Abstract

The vanishing of the conditional mutual information (CMI) characterizes quantum Markov chains (QMCs) in the standard setting based on the Umegaki relative entropy. By replacing the Umegaki entropy with the Belavkin-Staszewski (BS) relative entropy, one can define modified versions of CMIs (called BS-conditional mutual informations) and refer to quantum states with vanishing BS-CMIs as Belavkin-Staszewski quantum Markov chains (BS-QMCs). In this talk, I will present a framework that establishes a precise relationship between conventional quantum Markov chains and their BS counterparts. This framework yields a recovery map for the BS-relative entropy in the spirtit of the Petz recovery map. Additionally, we find a structural decomposition theorem for BS-quantum Markov chains and analyze the behavior of states with approximately vanishing BS-CMI. In addition, we show that vanishing of the BS-CMIs does not imply that the marginal in the system AC is separable. Thanks to this property we will be able to generate BS-QMCs which are QMCs . As an application, we demonstrate that for the Gibbs state of a quantum spin chain with finite-range, translation-invariant interactions at positive temperature, the CMI of some special states decays superexponentially fast with the size of the intermediate region.