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arXiv:2609.11930

Hierarchy of Rényi Coherent Information in Stabilizer Codes

Akash Vijay, Luis Colmenarez, Jong Yeon Lee

quant-phcond-mat.stat-mechSubmitted Sep 10, 2026• Announced Sep 11, 2026• 0 citations
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Abstract

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 $n \in \mathbb{Z}^+$. 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 $H_n(C)-H_n(T)$ 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.

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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-nnn coherent information is nondecreasing in n∈Z+n \in \mathbb{Z}^+n∈Z+. This follows from a general theorem: if independent random bits are mapped linearly to a fine label TTT and a coarse label CCC, then the Rényi entropy difference Hn(C)−Hn(T)H_n(C)-H_n(T)Hn​(C)−Hn​(T) is nondecreasing in nnn. For stabilizer codes, TTT is the joint syndrome--logical class and CCC is the syndrome, and the difference is the Rényi-nnn 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-nnn coherent information an operational meaning via postselection on matching syndromes between one data block and n−1n-1n−1 auxiliary blocks. We determine when this defines a quantum channel and show that saturation of the Rényi-nnn coherent information is equivalent to asymptotically perfect recovery of the postselected channel. Moreover, the Rényi-nnn coherent information also upper-bounds the ordinary coherent information achievable after any syndrome-conditioned recovery.
We investigate the stability of logical information in quantum stabilizer codes subject to coherent unitary errors. Beginning with a logical state, we apply a random unitary error channel and subsequently measure stabilizer checks, resulting in a syndrome-dependent post-measurement state. By examining both this \emph{syndrome state} and the associated syndrome distribution, we identify a phase transition in the behavior of the logical information. In the Clifford/stabilizer setting, the change of logical stabilizer structure in a syndrome branch determines the state-level MAP Pauli-frame return probability for a chosen logical-basis input. We separately use quantum coherent information to diagnose recoverability of arbitrary encoded inputs at the channel level. Below a critical error threshold pcp_cpc​, the syndrome branches remain compatible with the original logical sector, enabling a high state-return probability. Above pcp_cpc​, the data are consistent with effective scrambling of the encoded subspace relative to the stabilizer checks, where the syndrome-resolved logical maps approach a global-Clifford-induced stabilizer instrument. This common post-threshold picture organizes both the toric-code and finite-rate random-stabilizer-code results within the same scrambling mechanism. The syndrome distribution supplies complementary local and global structural diagnostics whose scaling depends on the code family. We refer to this phenomenon as a \emph{coherent error induced phase transition}. To illustrate this transition, we study two classes of quantum error-correcting codes, the toric code and finite-rate random stabilizer codes, thereby shedding light on the design and performance limits of quantum error correction under coherent errors.
We formulate stabilizer-code error correction as a code- and recovery-dependent transformation of the noise channel. For a fixed recovery map, syndrome projection produces a block-diagonal logical channel whose diagonal Pauli weights follow from code-induced error classes, with no physical Pauli twirl; concatenation is then iterated redistribution, with the dominant logical error guiding the next code. The bit-flip and phase-flip three-qubit codes with standard recoveries supply a closed-form benchmark, and logically inequivalent recoveries appear as a design degree of freedom for shaping the channel presented to the next decoder. For amplitude damping, the three-qubit rules select C2(σ^y)C1(σ^x)C_2(\hatσ_y)C_1(\hatσ_x)C2​(σ^y​)C1​(σ^x​) and give fl=1−O(γ2)f_l=1-\mathcal O(γ^2)fl​=1−O(γ2), with an ideal-gate crossover at γc≈0.30γ_c\approx0.30γc​≈0.30 (f=1−1.72755γ2f=1-1.72755γ^2f=1−1.72755γ2) and a favorable gate-count comparison with the five-qubit code under the same assumptions. Correlated noise requires joint class statistics, whereas coherent noise retains intra-class coherences that make the same transformation usable in reverse as a tomographic fingerprint; there the companion work proves local identifiability from a rank certificate evaluated inside the parameter domain, classifies the sole permutation-symmetry ambiguity as structural at any chain length, and closes it with one two-qubit marginal channel. The construction remains an analytic benchmark, not a universal prescription for an inner code beneath a surface code.
