Rényi Markov length in one-dimensional non-trivial mixed state phases and mixed state phase transitions

Discovering and classifying non-trivial mixed states and mixed state phase transitions are some of the most important current issues in condensed matter and quantum information. In this study, we investigate some non-trivial mixed states and phase transitions between them by using the second Rényi conditional mutual information (CMI). The CMI can measure mixed state gap'', estimated by the exponential decay rate of the second Rényi CMI under a tripartition of system, which provides the second Rényi version of the Markov length. We introduce an efficient numerical scheme for the calculation of the second Rényi CMI based on the doubled Hilbert space formalism, and study the classification of non-trivial mixed states and the emergence of mixed state phase transitions for (i) the cluster model under odd-site local $Z$ decoherence and (ii) transverse field Ising model under both $ZZ$ and $X$ decoherence. The second Rényi CMI is a powerful measure to study non-trivial mixed gapped" quantum matters and mixed phase transitions. In addition to this, the present study shows that the second Rényi CMI exhibits specific behavior for the transition to strong-to-weak spontaneous symmetry breaking mixed phase.