Commutation of Lindblad Operators

[Muthumanimaran]

Homework Statement

Let operator $$\mathcal{L}_{AD}(\rho)$$ and $$\mathcal{L}_{PD}(\rho)$$ is defined as

$$\mathcal{L}_{AD}(\rho)=2a\rho{a}^{\dagger}-a^{\dagger}a\rho-\rho{a^{\dagger}}a$$

and $$\mathcal{L}_{PD}(\rho)=2a^{\dagger}a\rho{a^{\dagger}}a-(a^{\dagger}a)^2\rho-\rho(a^{\dagger}a)^2$$

As we could see it is easy to show that $$\mathcal{L}_{AD}(\rho)$$ and $$\mathcal{L}_{PD}(\rho)$$ commute with each other. But if I do explicit calculation to show they commute with each other,
i.e, $$[\mathcal{L}_{AD}(\rho),\mathcal{L}_{PD}(\rho)]=0$$

Homework Equations

The Attempt at a Solution

on explicit calculation I got,
$$ [a{\rho}{a}^{\dagger} , a^{\dagger}a{\rho}a^{\dagger}a] - 2[a^{\dagger}a{\rho} , a^{\dagger}a{\rho}a^{\dagger}a] - 2[\rho{a}^{\dagger}a , a^{dagger}a{\rho}a^{dagger}a] - 2[a{\rho}a^{\dagger} , (a^{\dagger}a)^2{\rho}] + [a^{\dagger}a{\rho} , (a^{\dagger}a)^2{\rho}] + [{\rho}{a}^{\dagger}a , (a^{\dagger}a)^2{\rho}] - 2[a{\rho}a^{\dagger , {\rho}(a^{\dagger}a)^2}] + [a^{\dagger}a{\rho} , {\rho}(a^{\dagger}a)^2] + [{\rho}{a^{\dagger}a , {\rho}(a^{\dagger}a)^2] $$

But from here I don't know how to cancel the terms in these commutators, I don't how to proceed further.

 

[mark726]

I would first check if the expressions for $$\mathcal{L}_{AD}(\rho)$$ and $$\mathcal{L}_{PD}(\rho)$$ are correct. I would also check if the commutation relation is indeed supposed to be zero, as it seems unusual for two operators to commute with each other.

Assuming that the expressions and commutation relation are correct, I would then try to simplify the commutator by using known properties of operators and commutators. For example, I can use the fact that $$[AB,C]=A[B,C]+[A,C]B$$ to split up the commutator into smaller parts that might be easier to evaluate.

I might also try using the definition of commutator, $$[A,B]=AB-BA$$, to expand the commutators in terms of the operators involved. This could help identify terms that can be cancelled out.

If these methods do not yield a solution, I would consult with other colleagues or look for similar problems in textbooks or online resources to see if there are any common techniques or tricks that can be used to simplify the commutator. I might also try using a computer program or software to assist with the calculations.

Ultimately, as a scientist, I would approach this problem with a systematic and logical mindset, using known principles and techniques to try and simplify the commutator and determine if it is indeed equal to zero. If I am unable to solve it, I would seek help from others and continue to explore different methods until a solution is found.