2021/08/31 by Sam Cree, Jonathan Sorce · 5 citations
Computer Science · Mathematics · Physics and Astronomy · #Dimension (graph theory) #Entropy (arrow of time) #Geometry #Hermitian matrix #Hilbert space #Inverse #Mathematical analysis #Mathematics #Numerical methods in inverse problems #Physics #Pure mathematics #Quadratic equation #Quantum #Quantum Information and Cryptography #Quantum Mechanics and Non-Hermitian Physics #Quantum discord #Quantum entanglement #Quantum mechanics #Quantum relative entropy #Upper and lower bounds #math-ph #math.MP #quant-ph
paper · pdf · doi:10.1007/s00220-022-04357-2
published in Communications in Mathematical Physics 392(3), 907-919 (Springer Science+Business Media) · 10 pages; most recent version published in Commun. Math. Phys. (2022)
arxiv created 2022/03/18 · openalex publication_date 2022/03/18 · arxiv updated 2022/03/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We derive a new bound on the effectiveness of the Petz map as a universal recovery channel in approximate quantum error correction using the second sandwiched Rényi relative entropy D2. For large Hilbert spaces, our bound implies that the Petz map performs quantum error correction with order-ε accuracy whenever the data processing inequality for D2 is saturated up to terms of order ε2 times the inverse Hilbert space dimension. Conceptually, our result is obtained by extending arXiv:2011.03473, in which we studied exact saturation of the data processing inequality using differential geometry, to the case of approximate saturation. Important roles are played by (i) the fact that the exponential of the second sandwiched Rényi relative entropy is quadratic in its first argument, and (ii) the observation that the second sandwiched Rényi relative entropy satisfies the data processing inequality even when its first argument is a non-positive Hermitian operator.