We investigate the use of coalgebra to represent quantum systems, thus providing a basis for the use of coalgebraic methods in quantum information and computation. Coalgebras allow the dynamics of repeated measurement to be captured, and provide mathematical tools such as final coalgebras, bisimulation and coalgebraic logic. However, this application raises new challenges for coalgebra: how to accommodate the contravariance which arises naturally as we represent both the states and the properties of physical systems; and how to represent the symmetries of these systems, which account e.g. for their unitary dynamics. This motivates us to introduce a novel fibrational structure for coalgebra, and also to make new connections between coalgebras and Chu spaces.