Error Correction of Quantum Reference Frame Information

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.
Comment: 15 pages, 2 figures