We prove a strong converse for quantum communication over Pauli channels within the class of stabilizer codes. If a code whose code space is a full joint eigenspace of a stabilizer group transmits above the coherent information of its own input state, its entanglement fidelity decays exponentially in the block length; the encoder may be any isometry onto that space and the decoder any channel. For memoryless channels this determines the
ε-quantum capacity of the class for every
ε<1, so that tolerating a constant error buys no rate; for antidegradable channels, such as the depolarizing channel with error probability
p∈[1/4,3/4], that capacity is zero, while for
p∈[1/4,1/2) partial-transposition bounds provably cannot certify a strong converse. The proof uses neither additivity assumptions nor semidefinite relaxations: optimal decoding succeeds precisely on an event in a product probability space, so the blowing-up lemma of Ahlswede, Gács and Körner applies, and the side information it produces is charged against the coherent information. The argument also constrains near-deterministic decoding for codes of any kind, and we isolate the encoder-side statement that would extend it to all of them.
The performance of quantum error correction can be significantly improved if detailed information about the noise is available, allowing to optimize both codes and decoders. It has been proposed to estimate error rates from the syndrome measurements done anyway during quantum error correction. While these measurements preserve the encoded quantum state, it is currently not clear how much information about the noise can be extracted in this way. So far, apart from the limit of vanishing error rates, rigorous results have only been established for some specific codes. In this work, we rigorously resolve the question for arbitrary stabilizer codes. The main result is that a stabilizer code can be used to estimate Pauli channels with correlations across a number of qubits given by the pure distance. This result does not rely on the limit of vanishing error rates, and applies even if high weight errors occur frequently. Moreover, it also allows for measurement errors within the framework of quantum data-syndrome codes. Our proof combines Boolean Fourier analysis, combinatorics and elementary algebraic geometry. It is our hope that this work opens up interesting applications, such as the online adaptation of a decoder to time-varying noise.
This paper focuses on error thresholds for Pauli channels. We numerically compute lower bounds for the thresholds using the analytic framework of coset weight enumerators pioneered by DiVincenzo, Shor and Smolin in 1998. In particular, we study potential non-additivity of a variety of small stabilizer codes and their concatenations, and report several new concatenated stabilizer codes of small length that show significant non-additivity. We also give a closed form expression of coset weight enumerators of concatenated phase and bit flip repetition codes. Using insights from this formalism, we estimate the threshold for concatenated repetition codes of large lengths. Finally, for several concatenations of small stabilizer codes we optimize for channels which lead to maximal non-additivity at the hashing point of the corresponding channel. We supplement these results with a discussion on the performance of various stabilizer codes from the perspective of the non-additivity and threshold problem. We report both positive and negative results, and highlight some counterintuitive observations, to support subsequent work on lower bounds for error thresholds.
Abstract The classical noisy-channel coding theorem developed by Shannon establishes a fundamental limit on reliable information transmission. In quantum regime, this framework is extended by results such as the Lloyd-Shor-Devetak theorem, which gives the hashing limit on quantum channel capacity. In this work, we beat the hashing limit for low rate coding schemes over quantum Pauli channels. Our approach is based on a re-examination of Bacon-Shor codes and their generalizations, which we show are not merely simple constructions but instances of a broader mechanism that fundamentally alters achievable rate limits. Over depolarizing channels with depolarizing probability of p, these codes exceed the hashing bound with 0.1145 ≤ p ≤ 0.75, and further surpass the asymptotic quantum Singleton bound under bounded minimum-distance decoding for 0.3301 ≤ p ≤ 0.75. We further present explicit constructions based on quantum low-density parity-check codes, achieving frame error rates below 10−4 while exceeding the hashing bound. The advantage persists and is pronouncedly enhanced in the presence of biased noise, demonstrating robustness across realistic quantum Pauli channels. Our results challenge the prevailing understanding of fundamental limits in quantum coding and information theory.
Rényi coherent information, a computable proxy for the von Neumann coherent information, is widely used to study mixed-state phases of matter and decodability transitions in noisy quantum error-correcting codes. However, being a difference of two Rényi entropies, it need not be monotonic in the Rényi index, and lacks the operational meaning of its von Neumann counterpart. Here we address both issues for stabilizer codes. First, for Pauli noise generated by independent Bernoulli events, we prove that the Rényi-
n coherent information is nondecreasing in
n∈Z+. This follows from a general theorem: if independent random bits are mapped linearly to a fine label
T and a coarse label
C, then the Rényi entropy difference
Hn(C)−Hn(T) is nondecreasing in
n. For stabilizer codes,
T is the joint syndrome--logical class and
C is the syndrome, and the difference is the Rényi-
n coherent information up to a constant. The same theorem covers classical linear codes and independent detector error models. Second, for arbitrary stochastic Pauli noise, we give the Rényi-
n coherent information an operational meaning via postselection on matching syndromes between one data block and
n−1 auxiliary blocks. We determine when this defines a quantum channel and show that saturation of the Rényi-
n coherent information is equivalent to asymptotically perfect recovery of the postselected channel. Moreover, the Rényi-
n coherent information also upper-bounds the ordinary coherent information achievable after any syndrome-conditioned recovery.
Coherent errors in stabilizer codes are often correlated across qubits and QEC cycles. Having a general analytical treatment of such noise would thus be extremely valuable. We derive here the exact logical channel induced by repeated QEC cycles under correlated coherent
Z noise, and develop a broadly general cumulant-expansion framework that yields a tractable expression for the noise-averaged logical infidelity. Crucially, this expression is non-perturbative in the noise, and applies to arbitrary stabilizer codes and correlation structures. It reveals a feature with no analogue in standard stochastic Pauli error models: the induced channel depends on which stabilizer eigenspace is chosen as the codespace. Exploiting this, we introduce protected stabilizer eigenspace (PROSE) encoding, an error-suppression strategy that selects the optimal codespace. We show that this eigenspace can be efficiently identified in many relevant situations. Further, when combined with logical Pauli twirling, PROSE matches or outperforms standard error suppression techniques (dynamical decoupling, Pauli twirling of physical qubits). We also show that noise correlations, usually assumed to be harmful to QEC, can instead be a resource: with the right encoding, even positive correlations reduce the logical infidelity below the uncorrelated baseline. Our results offer a new, broadly applicable lens on correlated coherent noise in stabilizer codes.