  The existence of quantum error correcting codes is one of the most
counterintuitive and potentially technologically important discoveries of
quantum information theory. However, standard error correction refers to
abstract quantum information, i.e., information that is independent of the
physical incarnation of the systems used for storing the information. There
are, however, other forms of information that are physical - one of the most
ubiquitous being reference frame information. Here we analyze the problem of
error correcting physical information. The basic question we seek to answer is
whether or not such error correction is possible and, if so, what limitations
govern the process. The main challenge is that the systems used for
transmitting physical information, in addition to any actions applied to them,
must necessarily obey these limitations. Encoding and decoding operations that
obey a restrictive set of limitations need not exist a priori. We focus on the
case of erasure errors, and we first show that the problem is equivalent to
quantum error correction using group-covariant encodings. We prove a no-go
theorem showing that that no finite dimensional, group-covariant quantum codes
exist for Lie groups with an infinitesimal generator (e.g., U(1), SU(2), and
SO(3)). We then explain how one can circumvent this no-go theorem using
infinite dimensional codes, and we give an explicit example of a covariant
quantum error correcting code using continuous variables for the group U(1).
Finally, we demonstrate that all finite groups have finite dimensional codes,
giving both an explicit construction and a randomized approximate construction
with exponentially better parameters.
