Description
Protection
Given some algebra \(\mathcal{A}\), \(\mathcal{A}\) is \(\epsilon\)-correctable under noise channel \(\mathcal{N}\) if there exists some quantum channel \(\mathcal{R}\) such that \begin{align} ||(\mathcal{R}\circ\mathcal{N})-P_{\mathcal{A}}||_{\diamond}\leq\epsilon~, \tag*{(1)}\end{align} where \(P_{\mathcal{A}}\) is the projector onto algebra \(A\) and we use the diamond norm \(\diamond\) [3].
Let the minimal error for some algebra \(\mathcal{A}\) under noise channel \(\mathcal{N}\) be \begin{align} \epsilon_{\mathcal{A}}=\min_{\mathcal{R}} ||\mathcal{R}\circ\mathcal{N}-P_{\mathcal{A}}||_{\diamond}~. \tag*{(2)}\end{align} Let \(\delta_{\mathcal{A}}=||\mathcal{N}^C-\mathcal{N}^C\circ P_{\mathcal{A}'}||_{\diamond}\) for commutant \(\mathcal{A}'\) of algebra \(\mathcal{A}\) and complementary channel \(\mathcal{N}^C\) of noise channel \(\mathcal{N}\). Then [1], \begin{align} \delta_{\mathcal{A}}^2/4\leq \epsilon_{\mathcal{A}}\leq 2\delta_{\mathcal{A}}^{1/2}~. \tag*{(3)}\end{align}
Parent
Child
Cousin
- Holographic code — Properties of holographic codes are often quantified in the Heisenberg picture, i.e., in terms of operator algebras [4,5].
References
- [1]
- C. Bény, “Conditions for the approximate correction of algebras”, (2009) arXiv:0907.4207
- [2]
- C. Bény and O. Oreshkov, “General Conditions for Approximate Quantum Error Correction and Near-Optimal Recovery Channels”, Physical Review Letters 104, (2010) arXiv:0907.5391 DOI
- [3]
- D. Aharonov, A. Kitaev, and N. Nisan, “Quantum Circuits with Mixed States”, (1998) arXiv:quant-ph/9806029
- [4]
- A. Almheiri, X. Dong, and D. Harlow, “Bulk locality and quantum error correction in AdS/CFT”, Journal of High Energy Physics 2015, (2015) arXiv:1411.7041 DOI
- [5]
- F. Pastawski and J. Preskill, “Code Properties from Holographic Geometries”, Physical Review X 7, (2017) arXiv:1612.00017 DOI
Page edit log
- Milan Tenn (2023-06-26) — most recent
- Milan Tenn (2023-06-22)
Cite as:
“Approximate operator-algebra error-correcting code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/approximate_oaecc
Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/approximate_oaecc.yml.