R\'{e}nyi 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 R\'{e}nyi conditional mutual information (CMI). The CMI can measure mixed state ``gap'', estimated by the exponential decay rate of the second R\'{e}nyi CMI under a tripartition of system, which provides the second R\'{e}nyi version of the Markov length. We introduce an efficient numerical scheme for the calculation of the second R\'{e}nyi 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 R\'{e}nyi 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 R\'{e}nyi CMI exhibits specific behavior for the transition to strong-to-weak spontaneous symmetry breaking mixed phase.