Consider the following interaction Hamiltonian H=ℏμσx⊗σx=ℏμ(|01⟩⟨10|+|10⟩⟨01|)
acting on the joint states of qubits ρprim⊗ρaux for t=π2μ. It is stated that if the primary and auxiliary systems (respectively ρprim and ρaux) are in the state |0⟩ then the interaction doesn't change the primary but if the primary is in state |1⟩ and auxiliary in state |0⟩ then the primary flips to |0⟩.
For the first case my revised working is as follows: We have e−iπσx⊗σx2[|0⟩⟨0|⊗|0⟩⟨0|]eiπσx⊗σx2
where the state of the primary is e−πσx2|0⟩=(0−i−i0)(10)=(0−i)
Answer
Evolving a state ρ according to an Hamiltonian H does not work that way: Hρ is not the evolved state (nor, in general, even a state at all).
The evolution with the Hamiltonian H for time t is described by the unitary operator e−itH. To evolve a density matrix you have to compute e−itHρeitH.
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