  We propose practical and efficient protocols for verifying bipartite pure
states for any finite dimension, which can also be applied to fidelity
estimation. Our protocols are based on adaptive local projective measurements
with either one-way or two-way communications, which are very easy to implement
in practice. They can extract the key information much more efficiently than
known protocols based on tomography or direct fidelity estimation, and their
efficiencies are comparable to the best protocols based on entangling
measurements. These protocols highlight the significance of mutually unbiased
bases (MUBs) and complex projective 2-designs in quantum state verification and
fidelity estimation. Moreover, our protocols can be applied to the adversarial
scenario, in which states are controlled by a malicious adversary. In this
case, surprisingly, one of our protocols based on local measurements is even
optimal among protocols without locality restriction.
