4. Consider a quantum system Q described by a Hilbert space H. (a) Suppose we are given a subspace Ho of H and a linear map from kets in Ho to others in h. That is, Vo)), in a linear way. This map preserves inner products of vectors in Ho, so that (o)=(o). Show that there exists a unitary operator Von H such that |)=V) for all ) E Ho. (Hint: Use the fact that any orthonormal set of vectors may be extended to a complete basis.) This allows us to build up a "unitary-like" map on a subspace into a full unitary operator, and it can be used for the following: (b) Start with a map & on density operators for Q with Kraus representation () - AAk=1. k Append a system E in state [0) and consider the following linear map on kets: 10)(A)) |ek), k where {e}} is a fixed orthonormal basis of H(E). Show that there exists a unitary U(QE) such that E(p) TrE)(UQ)(p10) (0) (U(QE))) In other words, every & on Q that has a Kraus representation can be realized as the result of unitary evolution on a larger system QE. 1