Communications in Mathematical Physics

, Volume 340, Issue 2, pp 575–611 | Cite as

Quantum Conditional Mutual Information and Approximate Markov Chains

  • Omar FawziEmail author
  • Renato Renner
Open Access


A state on a tripartite quantum system \({A \otimes B \otimes C}\) forms a Markov chain if it can be reconstructed from its marginal on \({A \otimes B}\) by a quantum operation from B to \({B \otimes C}\). We show that the quantum conditional mutual information I(A : C|B) of an arbitrary state is an upper bound on its distance to the closest reconstructed state. It thus quantifies how well the Markov chain property is approximated.


Density Operator Relative Entropy Trace Distance Conditional Mutual Information Strong Subadditivity 
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Authors and Affiliations

  1. 1.Institute for Theoretical Physics, ETH ZurichZurichSwitzerland
  2. 2.LIP, UMR 5668 ENS Lyon-CNRS-UCBL-INRIAUniversite de Lyon, ENS de LyonLyonFrance

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