Quantum Conditional Mutual Information, Reconstructed States, and State Redistribution

We give two strengthenings of an inequality for the quantum conditional mutual information of a tripartite quantum state recently proved by Fawzi and Renner, connecting it with the ability to reconstruct the state from its bipartite reductions. Namely, we show that the conditional mutual information is an upper bound on the regularized relative entropy distance between the quantum state and its reconstructed version. It is also an upper bound for the measured relative entropy distance of the state to its reconstructed version. The main ingredient of the proof is the fact that the conditional mutual information is the optimal quantum communication rate in the task of state redistribution.

Figure 1

Counterexamples for the case of equality in Eq. (9): Conditional mutual information against the reconstructed relative entropy using the transpose channel. The sample consists of 10 000 random pure states of dimension $2\times 2\times 2$.