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(Solved): Quantum Computing For Problem 8 below, we Let A and B be two quantum systems with finite-dimensio ...
Quantum Computing
For Problem 8 below, we Let A and B be two quantum systems with finite-dimensional state spaces HA and HB, respectively. Recall that, given Hilbert spaces HC and HD and linear maps T : HA ? HC and S : HB ? HD, the tensor product T ? S is the linear map from HA ? HB into HC ? HD defined by:
(T ? S)((|?? ? |??) = T |?? ? S|??.
Finally, recall that an operator on HA is a linear map from HA into HA itself.
Problem 8. Let and be two operators on , and assume that . Let be a nonzero eigenvalue of , and assume that the corresponding eigenspace of is 1-dimensional. Prove that any nonzero eigenvector of is also an eigenvector of .