une période caractéristique pouvant aller jusqu'à 0.3 s, avec un impact modéré sur les erreurs de phase-flip. De plus, nous démontrons un contrôle universel de ce qubit en utilisant la dissipation à 2 photons pour réaliser des portes logiques X, Y et Z d'un angle arbitraire.
The quantum capacity of the bosonic thermal attenuator, which is given by the regularization of its coherent information, is unknown. The seminal work of Holevo and Werner established in 1999 the standard one-use lower bound obtained from input thermal states. We first prove that this long-standing lower bound is the exact supremum over all single-mode Gaussian states and then show that, crucially, a non-Gaussian state can do better. As a consequence, we prove positivity of the quantum capacity in a parameter region where the channel is not antidegradable, yet its coherent information optimized over single-mode Gaussian states vanishes. For example, with one thermal photon in the environment and at transmissivity η=0.8, the coherent information is non-positive for every single-mode Gaussian input. We give an explicit rank-two non-Gaussian state, supported on only six Fock levels, whose coherent information is certified to be at least 4.7×10−4 qubits per channel use. This short witness is far from numerically optimal: a numerical optimization over fixed non-Gaussian families reaches at least 8.4×10−3 qubits per channel use at the same point. More generally, at ν=1, using non-Gaussian inputs we certify positivity of the coherent information, and therefore of the quantum capacity, down to η=0.7841; by contrast, the channel is antidegradable, and hence has zero quantum capacity, for η≤0.75. Overall, our work identifies new high-noise regimes in which bosonic quantum communication is possible.