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A depolarizing choir sings in Gaussian harmony

Rabsan Galib Ahmed, Sujeet Bhalerao, Sungjai Lee, Felix Leditzky, Debbie Leung, Luke Schaeffer, Graeme Smith

arXiv:2609.39747v1quant-phcs.ITmath.IT

TL;DR

The qubit depolarizing channel’s positive-capacity noise threshold is not known exactly, despite existing lower and upper bounds. This work analyzes the channel on symmetric input subspaces, connects it to a bosonic Gaussian thermal attenuator, and uses translated codes to establish an improved lower bound.

  • Problem

    The depolarizing channel’s positive-capacity noise threshold is unknown exactly, and certifying positive quantum capacity is challenging.

    Source: p. 1

  • Method

    The authors analyze the depolarizing channel on symmetric subspaces in the many-use limit, identify an emergent Gaussian thermal attenuator, and translate codes between the two settings.

    Source: pp. 1, 3, 4

  • Results

    The depolarizing-channel threshold lower bound reaches pth ≥ 0.2029161, improving on the previous best-known bound of pth ≥ 0.19397.

    Source: p. 5

  • Takeaways & Limitations

    The link between depolarizing noise and Gaussian channels offers a new perspective on symmetric codes, and the reported codes outperform any Gaussian code for the stated Gaussian-channel parameter relation.

    Source: p. 2

    The reported code evaluation fixes η = 0.7294452, corresponding to p = 0.2029161.

    Source: p. 5

Abstract

from arXiv · show

We study the noise threshold for positive quantum capacity for the qubit depolarizing channel. We explore analytically the action of the qubit depolarizing channel on the symmetric subspaces of the input qubits, in the limit of asymptotically many uses of the channel. We observe the emergence of a bosonic Gaussian channel. Furthermore, the codes previously developed for the depolarizing channel can be translated to codes for the emergent Gaussian channel, and it is easier to further optimize these codes for the simpler emergent Gaussian channel. Translating these codes back to the depolarizing channel leads to extremely good input states for the coherent information of the depolarizing channel producing new lower bounds on the noise threshold for positive capacity. In addition to improved lower bounds on the threshold, this newly found link between depolarizing noise and Gaussian channels offers a novel perspective contributing to our understanding of these symmetric codes.

Supplemental Material for “A depolarizing choir sings in Gaussian harmony”

The supplemental material lists the paper’s authors and their institutional affiliations.

A. Representation theory

This section sets up representation-theoretic notation for qubit tensor powers and their symmetric subspaces. It relates permutation symmetry, spin notation, and Dicke states.

A. Representation theory

  • Λ(M, 2) denotes partitions of M into at most two parts, and GL(2) is the set of invertible linear operators on C^2.Source: A. Representation theory · p. 7
  • The permutation action and the tensor-power action commute and span each other’s commutants, a relation known as Schur-Weyl duality.Source: A. Representation theory · p. 7
  • Spin notation labels a partition by N = λ1 − λ2, with λ1 = (M + N)/2 and λ2 = (M − N)/2.Source: A. Representation theory · p. 8
  • For each partition, the section defines the corresponding GL(2) and symmetric-group representations and the projector onto their isotypical component.Source: A. Representation theory · p. 8
  • The symmetric subspace is invariant under tensor powers of a linear operator and has a Dicke-state basis indexed by excitation number.The section also states that the symmetric subspace on n qubits has dimension n + 1.Source: A. Representation theory · p. 8
  • The map A ↦ S_n(A) is a representation, and the section gives an explicit diagonal-operator action on Dicke states.Source: A. Representation theory · p. 8

B. Quantum capacity

This section introduces coherent information and quantum capacity, whose regularized optimization is difficult. It also places the depolarizing-channel threshold bounds in context.

B. Quantum capacity

  • The quantum capacity is defined through a regularized optimization of coherent information over channel uses, and that regularization cannot generally be removed because coherent information can be superadditive.Source: B. Quantum capacity · p. 8
  • When a channel is antidegradable, its quantum capacity is zero by a no-cloning argument.Source: B. Quantum capacity · p. 8
  • The maximally mixed input supplies an achievable rate for the depolarizing channel.Source: B. Quantum capacity · p. 9
  • The hashing bound vanishes at p ≈ 0.18929 (η ≈ 0.7476); superadditivity pushes the threshold beyond it, while optimizing rank-two states in the n-qubit symmetric subspace yields a lower bound p ≈ 0.1940 (η ≈ 0.7414).Source: B. Quantum capacity · p. 9

C. Gaussian channels

This section introduces bosonic Gaussian states and operations, then describes loss and amplification channels. Their composition yields effective Gaussian-channel types determined by transmissivity and gain.

