Amplitude damping with harmonic oscillators

[haxel]

Hi

I am new to this community, so don't beat me up too hard :).

I have a question about the Hamiltonian when it will simulate the principal system as a harmonic oscillator interacting with the environment which is also an harmonic oscillator (page 291 in "Quantum computation and Quantum information" Nielsen, Chuang).
Is the communtator [a,b] = 0 ? It seems to me that the operators can't affect each other but I am not sure, should there be a tensor between "a" and "b" in the Hamiltonian?
Do you have any source to back up your statements?

Sincerely
Axel

 

[Jackson Tan]

Okay, I don't have Nielson and Chuang, so I'm not certain what you're talking about, but by conventional notation, it seems that $$a$$ and $$b$$ are annihilation operators acting on the system and the environment respectively.

If that is so, then the $$[a,b] = 0$$ because they act on different Hilbert space. $$a$$ is an operator in the system Hilbert space; $$b$$ is an operator in the environment Hilbert space.