Stochastic dissipative quantum spin chains (I) : Quantum fluctuating discrete hydrodynamics

- Interesting and timely topic
- Novel aspects on dissipative dynamics of XXZ chain

- Introduction poorly written and hard to understand
- Better comparison to standard Lindblad formalism should be provided

The authors consider stochastic dynamics of quantum chains,
using the methods of quantum stochastic calculus.
In particular, they are interested in the limit of large friction,
i.e. when the coefficient of the dissipative coupling to an environment
is much larger in magnitude then the couplings generating the
coherent part of the time evolution. After introducing the
general formalism of stochastic calculus, they study the slow-modes
of the dynamics in a perturbative expansion w.r.t. the inverse friction
coefficient. The concrete example of an XXZ chain with a dephasing
noise is considered afterwards, where the slow-mode observables
are explicitly constructed and the evolution equations for the
density matrix and observables are presented.

In my opinion, although the paper is rather technical and thus somewhat
difficult to read, it contains some interesting novel results which
are worth publishing. There are, however, certain amendments that
should be considered in order to improve the quality of the
manuscript.

1)
The introduction is not well written. In particular, I do not
understand why the various evolution equations are presented
here. On one hand, these are impossible to comprehend without
any clue about the notation and the underlying methods.
On the other hand, this only makes the manuscript repetitive,
the introduction lengthy, without providing a better insight
to the reader. Some more detailed general introduction to
quantum stochastic differential equations (e.g. what are the
main assumptions to their applicability? where are they used?)
would be much more instructive at this point than quoting the
results for the XXZ chain. I would suggest the authors to
extensively revise their introductory chapter.

2)
I found the notion of “stochastic Lindblad equation” rather confusing.
The Lindblad master equation describes the evolution of a system
density matrix interacting with its environment in a Markovian
approximation, after integrating out the bath degrees of freedom.
The Lindblad dynamics describes coherent as well as stochastic
processes, resulting in a $non$-$unitary$ time-evolution of the
density matrix. In contrast, the stochastic differential equations
the authors consider describe a $unitary$ time evolution of the
density matrix. After an averaging over the noise terms they
indeed yield the Lindblad equation. However, this difference should
be made clear from the very beginning, or one should rather stick to
“stochastic differential equation” when referring to the noisy
dynamics.

3)
It did not become completely clear to me what the advantage of working
with the stochastic differential equation formalism really is?
If I understood it correctly, the Lindblad generators completely fix
the form of the noise term. On the other hand, when considering
expectation values, the noise has to be averaged over anyway.
What extra information can one expect to have on the dynamics of the
density operator and physical observables which is not there when
working with a Lindblad master equation? I believe this issue
deserves some more clarification.

4)
The word “hopping” is misspelled throughout the text.