Rényi and Shannon mutual information in critical and decohered critical system

We investigate a critical many-body system by introducing a Rényi generalized mutual information, connecting between Rényi mutual information and Rényi Shannon mutual information. This Rényi generalized mutual information can offer more experimentally accessible alternative than the conventional entanglement entropy. As a critical many-body state, we focus on the critical transverse-field Ising model (TFIM) described by the Ising conformal field theory (CFT). We show that even if we modify the non-selective projective measurement assumed in Rényi Shannon mutual information by replacing the measurement into decoherence by environment, the Rényi generalized Shannon mutual information maintains the CFT properties such as subsystem CFT scaling law and its central charge observed through both the conventional Rényi Shannon mutual information and Rényi mutual information. Furthermore, we apply a local decoherence to the critical ground state of the TFIM and numerically observe the Rényi generalized mutual information by changing the parameter controlling environment effect (corresponding to the strength of measurement) in the Rényi generalized mutual information and the strength of the decoherence to which the entire system subjects. We find that Rényi-$2$ type central charge connected to the central charge is fairly robust, indicating the strong robustness of the Ising CFT properties against local decoherence by environment.