C. Gaussian channels

  • Gaussian states are determined by their first moments and quantum covariance matrices, which must satisfy the uncertainty principle.Source: C. Gaussian channels · p. 9
  • Gaussian operations preserve Gaussianity, and linear bosonic Gaussian channels arise when measurement outcomes are discarded.Source: C. Gaussian channels · p. 9
  • Pure loss mixes a mode with vacuum on a beam splitter of transmissivity T, while quantum-limited amplification uses a two-mode squeezer with gain G.In the Fock basis, u and t count photons lost to or added by the environment, respectively.Source: C. Gaussian channels · p. 9
  • Composing amplification after loss yields a Gaussian channel whose effective type depends on GT: thermal attenuation for GT < 1, additive Gaussian noise for GT = 1, and thermal amplification for GT > 1.Source: C. Gaussian channels · p. 10

II. DEPOLARIZING NOISE ON THE SYMMETRIC SUBSPACE

For symmetric-subspace inputs, the depolarizing channel’s output decomposes into Schur-Weyl blocks, and the probability of each block is independent of the input state. The block channels are normalized using these fixed probabilities.

II. DEPOLARIZING NOISE ON THE SYMMETRIC SUBSPACE

  • For a symmetric-subspace input, the IID channel preserves permutation invariance, enabling decomposition of its output into irreducible blocks.The decomposition is described using Schur-Weyl duality.Source: §II · pp. 10–11
  • Each output block occurs with an input-independent probability, so the channel’s symmetric-subspace output admits a normalized block decomposition.The probabilities form a fixed distribution over N for each M.Source: §II · pp. 10–11
  • The block probability can be evaluated using |0⟩⟨0|^⊗M, whose depolarized output is diagonal.The proof uses input independence to choose this state for evaluating the probability.Source: §II · pp. 10–11

A. Decomposition of the channel on irreps

Each conditional channel on an irreducible block is expressed through loss and symmetric cloning operations. The theorem represents the channel as a mixture of their compositions.

A. Decomposition of the channel on irreps

III. CONCENTRATION LEMMAS

The section develops concentration estimates for the block probabilities pN and the mixing weights πs. It introduces sums of independent Bernoulli variables in the analysis of these quantities.

III. CONCENTRATION LEMMAS

  • The concentration estimates address both block probabilities pN and mixing weights πs.The section treats pN first, then turns to πs.Source: §III · pp. 14–15
  • The analysis represents XM as a sum of IID Bernoulli variables with success probability (1 + η)/2.Source: §III · p. 14
  • The analysis represents YN+1 as a sum of IID Bernoulli variables with success probability 2η/(1 + η).Source: §III · p. 15

A. Gaussian limit of the random loss-cloning channels

For a fixed excitation cutoff and fixed ratio of sector size to input size, the random loss-cloning channels approach Gaussian channels as the number of input qubits grows. The approximation is established using typical conditional channels and comparisons of their excitation-basis distributions.

A. Gaussian limit of the random loss-cloning channels

  • For fixed cutoff K and fixed ratio q = N/M, the approximation error εM,K,q tends to zero as M grows.The theorem concerns the restricted channels at the fixed excitation cutoff.Source: A. Gaussian limit of the random… · p. 16
  • The convergence follows from excitation-basis Kraus coefficients resembling hypergeometric and negative hypergeometric distributions, which approach their binomial counterparts.The loss and attenuation pair, and the cloning and amplification pair, are compared separately.Source: A. Gaussian limit of the random… · p. 17

B. Convergence among the typical Gaussian channels

For a fixed excitation cutoff, the typical conditional attenuation channels approach one another in diamond norm as the number of uses grows. The proof controls the bound at the cutoff excitation.

B. Convergence among the typical Gaussian channels

C. Convergence of coherent information

For states supported within a fixed Dicke excitation cutoff, the coherent information difference in Theorem IV.6 vanishes as M grows. The proof handles typical and atypical sectors separately.

C. Convergence of coherent information

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