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We investigate the static and the dynamical behavior of localizableentanglement and its lower bounds on nontrivial loops of topological quantumcodes with parallel magnetic field. Exploiting the connection between thestabilizer states and graph states in the absence of the parallel field andexternal noise, we identify a specific measurement basis, referred to as thecanonical measurement basis, that optimizes localizable entanglement whenmeasurement is restricted to single-qubit Pauli measurements only, therebyproviding a lower bound. We also propose an approximation of the lower boundthat can be computed for larger systems according to the computational resourcein hand. Additionally, we compute a lower bound of the localizable entanglementthat can be computed by determining the expectation value of an appropriatelydesigned witness operator. We study the behavior of these lower bounds in thevicinity of the topological to nontopological quantum phase transition of thesystem, and perform a finite-size scaling analysis. We also investigate thedynamical features of these lower bounds when the system is subjected toMarkovian or non-Markovian single-qubit dephasing noise. We find that in thecase of the non-Markovian dephasing noise, at large time, the canonicalmeasurement-based lower bound oscillates with a larger amplitude when theinitial state of the system undergoing dephasing dynamics is chosen from thenontopological phase, compared to the same for an initial state from thetopological phase. These features can be utilized to distinguish thetopological phase of the system from the nontopological phase in the presenceof dephasing noise.

Distinguishing Phases via Non-Markovian Dynamics of Entanglement in Topological Quantum Codes under Parallel Magnetic Field