Enhanced quantum capacity thresholds from symmetry

Abstract: 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.

• The paper presents a new lower bound for the depolarizing channel threshold, achieving p=0.064657 at n=45, marking the first improvement in nearly 20 years.
• It employs a representation-theoretic framework using symmetric subspaces and Schur-Weyl duality to optimize coherent information and reduce environmental entropy.
• The study demonstrates significant threshold enhancements for depolarizing, independent X-Z, and 2-Pauli channels with moderate blocklengths, indicating practical gains in quantum coding.

Enhanced Quantum Capacity Thresholds from Symmetry

Introduction

This work addresses one of the foundational problems in quantum information theory: determining the quantum capacity of noisy quantum channels, particularly Pauli channels such as the qubit depolarizing channel. The key technical obstacle is the non-additivity of the channel coherent information, which undermines the regular structure observed in classical information theory and mandates highly nontrivial, non-asymptotic analysis to bound channel capacities.

The paper provides a new lower bound for the depolarizing channel threshold—the first improvement in nearly two decades—and improves thresholds for other Pauli noise models. The advance is achieved by leveraging a generalization of the representation-theoretic analysis of permutation-invariant input states, which allows for effective and tractable optimization of coherent information in the symmetric subspace. This is underpinned by explicit analysis of how symmetry and degeneracy in code construction yield exponential decreases in the environment entropy, thus directly enhancing achievable quantum communication rates.

Background and Framework

In quantum information transmission, the key rate-determining quantity is the quantum capacity $\mathcal{Q}(\mathcal{N})$, given by the regularized coherent information:

$$\mathcal{Q}(\mathcal{N}) = \lim_{n\rightarrow \infty} \frac{1}{n} \max_{\rho} I_c(\mathcal{N}^{\otimes n}, \rho)$$

where $I_c$ denotes the coherent information, and $\rho$ is an input state.

For highly symmetric noise models, prior works largely failed to close the gap between existing lower and upper bounds. Traditionally, explicit code constructions—namely concatenated repetition codes—have been used, but improvements have become increasingly incremental and computationally demanding. The current paper turns to group-theoretic methods, using the machinery of Schur-Weyl duality and the block structure of symmetric and unitary representations to analyze the channel and its codespace in the symmetric subspace over $n$ qubits.

The block-diagonalization of tensor power matrices under the Schur transform leads to substantial computational simplifications, enabling the explicit calculation and optimization of coherent information over symmetric states, and, crucially, arbitrary rank-two states within the symmetric subspace.

Main Results

Improved Thresholds for Depolarizing and Pauli Channels

By optimizing coherent information within this enlarged code space, the authors obtain significantly stronger threshold lower bounds:

• Depolarizing channel: Lower bound $p = 0.064657$ for the noise threshold at blocklength $n=45$, improving on the longstanding Fern-Whaley result at $p = 0.06376$. Notably, comparable or better thresholds are obtained for $n \geq 24$, a blocklength orders of magnitude smaller than used in previous Monte Carlo approaches.

Figure 1: Error thresholds for the depolarizing channel, comparing new and prior results.

• Independent X-Z channel: Lower bound $p = 0.118371$ at $$\mathcal{Q}(\mathcal{N}) = \lim_{n\rightarrow \infty} \frac{1}{n} \max_{\rho} I_c(\mathcal{N}^{\otimes n}, \rho)$$0, surpassing all previous constructive thresholds.

Figure 2: Error thresholds for the independent X-Z channel; significant improvements are achieved via the symmetric subspace approach.

• 2-Pauli channel: Lower bound $$\mathcal{Q}(\mathcal{N}) = \lim_{n\rightarrow \infty} \frac{1}{n} \max_{\rho} I_c(\mathcal{N}^{\otimes n}, \rho)$$1 at $$\mathcal{Q}(\mathcal{N}) = \lim_{n\rightarrow \infty} \frac{1}{n} \max_{\rho} I_c(\mathcal{N}^{\otimes n}, \rho)$$2, with nontrivial threshold increases at much smaller encoding lengths than required for previous degenerate stabilizer constructions.

Figure 3: Error thresholds for the 2-Pauli channel, indicating marked gains at modest blocklength.

These results surpass the so-called hashing bound by more than all previous incremental improvements combined for the depolarizing channel. Importantly, the new symmetry-based code classes allow significant threshold gains even at moderate blocklength $$\mathcal{Q}(\mathcal{N}) = \lim_{n\rightarrow \infty} \frac{1}{n} \max_{\rho} I_c(\mathcal{N}^{\otimes n}, \rho)$$3.

Representation Theory and Coherent Information Computation

The authors generalize the representation-theoretic calculation of coherent information for permutation-invariant states to all states in the symmetric subspace. By expressing arbitrary symmetric states in the Dicke and tensor power basis, and utilizing the Schur transform framework, the density matrix outputs under the channel action are block-diagonal in irrep labels. This enables substantial computational efficiency and scaling.

Numerical optimization over these parameters (with gradient-based heuristics) identifies symmetric rank-two input mixtures that yield optimal threshold values. Through careful numerical implementation and error handling, the methods yield stable and highly accurate threshold estimates at feasible computational cost.

Degeneracy and the Mechanism of Threshold Enhancement

The paper provides a rigorous representation-theoretic explanation for why permutation-invariant codes achieve enhanced thresholds. The key result is a calculation showing that, in the symmetric subspace, exponentially many Kraus operators (i.e., errors) act trivially ("annihilate") the codespace. This degeneracy severely restricts the entropy of the environment for channel outputs, directly improving the coherent information:

• The number of non-annihilating Schur basis Kraus operators is $$\mathcal{Q}(\mathcal{N}) = \lim_{n\rightarrow \infty} \frac{1}{n} \max_{\rho} I_c(\mathcal{N}^{\otimes n}, \rho)$$4, which is exponentially smaller (in $$\mathcal{Q}(\mathcal{N}) = \lim_{n\rightarrow \infty} \frac{1}{n} \max_{\rho} I_c(\mathcal{N}^{\otimes n}, \rho)$$5) than the number of nonzero probability typical errors for generic codes. This dramatic reduction underpins the significant threshold enhancement.
• The degeneracy framework generalizes the intuition behind concatenated repetition codes but makes it systematic—and, via hook length calculations and Schur polynomials, supports explicit enumeration and scaling bounds.

Implications and Future Directions

The results point to a compelling lesson: highly symmetric code design—beyond the stabilizer formalism—can offer practical and tractable routes to quantum capacity at higher noise thresholds in prototypical channels. This reinforces the view that group-theoretical and representation-theoretic tools can drive future advances, both numerical and conceptual, in quantum coding theory.

Practically, these methods enable the efficient search for capacity-achieving codes in the symmetric (and possibly other invariant) subspaces at moderate blocklength. The theoretical framework makes explicit the entropic reductions available via code degeneracy—suggesting new directions for code constructions in high-noise regimes and for channels with significant symmetries.

Future work may focus on extending these methods:

• To non-Pauli and higher-dimensional (qudit) channels.
• To structured non-symmetric subspaces and many-body noise environments.
• To asymptotic analysis at even longer blocklengths, possibly via further optimization or hybrid analytical-numerical techniques.

Conclusion

By unifying group representation theory and quantum information, the paper achieves substantial progress in the study of quantum capacity thresholds for Pauli channels. The synergy of symmetry, degeneracy, and explicit block-diagonalization has not only raised achievable bounds but clarified the mechanisms behind threshold enhancement—pointing toward more systematic, scalable, and ultimately optimal code design paradigms for quantum communication.