  In order to investigate the quantum phase transition in the one-dimensional
quantum compass model, we numerically calculate non-local string correlations,
entanglement entropy, and fidelity per lattice site by using the infinite
matrix product state representation with the infinite time evolving block
decimation method. In the whole range of the interaction parameters, we find
that the four distinct string orders characterize the four different Haldane
phases and the topological quantum phase transition occurs between the Haldane
phases. The critical exponents of the string order parameters {\beta}=1/8 and
the cental charges c=1/2 at the critical points show that the topological phase
transitions between the phases belong to an Ising type of universality classes.
In addition to the string order parameters, the singularities of the second
derivative of the ground state energies per site, the continuous and singular
behaviors of the von Neumann entropy, and the pinch points of the fidelity per
lattice site manifest that the phase transitions between the phases are of the
second-order, in contrast to the first-order transition suggested in pervious
studies.
