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Resource State Distillation via Stabilizer Channels

Christopher Popp, Tobias C. Sutter, Beatrix C. Hiesmayr

TL;DR

This work introduces a unified framework for stabilizer-based resource distillation in systems of prime local dimension by formulating stabilizer routines as quantum channels and deriving closed-form expressions for their output, and introduces several protocols that effectively tailor stabilizer channels to specific operational tasks.

Abstract

Quantum technologies rely on high-quality resource states, such as maximally entangled or private states, which are indispensable for quantum communication and cryptography. In practice, however, these states are inevitably degraded by noise. Distillation protocols aim to recover high-resource states from multiple imperfect copies, and while stabilizer-based methods have demonstrated high performance in entanglement purification, they have yet to be established for broader tasks such as secret-key distillation. This work introduces a unified framework for stabilizer-based resource distillation in systems of prime local dimension. By formulating stabilizer routines as quantum channels and deriving closed-form expressions for their output, we enable the application of stabilizer operations to general input states and diverse distillation objectives. We identify key invariances in resource measures, such as coherent and private information, and demonstrate how they can be leveraged to significantly reduce the numerical complexity of channel optimization. To illustrate the framework's versatility, we introduce several protocols: gF-IMAX for general fidelity optimization, and (S)CI-IMAX and (S)PI-IMAX for optimizing (smooth) coherent and private information in both asymptotic and one-shot regimes. Our numerical results confirm that these protocols effectively tailor stabilizer channels to specific operational tasks, establishing them as a robust and flexible tool for quantum resource distillation.

Resource State Distillation via Stabilizer Channels

TL;DR

This work introduces a unified framework for stabilizer-based resource distillation in systems of prime local dimension by formulating stabilizer routines as quantum channels and deriving closed-form expressions for their output, and introduces several protocols that effectively tailor stabilizer channels to specific operational tasks.

Abstract

Quantum technologies rely on high-quality resource states, such as maximally entangled or private states, which are indispensable for quantum communication and cryptography. In practice, however, these states are inevitably degraded by noise. Distillation protocols aim to recover high-resource states from multiple imperfect copies, and while stabilizer-based methods have demonstrated high performance in entanglement purification, they have yet to be established for broader tasks such as secret-key distillation. This work introduces a unified framework for stabilizer-based resource distillation in systems of prime local dimension. By formulating stabilizer routines as quantum channels and deriving closed-form expressions for their output, we enable the application of stabilizer operations to general input states and diverse distillation objectives. We identify key invariances in resource measures, such as coherent and private information, and demonstrate how they can be leveraged to significantly reduce the numerical complexity of channel optimization. To illustrate the framework's versatility, we introduce several protocols: gF-IMAX for general fidelity optimization, and (S)CI-IMAX and (S)PI-IMAX for optimizing (smooth) coherent and private information in both asymptotic and one-shot regimes. Our numerical results confirm that these protocols effectively tailor stabilizer channels to specific operational tasks, establishing them as a robust and flexible tool for quantum resource distillation.
Paper Structure (23 sections, 6 theorems, 51 equations, 3 figures)

This paper contains 23 sections, 6 theorems, 51 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Setting and Notation
  3. Results
  4. The stabilizer routine as a quantum channel
  5. The stabilizer distillation routine
  6. The bipartite stabilizer channel
  7. The tripartite stabilizer channel
  8. Principle and relevant objects for stabilizer-based resource distillation
  9. Relevant states and measures for entanglement distillation
  10. Relevant states and measures for secret-key distillation
  11. Simplifications for local isometric and unitary invariance
  12. Local unitary equivalence of encodings
  13. Local isometric equivalence of purifications
  14. Defining stabilizer-based resource distillation protocols
  15. Entanglement distillation
  16. ...and 8 more sections

Key Result

Lemma 1

Let $U$ be an encoding of a stabilizer $S$ with generating elements $\lbrace g_1,\dots,g_p\rbrace$. For each codespace $\mathcal{Q}(x) \subset \mathcal{H}_A^{\otimes N}$ and for each $W(e) \in \mathcal{E}_N$ with $(\langle g_1, e \rangle,\dots,\langle g_p, e \rangle) = (s_1,\dots,s_p)$, there exist

Figures (3)

  • Figure 1: Hilbert space $\mathcal{H}^{\otimes N}_{AB}$ of $N$-copies of a bipartite system $\mathcal{H}_{AB} = \mathcal{H}_A \otimes \mathcal{H}_B$.
  • Figure 2: Protocol comparison of mean distillation efficiencies to reach a target fidelity of $0.9$ depending on the fidelity of pure states. Each data point represents the mean efficiency for a group of $1000$ bipartite entangled input states with fidelities rounded to one digit for local dimensions $d=2$ (left) and $d=3$ (right). The efficiency is defined as the inverse of the required number of input states to produce one output state with fidelity larger than the target fidelity. To generate these results, we used the methods implemented in Ref.popp_belldiagonalqudits_2023.
  • Figure 3: Resource state distillation based on three optimization quantities for $d=2$. Starting with the isotropic state $\rho(p) = p~|\Omega_{0,0}\rangle\langle \Omega_{0,0}| + (1-p)~\pi_{mm}$ for $p=0.7$ and where $\pi_{mm}$ is the maximally mixed state, different optimization quantities are iteratively increased. For the evaluation of the smooth coherent information $(-H^\varepsilon_{max})$ (used for SCI-IMAX) we use the semidefinite program (SDP) presented in nuradha_fidelity-based_2024. For the smooth private information (used for SPI-IMAX), $I^\varepsilon_H$ can also be evaluated by an SDP khatri_principles_2024. The smooth max-mutual information $I^{\epsilon}_{max}$ is approximated by the SDP-based algorithm given in Ref.popp_computation_2026. The smoothing parameters $\varepsilon, \epsilon$ are both set to $0.1$. The protocols are iterated until the optimization quantity is larger than $90 \%$ of the maximally achievable value. For all protocols we use the implementation of Ref.popp_belldiagonalqudits_2023.

Theorems & Definitions (24)

  • Definition 1: Weyl(-Heisenberg) operators, Weyl(-Heisenberg) errors
  • Definition 2: Bell states
  • Definition 3: Stabilizer group, generating operators, generating elements
  • Definition 4: Error coset
  • Definition 5: Codeword
  • Definition 6: Encoding
  • Definition 7: Stabilizer measurement
  • Definition 8: Symplectic product decomposition
  • Lemma 1: popp_novel_2025
  • Proposition 2: popp_novel_2025
  • ...and 14 more