  Using symmetry properties, we determine the mixing pattern of a class of
nonlocal quark bilinear operators containing a straight Wilson line along a
spatial direction. We confirm the previous study that mixing among the lowest
dimensional operators, which have mass dimension equals three, can occur if
chiral symmetry is broken in the lattice action. For higher dimensional
operators, we find that the dimension three operators will always mix with
dimension four operators even if chiral symmetry is preserved. Also, the number
of dimension four operators involved in the mixing is large hence it is
impractical to remove the mixing by the improvement procedure. Our result is
important to determining the Bjorken-$x$ dependence parton distribution
functions using the quasi-distribution method on a Euclidean lattice. The
requirement of using large hadron momentum in this approach makes the control
of errors from dimension four operators even more important.
