  We show that every three-dimensional subspace of qutrit-qudit complex or real
systems has a distinguishable basis under one-way local operations and
classical communication (LOCC). In particular this solves an open problem
proposed in [J. Phys. A, 40, 7937, 2007]. We construct a three-dimensional
space whose locally distinguishable basis is unique and apply the uniqueness
property to the task of state transformation. We also construct a
three-dimensional locally distinguishable multipartite space assisted with
entanglement. On the other hand, we show that four-dimensional
indistinguishable bipartite subspaces under one-way LOCC exist. Further, we
show that the environment-assisted classical capacity of every channel with a
three-dimensional environment is at least $\log_2 3$, and the
environment-assisting classical capacity of any qutrit channel is $\log_2 3$.
We also show that every two-qutrit state can be converted into a generalized
classical state near the quantum-classical boundary by an entanglement-breaking
channel.
