Monotonicity of quantum relative entropy and recoverability

Understand

The relative entropy is a principal measure of distinguishability in quantum information theory, with its most important property being that it is non-increasing with respect to noisy quantum operations.

• Here, we establish a remainder term for this inequality that quantifies how well one can recover from a loss of information by employing a rotated Petz recovery map.
• The main approach for proving this refinement is to combine the methods of [Fawzi and Renner, arXiv:1410.0664] with the notion of a relative typical subspace from [Bjelakovic and Siegmund-Schultze, arXiv:quant-ph/0307170].
• Our paper constitutes partial progress towards a remainder term which features just the Petz recovery map (not a rotated Petz map), a conjecture which would have many consequences in quantum information theory.