In $2\ensuremath{\bigotimes}2$, more than two orthogonal Bell states with a single copy can never be discriminated with certainty if only local operations and classical communication (LOCC) are allowed. We show here that more than
d numbers of pairwise orthogonal maximally entangled states in $d\ensuremath{\bigotimes}d$, which are in canonical form, used by Bennett et al. [Phys. Rev. Lett. 70, 1895 (1993)], can never be discriminated with certainty by LOCC, when single copies of the states are provided. Interestingly we show here that all orthogonal maximally entangled states, which are in canonical form, can be discriminated with certainty by LOCC if and only if two copies of each of the states are provided. We provide here a conjecture regarding the highly nontrivial problem of local distinguishability of any
d or fewer numbers of pairwise orthogonal maximally entangled states in $d\ensuremath{\bigotimes}d$ (in the single copy case).
We prove that the entangling capacity of a two-qubit unitary operator without local ancillas, both with and without the restriction to initial product states, as quantified by the maximum attainable concurrence, is directly related to the distinguishability of a closely related pair of two-qubit unitary operators. These operators are the original operator transformed into its canonical form and the adjoint of this canonical form. The distinguishability of these operators is quantified by the minimum overlap of the output states over all possible input probe states. The entangling capacity of the original unitary operator is therefore directly related to the degree of non-Hermiticity of its canonical form, as quantified in an operationally satisfactory manner in terms of the extent to which it can be distinguished, by measurement, from its adjoint. Furthermore, the maximum entropy of entanglement, again without local ancillas, that a given two-qubit unitary operator can generate is found to be closely related to the classical capacities of certain quantum channels.
We find that the mixed maximally entangled states exist and prove that the form of the mixed maximally entangled states is unique in terms of the entanglement of formation. Moreover, even if the entanglement is quantified by other entanglement measures, this conclusion is still proven right. This result is a supplementary to the generally accepted fact that all maximally entangled states are pure. These states possess important properties of the pure maximally entangled states, for example, these states can be used as a resource for faithful teleportation and they can be distinguished perfectly by local operations and classical communication.