Quantum geometric bounds at finite temperature for one-dimensional chiral systems

The geometry and topology of quantum states are intimately related at zero temperature through exact bounds that constrain geometric quantities from below by topological invariants. At finite-temperature, however, the analogous relations remain unclear. Here we establish rigorous geometric lower bounds for one-dimensional (1D) chiral-symmetric systems at finite temperature within the Uhlmann's framework for mixed states. We show that the Bures length is bounded by a continuous geometric phase angle. We further derive a temperature-dependent bound that interpolates between the zero-temperature limit and a trivial high-temperature regime. Our results are verified analytically and numerically using the Su-Schrieffer-Heeger (SSH) model and the spinless Kitaev chain model. Finally, we discuss potential ways to detect the geometry of the density matrix in quantum circuits, with the quantum imaginary time evolution (QITE) method.