# What kinds of objects are Liouvillian, Lindbladian, and Davies generator?

I have a rather basic question. I'm starting to read papers such as Chen–Brandao, Chen–Kastoryano–Brandao–Gilyen, and I'm having trouble parsing even what kind of objects a Liouvillian, Lindbladian, and Davies generator are.

It seems that the Davies generator $$D$$ might be a Hamiltonian, which gives rise to some semigroup via exponentiation $$e^{itD}$$ (though I'm not certain that this is correct, and I don't know why this particular semigroup is of interest).

The Liouvillian is given in places as not even self-adjoint, so I don't know what role it plays.

I see "Lindbladian form" mentioned a number of times, so does "a Lindbladian" just refer to an operator, which is a sum of operators of the prescribed form?

Do all of these objects act only on density matrices?

• I think they are all the same actually. Liouvillian and Lindbladian are more common in the physics literature and are used Interchangeably, while the Davies generator is apparently the mathematicians' preferred term. Commented Apr 19, 2023 at 14:14
• @NikitaNemkov What is confusing then is that these papers do not use the terms interchangeably; for instance the Liouvillians in [CKBG] are not Hermitian while the Davies generators in [CB] are (in fact they are used as Hamiltonians). So for this reason I do think they are all meaningfully distinct concepts.
– zjs
Commented Apr 19, 2023 at 20:05
• Judging from App. A of [CB], the Davies generator appears to be the dissipative part of a special Liouvillian, that governs a system weakly interacting with a thermal bath. Could you please clarify what you mean by "Davies generators are used as Hamiltonians"? Commented Apr 20, 2023 at 8:45

Much like how the Hamiltonian $$H$$ determines the dynamics of a closed system via $$\dot\rho=-i[H,\rho]$$, the Lindbladian (or Liouvillian$${}^1$$) is an extension of Hamiltonian generators to model the interaction of a system with the environment. More precisely, every Lindbladian $$L$$ consists of the closed system generator $$-i[H,\cdot]$$ and adds another linear map $$-\Gamma$$ onto it. In other words---to connect this to system dynamics---the system evolves according to $$\dot\rho=-i[H,\rho]-\Gamma(\rho)$$ so the solution is no longer $$\rho(t)=e^{-i[H,\cdot]t}\rho_0=e^{-iHt}\rho_0e^{iHt}$$ but now reads $$\rho(t)=e^{tL}(\rho_0)$$ and has no nice closed form anymore. To summarize the basics: a Lindbladian is not a Hamiltonian, it extends the Hamiltonian (more precisely: the Liouville-von Neumann generator $$-i[H,\cdot]$$) to model dissipative effects.

Either way, to ensure that $$e^{tL}$$ maps quantum states to quantum states the map $$\Gamma$$ of course has to satisfy additional conditions. In their famous seminal paper Gorini et al. proved that $$L$$ gives rise to "valid" dynamics if and only if there exist square matrices $$\{V_j\}_j$$ such that $$\Gamma(\rho)=\sum_j\frac12(V_j^*V_j\rho+\rho V_j^*V_j)-V_j\rho V_j^*$$ for all inputs $$\rho$$; to add to the confusion, the $$V_j$$ are sometimes called Lindblad operators (in contrast to the Lindbladian $$L$$ for which the $$V_j$$ are the "building blocks"). Equivalently, there has to exist a completely positive map $$\Phi$$ such that $$\Gamma=\frac12\{\Phi^*({\bf1}),\cdot\}-\Phi$$ (where $$\Phi^*$$ is the adjoint of $$\Phi$$).

Having recapped all of this, a Davies generator is a special type of a Lindbladian, i.e. a Davies generator $$D$$ is of the form $$-i[H,\cdot]-\Gamma$$ and it satisfies some additional conditions. These read as follows:

• $$[H,\cdot]$$ and $$\Gamma$$ have to commute, that is, $$\Gamma(H\rho)-\Gamma(\rho H)=H\Gamma(\rho)-\Gamma(\rho)H$$ for all $$\rho$$
• It satisfies the detailed balance condition, i.e. $${\rm tr}(\Gamma(\rho)e^{-H/T}\omega)={\rm tr}(\rho\Gamma(e^{-H/T}\omega))$$ for all $$\rho,\omega$$. This is basically self-adjointness with respect to a weighted inner product which guarantees that $$\Gamma$$ has only real eigenvalues.

Depending on the context sometimes one imposes yet another condition which is ergodicity meaning the kernel of $$L$$ has to be one-dimensional and spanned by $$e^{-H/T}$$. Anyway the reasons Davies generators come up are that they come from physical considerations ("system weakly interacting with a thermal bath") and that, on top, the additional conditions make them a handy and easy-to-analyze subclass of Lindbladians.

$${}^1$$ Strictly speaking Liouvillian is the general term for the generator $$\mathcal L$$ of a master equation $$\dot X=\mathcal L(X)$$, but in the case of open quantum systems---due to the requirement of complete positivity---this boils down to the Lindbladian which is why these names are often used interchangeably in the quantum literature

• What is the relation between $\Phi$ and $\Phi^*$? I haven't come across the concept of Hilbert-Schmidt dual before. Just hermitian conjugation of all Kraus operators? Also, should all the $\{V_j^*\}$ operators in the $\Gamma(\rho)$ equation have a $\dagger$ sign instead of $*$? Commented Mar 16 at 15:26
• $\Phi^*$ has many names, maybe you know it under "adjoint" or "dual channel" (i.e. channel translated to the Heisenberg picture), I edited my answer accordingly. Strictly speaking these two concepts are not the same, but they coincide for maps which preserve Hermiticity so to not overcomplicate things I'll pretend like they're equivalent Commented Mar 16 at 15:30