Approximate Petz recovery from the geometry of density operators

Abstract

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 ${\tilde{D}}_2$. For large Hilbert spaces, our bound implies that the Petz map performs quantum error correction with order-$\epsilon$ accuracy whenever the data processing inequality for ${\tilde{D}}_2$ is saturated up to terms of order ${\epsilon }^2$ 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.

arXiv comment10 pages; most recent version published in Commun. Math. Phys. (2022)

References in corpus (13)

• On quantum Renyi entropies: a new generalization and some properties
• Strong converse for the classical capacity of entanglement-breaking and Hadamard channels via a sandwiched Renyi relative entropy
• Reversing quantum dynamics with near-optimal quantum and classical fidelity
• Sandwiched Rényi Divergence Satisfies Data Processing Inequality
• Monotonicity of a relative Rényi entropy
• Quantum f-divergences and error correction
• Universal recovery maps and approximate sufficiency of quantum relative entropy
• Generating random quantum channels
• $\alpha$-Logarithmic negativity
• Gauss' Law and String-Localized Quantum Field Theory
• Data processing for the sandwiched Rényi divergence: a condition for equality
• Recoverability for optimized quantum $f$-divergences
• Geometric conditions for saturating the data processing inequality

Cited by in corpus (2)

• Optimality Condition for the Petz Map
• Geometric conditions for saturating the data processing inequality