  It has been known that all bipartite pure quantum states can be self-tested,
i.e., any such state can be certified completely by initially measuring both
subsystems of this state by proper local quantum measurements and subsequently
verifying that the correlation between the measurement choices and the outcomes
satisfies a specific condition. In such a protocol, a key feature is that the
conclusion can still be reliable even if involved quantum measurements are
untrusted, where quantum nonlocality is crucial and plays a central role, and
this means that each party has to conduct at least two different quantum
measurements to produce a desirable correlation. Here, we prove that when the
underlying Hilbert space dimension is known beforehand, an arbitrary $d\times
d$ bipartite pure state can be certified completely (up to local unitary
transformations) by a certain correlation generated by a single measurement
setting on each party, where each measurement yields only $3d$ outcomes.
Notably, our protocols do not involve any quantum nonlocality. We believe that
our result may provide us a remarkable convenience when certifying bipartite
pure quantum states in quantum labs.