Quantum low-density parity-check (qLDPC) codes have a canonical mixed state---the normalized projector onto the code space---that lies at the intersection of quantum error correction and quantum many-body physics, yet the stability of its information-theoretic structure under local noise remains poorly understood. Recent progress on mixed-state phases of matter has identified the spatial decay of conditional mutual information (CMI) I(A:C∣B)I(A:C|B)I(A:C∣B) as a key diagnostic of such stability, analogous to the role of the spectral gap for quantum ground states. We prove that, for the normalized code-projector state of a Zd\mathbb{Z}_dZd​ qLDPC stabilizer code under local Pauli noise, the CMI across a buffer BBB separating AAA from CCC decays exponentially with the width of BBB in two complementary noise regimes independent of system size. At weak noise, this holds for full-system tripartitions, where correlations between the two sides require an error cluster to cross the buffer, an exponentially unlikely event. At strong noise, exponential decay holds whenever the buffer supports no logical operator, since correlations can persist only along rare chains of buffer qudits that remain informative. Under an additional space-uniformity assumption, we establish the robustness of Levin--Wen topological entropy. On finite-dimensional lattices of size nnn, the weak-noise bound further implies a constant-layer quasi-local Petz recovery channel of range $O(\poly(\log n)/\varepsilon)$ that recovers the noiseless state within trace-distance error ε\varepsilonε. Both CMI bounds apply to general qLDPC interaction graphs, including geometries where such constant-depth quasi-local recovery need not exist. Thus, the exponential decay of the noise-induced CMI provides a general criterion for mixed-state phase stability.
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\varepsilonε-quantum capacity of the class for every ε<1\varepsilon < 1ε<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]p \in [1/4, 3/4]p∈[1/4,3/4], that capacity is zero, while for p∈[1/4,1/2)p \in [1/4,1/2)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.
What must a quantum error-correction decoder know about temporally correlated noise, and what does it cost to learn? We study this question for fixed stabilizer memory experiments with known system-environment interactions. We formulate calibration in terms of the fidelity differences that select a Pauli recovery and relate missing information to the resulting fidelity loss. Our main example uses weak coherent XXX rotations in a class of CSS codes including Shor, Steane, and odd-distance rotated surface codes. All physical qubits share a sign either fixed throughout a run or independently redrawn each interval. The two models have identical syndrome statistics and the same spatiotemporal Pauli process (SPP), obtained by Pauli-twirling every noise interval. Yet at suitable storage times, they require different logical corrections. In the weak-noise limit, each model is almost perfectly correctable when known. Without model information, even the best final recovery has a worst-case fidelity loss approaching 1/21/21/2 relative to the noise-aware optimum, despite access to the full syndrome record. For this pair of models, at fixed code and fixed positive target loss below 1/21/21/2, we prove matching calibration bounds. With logical preparation and final readout, complete QEC cycles require a number of noise intervals scaling as the inverse rotation angle raised to the code's XXX distance. Allowing two uninterrupted intervals before one extraction reduces the optimal cost to inverse-square scaling. Physical Pauli twirling of every noise interval changes both recovery performance and its information requirements. Discarding the record of applied random Pauli operations makes the SPP sufficient for optimal recovery. Keeping this record can improve the attainable fidelity, but reaching that optimum may require additional noise information.
The codespace of a quantum error-correcting code can often be identified with the degenerate ground space within a gapped phase of quantum matter. We argue that the stability of such a phase is directly related to a set of coherent error processes against which this quantum error-correcting code (QECC) is robust: such a quantum code can recover from , corresponding to random adiabatic drift of code states through the phase, with asymptotically perfect fidelity in the thermodynamic limit, as long as this adiabatic evolution keeps states sufficiently “close” to the initial ground space. We further argue that when specific decoders—such as minimum-weight perfect matching—are applied to recover this information, an error-correcting threshold is generically encountered the gapped phase. In cases where the adiabatic evolution is known, we explicitly show examples in which quantum information can be recovered by using stabilizer measurements and Pauli feedback, even up to a phase boundary, though the resulting decoding transitions are in different universality classes from the optimal decoding transitions in the presence of incoherent Pauli noise. This provides examples where nonlocal, coherent noise effectively decoheres in the presence of syndrome measurements in a stabilizer QECC.