???????Quantum Computing

For Problem 4 below, we Let A and B be two quantum systems with finite-dimensional state spaces H_A and H_B, respectively. Recall that, given Hilbert spaces H_C and HD and linear maps T : H_A ? H_C and S : H_B ? H_D, the tensor product T ? S is the linear map from H_A ? H_B into H_C ? H_D defined by: (T ? S)((|?? ? |??) = T |?? ? S|??. Finally, recall that an operator on H_A is a linear map from H_Ainto H_Aitself.

Problem 4. Let Jo), ) E HA. For each of the following statements, either give a proof or a counterexample: (a) The interior product () belongs to R. (b) The interior product (@) belongs to R. (c) If () = 1, then (1)) (@))= )(). In this case, what el- ementary geometric operation does the operator ()(| repre- sent