  We consider the problem of implementing mutually unbiased bases (MUB) for a
polarization qubit with only one wave plate, the minimum number of wave plates.
We show that one wave plate is sufficient to realize two MUB as long as its
phase shift (modulo $360^\circ$) ranges between $45^\circ$ and $315^\circ$. {It
can realize} three MUB (a complete set of MUB for a qubit) if the phase shift
of the wave plate is within $[111.5^\circ, 141.7^\circ]$ or its symmetric range
with respect to 180$^\circ$. The systematic error of the realized MUB using a
third-wave plate (TWP) with $120^\circ$ phase is calculated to be a half of
that using the combination of a quarter-wave plate (QWP) and a half-wave plate
(HWP). As experimental applications, TWPs are used in single-qubit and
two-qubit quantum state tomography experiments and the results show a
systematic error reduction by $50\%$. This technique not only saves one wave
plate but also reduces the systematic error, which can be applied to quantum
state tomography and other applications involving MUB. The proposed TWP may
become a useful instrument in optical experiments, replacing multiple elements
like QWP and HWP.

Quantum Physics