  In a measurement-device-independent or quantum-refereed protocol, a referee
can verify whether two parties share entanglement or Einstein-Podolsky-Rosen
(EPR) steering without the need to trust either of the parties or their
devices. The need for trusting a party is substituted by a quantum channel
between the referee and that party, through which the referee encodes the
measurements to be performed on that party's subsystem in a set of
nonorthogonal quantum states. In this Letter, an EPR-steering inequality is
adapted as a quantum-refereed EPR-steering witness, and the trust-free
experimental verification of higher dimensional quantum steering is reported
via preparing a class of entangled photonic qutrits. Further, with two
measurement settings, we extract $1.106\pm0.023$ bits of private randomness per
every photon pair from our observed data, which surpasses the one-bit limit for
projective measurements performed on qubit systems. Our results advance
research on quantum information processing tasks beyond qubits.
