  To perform reliable quantum computation, quantum error correction is
indispensable. In certain cases, continuous covariance symmetry of the physical
system can make exact error correction impossible. In this work we study the
approximate error correction and covariance symmetry from the
information-theoretic perspective. For general encoding and noise channels, we
define a quantity named infidelity to characterize the performance of the
approximate quantum error correction and quantify the noncovariance of an
encoding channel with respect to a general Lie group from the asymmetry measure
of the corresponding Choi state. In particular, when the encoding channel is
isometric, we derive a trade-off relation between infidelity and noncovariance.
Furthermore, we calculate the average infidelity and noncovariance measure for
a type of random code.
