On errors generated by unitary dynamics of bipartite quantum systems

Given a quantum channel it is possible to define the non-commutative operator graph whose properties determine a possibility of error-free transmission of information via this channel. The corresponding graph has a straight definition through Kraus operators determining quantum errors. We are discussing the opposite problem of a proper definition of errors that some graph corresponds to. Taking into account that any graph is generated by some POVM we give a solution to such a problem by means of the Naimark dilatation theorem. Using our approach we construct errors corresponding to the graphs generated by unitary dynamics of bipartite quantum systems. The cases of POVMs on the circle group ${\mathbb Z}_n$ and the additive group $\mathbb R$ are discussed. As an example we construct the graph corresponding to the errors generated by dynamics of two mode quantum oscillator.