On non-commutative operator graphs for two-mode quantum oscillator.

It is possible to formulate in terms of non-commutative operator graphs a sufficient and necessary condition of the existence of zero-error coding for a given quantum system. In this framework, an approach in which an operator graph is linearly generated by a covariant resolution of identity is very fruitful. We will discuss the example of a non-commutative operator graph generated by a resolution of identity covariant with respect to the unitary group giving a solution to the Schroedinger equation describing the dynamics of a two-mode quantum oscillator. For this operator graph, it is possible to show the existence of zero-error coding.