  Maximally entangled Bell states are of crucial importance for entanglement
based methods in quantum information science. Typically, a standard
construction of a complete orthonormal Bell-basis by Weyl-Heisenberg operators
is considered. We show that the group structure of these operators has strong
implication on error correction schemes and on the entanglement structure
within Bell-diagonal states. In particular, it implies a equivalence between a
Pauli channel and a twirl channel. Interestingly, other complete orthonormal
Bell-bases do break the equivalence and lead to a completely different
entanglement structure, for instance in the share of PPT-entangled states. In
detail, we find that the standard Bell basis has the highest observed share on
PPT-states and PPT-entangled states compared to other Bell bases. In summary,
our findings show that the standard Bell basis construction exploits a very
special structure with strong implications to quantum information theoretic
protocols if a deviation is considered.
