Lindblad Equation is differential form of CP map

  By

The Master equation in Lindblad form (aka the Lindblad equation) is the most general possible evolution of an open quantum system that is Markovian and time-homogeneous. Markovian means that the way in which the density matrix evolves is determined completely by the current density matrix. This is the assumption that there are no memory effects, i.e. that the environment does not store information about earlier state of the system that can influence the system in the future.Here’s an example of a memory effect: An atom immersed in an electromagnetic field can be in one of two states, excited or ground. If it is in an excited state then, during a time interval, it has a certain probability of decaying to the ground state by emitting a photon. If it is in the ground state then it also has a chance of becoming excited by the ambient field. The situation where the atom is in a space of essentially infinite size would be Markovian, because the emitted photon (which embodies a record of the atom’s previous state of excitement) would travel away from the atom never to interact with it again. It might still become excited because of the ambient field, but its chance of doing so isn’t influenced by its previous state. But if the atom is in a container with reflecting walls, then the photon might be reflected back towards the atom, changing the probability that it becomes excited during a later period.a   Time-homogeneous just means that the rule for stochastically evolving the system from one time to the next is the same for all times.

Given an arbitrary orthonormal basis L_n of the space of operators on the N-dimensional Hilbert space of the system (according to the Hilbert-Schmidt inner product \langle A,B \rangle = \mathrm{Tr}[A^\dagger B]), the Lindblad equation takes the following form:

(1)   \begin{align*} \frac{\mathrm{d}}{\mathrm{d}t} \rho=- i[H,\rho]+\sum_{n,m = 1}^{N^2-1} h_{n,m}\left(L_n\rho L_m^\dagger-\frac{1}{2}\left(\rho L_m^\dagger L_n + L_m^\dagger L_n\rho\right)\right) , \end{align*}

with \hbar=1. Here, H generates the unitary component of the evolution and the positive matrix h_{n,m} describes the non-unitary component (which is still completely positive and trace-preserving).

It turns out that the Lindblad equation is just the differential version of the most general smooth, 1-parameter family of completely positive (CP) maps on the space of density matrices. It is known that complete positivity is essentially a requirement derived from the assumption that the system is not initially correlated with the environment (i.e. that the environment has no memory at the beginning of the discrete step of evolution described by the CP map) and that quantum systems may evolve without complete positivity otherwisePhilip Pechukas, “Reduced Dynamics Need Not Be Completely Positive”, Phys. Rev. Lett. 73, 1060 (1994).b  . One can derive the Lindblad equation, then, by constructing a smooth family of CP map indexed by time, and then taking the time parameter to be infinitesimal. A satisfyingly detailed description of this process can be found in some notes by Caves.C. Caves, “Completely positive maps, positive maps, and the Lindblad form [PDF]”.c   The full Markovian and time-homogeneous evolution is obtained by exponentiating this map or, equivalently, by integrating the Lindblad equation.

Of course, the requirement that the family of CP maps be smooth around t=0 is essential. There are “indivisible” CP maps that cannot be decomposed into a non-trivial composition of two or more CP mapsMichael M. Wolf and J. Ignacio Cirac, “Dividing Quantum Channels,” Communications in Mathematical Physics, 279, 147-168 (2008).d  , and therefore have no differential structure. The simplest example is the transpose-operation combined with a partial dephasing operation. (The dephasing is essential to ensure that the total map is CP, as the transpose operation on its own is not.) This prompts the question of whether there is a cool way to think about all this in terms of Lie semigroups and their connected components. It appears that the answer is “yes”, and a bunch of interesting looking references are available in a work by Dirr et al.G. Dirr , U. Helmke, I. Kurniawan, and T. Schulte-Herbrüggen, “Lie-semigroup structures for reachability and control of open quantum systems: kossakowski-lindblad generators form lie wedge to markovian channels,” Reports on Mathematical Physics, 64, 93-121 (2009)e  

Now, the Lindblad equation can be rewritten by diagonalizing the matrix h \ge 0 as

