Quantum Error Correction and Young Tableaux

A quantum channel is a completely positive trace preserving map which acts on the set of operators for the Hilbert space associated with a given quantum system. Analysis of such channels is central to quantum computing and quantum information theory. We present and investigate a new class of quantum channels that includes the class of collective rotation channels as a special case. We use the phrase ‘universal collective rotation channels’ for this class. The fixed point set and noise commutant coincide for a channel in this class. Computing the precise structure of this operator algebra is a core problem in a particular noiseless subsystem method of quantum error correction. We apply classical representation theory of the symmetric group via Young tableaux and give a computationally amenable method for explicitly finding this structure for the class of universal collective rotation channels.