  Two protocols are proposed for two closely linked but different variants of
remote implementation of quantum operators of specific forms. The first
protocol is designed for the remote implementation of the single qubit hidden
quantum operator, whereas the second one is designed for the remote
implementation of the partially unknown single qubit quantum operator. In both
cases two-qubit maximally entangled state, which is entangled in the spatial
degree of freedom is used. The quantum resources used here are optimal and easy
to realize and maintain in comparison to the multi-partite or multi-mode
entangled states used in earlier works. The proposed protocols are also
generalized to their controlled, bidirectional, cyclic, controlled cyclic, and
controlled bidirectional versions and it is shown that either Bell state alone
or products of Bell states will be sufficient to perform these tasks with some
additional classical communications in the controlled cases only. This is in
sharp contrast to the earlier proposals that require large entangled states. In
addition, it's noted that remote implementation of hidden or partially unknown
operators involving multiple controllers and/or multiple players who jointly
apply the desired operator(s) would require quantum channels more complex than
the Bell states and their products. Explicit forms of such quantum channels are
also provided.
