  The matrix product state (MPS) is utilized to study the ground state
properties and quantum phase transitions (QPTs) of the one-dimensional quantum
compass model (QCM). The MPS wavefunctions are argued to be very efficient
descriptions of QCM ground states, and are numerically determined by imaginary
time projections. The ground state energy, correlations, quantum entanglement
and its spectrum, local and nonlocal order parameters, etc., are calculated and
studied in details. It is revealed that the bipartite and block entanglement
entropies, as well as the nearest neighbor correlation functions can be used to
detect the second-order QPTs, but not the first-order ones, while fidelity
detections can recognize both. The entanglement spectrum is extracted from the
MPS wavefunction, and found to be doubly degenerate in disordered phases of
QCM, where non-local string order parameters exist. Moreover, with linearized
tensor renormalization group method, the specific heat curves are evaluated and
their low temperature behaviors are investigated.