(2)   \begin{align*} d_{n,m} = \delta_{n,m} d_n = (u^\dagger h u)_{n,m} = \sum_{n',m'} u^*_{n',n} h_{n',m'} u_{m',m} \end{align*}

with the unitary matrix u, so that

(3)   \begin{align*} \frac{\mathrm{d}}{\mathrm{d}t} \rho=- i[H,\rho]+\sum_{n=1}^{N^2-1} \left(A_n \rho A_n^\dagger-\frac{1}{2} \rho A_n^\dagger A_n + A_n^\dagger A_n \rho\right) \end{align*}

with new operators A_n = \sqrt{d_n} \sum_{m=1}^{N^2-1} u_{m,n} L_m. What I had not appreciated beforeOf course, the following elegant form was well known to Lindblad when he originally derived the equation that bears his name. The uglier form of the equation seems to be yet another case where significant beauty is lost in the transition from the original literature to review articles because people appropriate results without understanding them.f   reading Caves, and AlickiRobert Alicki, “Invitation to quantum dynamical semigroups”, [arXiv:quant-ph/0205188].g  , is that the non-unitary terms on the right hand side can be understood in terms of a (non-trace-preserving) CP map

(4)   \begin{align*} \Phi[\rho]=\sum_{n=1}^{N^2-1} A_n \rho A_n^\dagger \end{align*}

and a positive operator

(5)   \begin{align*} G=\sum_{n=1}^{N^2-1} A_n^\dagger A_n . \end{align*}

Using these objects, we can writeNote that [A,B]=AB-BA is the commutator and \{A,B\}=AB+BA is the anti-commutator.h  

(6)   \begin{align*} \frac{\mathrm{d}}{\mathrm{d}t} \rho=- i[H,\rho]+\Phi[\rho]-\frac{1}{2} \{ G,\rho \}. \end{align*}

The CP map \Phi is naturally called the “quantum transition map”, and it describes the non-unitary stochastic transitions. It is balanced by the term \{G,\rho\} which essentially grabs the part of the density matrix that is transitioning due to \Phi and deletes it; this ensures that the overall evolution is trace-preserving (i.e. that \mathrm{Tr}[\dot{\rho}] = 0). The difference between the unitary transitions induced by H and the non-unitary ones induced by \Phi are still a bit subtle to me, and have to do with whether the Kraus operators of the one-parameter family of CP maps being differentiated have time dependence that goes like 1-t or like \sqrt{t}. (See Caves.) My intuition is that the transitions induced by \Phi could not be prevented with the quantum Zeno effect (i.e. by inserting repeated orthogonal measurements), unlike those for H which can be.

(For a related derivation using the global wavefunction rather than reduced states, see Sec 1.1 of Onodera’s thesis.)

[Updated: 2020-2-19]

Footnotes

(↵ returns to text)

  1. Here’s an example of a memory effect: An atom immersed in an electromagnetic field can be in one of two states, excited or ground. If it is in an excited state then, during a time interval, it has a certain probability of decaying to the ground state by emitting a photon. If it is in the ground state then it also has a chance of becoming excited by the ambient field. The situation where the atom is in a space of essentially infinite size would be Markovian, because the emitted photon (which embodies a record of the atom’s previous state of excitement) would travel away from the atom never to interact with it again. It might still become excited because of the ambient field, but its chance of doing so isn’t influenced by its previous state. But if the atom is in a container with reflecting walls, then the photon might be reflected back towards the atom, changing the probability that it becomes excited during a later period.↵
  2. Philip Pechukas, “Reduced Dynamics Need Not Be Completely Positive“, Phys. Rev. Lett. 73, 1060 (1994).↵
  3. C. Caves, “Completely positive maps, positive maps, and the Lindblad form [PDF]“.↵
  4. Michael M. Wolf and J. Ignacio Cirac, “Dividing Quantum Channels,” Communications in Mathematical Physics, 279, 147-168 (2008).↵
  5. G. Dirr , U. Helmke, I. Kurniawan, and T. Schulte-Herbrüggen, “Lie-semigroup structures for reachability and control of open quantum systems: kossakowski-lindblad generators form lie wedge to markovian channels,” Reports on Mathematical Physics, 64, 93-121 (2009)↵
  6. Of course, the following elegant form was well known to Lindblad when he originally derived the equation that bears his name. The uglier form of the equation seems to be yet another case where significant beauty is lost in the transition from the original literature to review articles because people appropriate results without understanding them.↵
  7. Robert Alicki, “Invitation to quantum dynamical semigroups”, [arXiv:quant-ph/0205188].↵
  8. Note that [A,B]=AB-BA is the commutator and \{A,B\}=AB+BA is the anti-commutator.↵
Tagged  .
Bookmark the permalink.

Leave a Reply

Required fields are marked with a *. Your email address will not be published.

Contact me if the spam filter gives you trouble.

Basic HTML tags like ❮em❯ work. Type [latexpage] somewhere to render LaTeX in $'s. (Details.)