Abstract
A tripartite state \(\rho _{ABC}\) forms a Markov chain if there exists a recovery map \(\mathcal {R}_{B \rightarrow BC}\) acting only on the B-part that perfectly reconstructs \(\rho _{ABC}\) from \(\rho _{AB}\). To achieve an approximate reconstruction, it suffices that the conditional mutual information \(I(A:C|B)_{\rho }\) is small, as shown recently. Here we ask what conditions are necessary for approximate state reconstruction. This is answered by a lower bound on the relative entropy between \(\rho _{ABC}\) and the recovered state \(\mathcal {R}_{B\rightarrow BC}(\rho _{AB})\). The bound consists of the conditional mutual information and an entropic correction term that quantifies the disturbance of the B-part by the recovery map.
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Acknowledgements
We thank Mario Berta and Marco Tomamichel for advice on Lemma 2.1. We further thank Aram W. Harrow, Franca Nester, Joseph M. Renes, and Volkher B. Scholz for inspiring discussions. We acknowledge support by the Swiss National Science Foundation (SNSF) via the National Centre of Competence in Research “QSIT” and Project No. \(200020\_165843\). We further acknowledge support by the Air Force Office of Scientific Research (AFOSR) via grant FA9550-16-1-0245.
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Communicated by David Pérez-García.
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Sutter, D., Renner, R. Necessary Criterion for Approximate Recoverability. Ann. Henri Poincaré 19, 3007–3029 (2018). https://doi.org/10.1007/s00023-018-0715-1
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DOI: https://doi.org/10.1007/s00023-018-0715-1