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

The existence of distinguishable bases in three-dimensional subspaces of  qutrit-qudit systems under one-way local operations and classical  communication