1. Lecture 12: the time-dependent variational principle for dissipative dynamics
The time-dependent variational principle of Dirac is a powerful method to simulate the real and imaginary time dynamics of strongly correlated quantum systems. The original formulation has, as far as I can tell, only been formulated in the case of pure states. The generalisation to quantum systems undergoing dissipative dynamics is nontrivial and appears not to have been attempted. Here we describe a natural generalisation.
The pdf version can be found here.
We denote by the set of all complex matrices with entries in . There is a natural inner product on provided by
The state space of a -dimensional quantum space is given by the set of all density operators, defined by
Throughout this lecture we regard as a -dimensional differentiable real manifold (with boundary) in the natural way, i.e., with a single coordinate chart provided by, e.g., the map
where , with
and is an orthonormal basis of traceless hermitian operators, i.e., .
The tangent space to at may be straightforwardly identified with the set
of traceless hermitian matrices. We give the structure of a Riemannian manifold by choosing a positive bilinear form on for all .
Throughout we define a variational class simply to be a submanifold of . We assume that the manifold can be parametrised as
where we assume the dependence on the parameters to be analytic.
It is convenient to introduce the left and right multiplication operators and , , respectively. The modular operator is then defined to be
for all . We also define
3. Monotone Riemannian metrics
There is no canonical choice of Riemannian metric on . However, there are several canonical families of Riemannian metrics which naturally arise from information-theoretic considerations.
Here the natural condition is that the metric is monotone, meaning that , where is a map. The reasoning here is that the distinguishability of two states infinitesimally close to should only be decreased under the action of a channel. Petz showed there is a one-to-one correspondence between the set of monotone metrics and a special class of convex operator functions. A complete classification is now well understood (see, e.g., this paper for a quick overview). We do not express our results in the most general way available (although this is entirely straightforward once we understand a couple of examples), but instead focus on two special metrics defined as follows.
Suppose for any given , and , the bilinear form defining the metric is given by
(It turns out that all monotone metrics have this form.) The two examples we study are furnished by the Bures metric
4. The time-dependent variational principle for dissipative dynamics
The setup is identical to the pure-state case: we aim to find the optimal path generated by the vector field induced by finding the optimal element which is closest to the RHS of (13), where we use the quadratic form to measure the distance, i.e., we aim to solve
This is equivalent to minimising
Writing this in terms of the explicit parametrisation , we see that we need to minimise
This may be rewritten as
is the Gram matrix and
The minimum is easily found to satisfy
5. The TDVP applied to a Lindblad equation
In this section we focus on the TDVP applied to the specific example
using the monotone metric based on . In this case we find that the Gram matrix is given by (we suppress the arguments of for clarity)
We also find that