Discovering and classifying nontrivial mixed states are the most important current issues in condensed matter and quantum information. In this study, we characterize some nontrivial mixed states by numerically observing 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. In this work, we show that the second Rényi CMI is a powerful measure to characterize and detect nontrivial mixed gapped quantum matters. Concretely, we study two typical systems: (i) the cluster model under odd-site local $Z$ decoherence and (ii) transverse field Ising model under decoherence of $ZZ+X$ type. There, we numerically observe how the second Rényi CMI and its Markov length behave along the change of mixed states. In addition, in the present study, we provide numerical evidence that the second Rényi CMI changes across the crossover between the strong and weak spontaneous symmetry-breaking mixed regimes.

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• Open quantum systems
• Open quantum systems & decoherence
• Quantum correlations in quantum information
• Quantum phase transitions
• Spontaneous symmetry breaking
• Symmetry protected topological states