Wednesday, 8 August 2018

Interaction between joint qubit quantum system



Consider the following interaction Hamiltonian H=μσxσx=μ(|0110|+|1001|)

acting on the joint states of qubits ρprimρaux 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 eiπσxσx2[|00||00|]eiπσxσx2

where the state of the primary is eπσx2|0=(0ii0)(10)=(0i)



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 eitH. To evolve a density matrix you have to compute eitHρeitH.



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