I'm wondering whether or not we can construct quantum deletion error correcting codes. The quantum deletion error is defined by the partial trace. If we can, could anyone give an example?
It is well known that quantum error correction can also correct erasure errors. In fact, a code which can correct $k$ general errors (in arbitrary locations) can correct erasure errors in $2k$ locations. Thus, any quantum error correction code serves as an example.
To learn more about that, you could e.g. consult Preskill's lecture notes or the book by Nielsen and Chuang.