Covariant Quantum Error correction: an approximate Eastin-Knill theorem, reference frame encoding, and continuous symmetries in AdS/CFT

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In the usual paradigm of quantum error correction, the information to be protected can be encoded in a system of abstract qubits or modes.  But how does this work for physical information, which cannot be described in this way?  Just as direction information cannot be conveyed using a sequence of words if the parties involved do not share a reference frame, physical quantum information cannot be conveyed using a sequence of qubits or modes without a shared reference frame.  Covariant quantum error correction is a procedure for protecting such physical information against noise in such a way that the encoding and decoding operations transform covariantly with respect to an external symmetry group. In this talk, we'll study covariant QEC, and we will see that there do not exist finite dimensional quantum codes that are covariant with respect to continuous symmetries.  Conversely, we'll see that there do exist finite codes for finite groups, and continuous variable (CV) codes for continuous groups.  This leads to a CV method of circumventing the Eastin-Knill theorem.  By relaxing our requirements to allow for only approximate error correction and covariance, we'll find a fundamental tension between a code's ability to approximately correct errors and covariance with respect to a continuous symmetry.  In this way, we'll arrive at an approximate version of the Eastin-Knill theorem, and we'll end by learning what covariant QEC tells us about continuous symmetries in AdS/CFT, among other applications.