SciPost Submission Page
An introduction to Markovian open quantum systems
by Shovan Dutta
Submission summary
| Authors (as registered SciPost users): | Shovan Dutta |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2510.26530v2 (pdf) |
| Date submitted: | Nov. 5, 2025, 5:42 a.m. |
| Submitted by: | Shovan Dutta |
| Submitted to: | SciPost Physics Lecture Notes |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
|
| Approach: | Theoretical |
Abstract
This is a concise, pedagogical introduction to the dynamic field of open quantum systems governed by Markovian master equations. We focus on the mathematical and physical origins of the widely used Lindblad equation, its unraveling in terms of stochastic pure-state trajectories and the corresponding continuous measurement protocols, the structure of steady states with emphasis on the role of symmetry and conservation laws, and a sampling of the novel physical phenomena that arise from nonunitary dynamics (dissipation and measurements). This is far from a comprehensive summary of the field. Rather, the objective is to provide a conceptual foundation and physically illuminating examples that are useful to graduate students and researchers entering this subject. There are exercise problems and references for further reading throughout the notes.
Current status:
Awaiting resubmission
Reports on this Submission
Strengths
1)Clear and pedagogical introduction to Markovian open quantum systems, Lindblad master equation and quantum trajectories.
2)Exercise Sessions are useful for students
2)Exercise Sessions are useful for students
Weaknesses
1)Substantial overlap with Ref[2], in the same Lecture Notes Series.
2)The discussion in Sec. 6 (novel physical phenomena) could be improved and made more unique and distinct (see comments below)
3)Exercise Sessions lack solution
2)The discussion in Sec. 6 (novel physical phenomena) could be improved and made more unique and distinct (see comments below)
3)Exercise Sessions lack solution
Report
I have read the manuscript by S. Dutta titled an “Introduction to Markovian Open Quantum Systems”.
In these lecture notes aimed at graduate students the Author discusses the theoretical frameworks related to the Lindblad master equation and its stochastic unravelling. Through a series of exercises it provides the reader
with a practical approach to enter the field. The vast majority of this work focuses
on single or few-body physics, while the last section discusses some extension to many-body case.” I think overall the manuscript is well written and useful.
My major comment on this manuscript regards its substantial overlap with Ref[2], also covering lectures notes on open quantum systems and published on the same journal. Although the scope of the present manuscript is less broad, I cannot avoid noticing the large overlaps between the two lecture notes.
To give some concrete example: beside the formalism (which is obviously the same) also the few examples chosen on the many-body side are strongly overlapping (see discussion on Sec. 6.1 on Mott insulators via reservoir engineering or Sec 6.3 on Dissipative phase transitions, where even Fig. 8 is taken from Ref [2] or the discussion on MIPT).
I would encourage the Author to modify the manuscript to make the angle into the field more unique. One possibility could be to modify Section 6 and discuss other types of Many-body applications, for example taken from the research experience of the Author himself. I can for example notice that dissipative impurities, such as Ref 33, 96 from the Author, are not properly covered in Ref. 2.
Similarly it is the case of boundary driven open quantum systems.
Once the final section is improved and made stronger I think the Lecture notes could provide a nice resource to enter the field.
In these lecture notes aimed at graduate students the Author discusses the theoretical frameworks related to the Lindblad master equation and its stochastic unravelling. Through a series of exercises it provides the reader
with a practical approach to enter the field. The vast majority of this work focuses
on single or few-body physics, while the last section discusses some extension to many-body case.” I think overall the manuscript is well written and useful.
My major comment on this manuscript regards its substantial overlap with Ref[2], also covering lectures notes on open quantum systems and published on the same journal. Although the scope of the present manuscript is less broad, I cannot avoid noticing the large overlaps between the two lecture notes.
To give some concrete example: beside the formalism (which is obviously the same) also the few examples chosen on the many-body side are strongly overlapping (see discussion on Sec. 6.1 on Mott insulators via reservoir engineering or Sec 6.3 on Dissipative phase transitions, where even Fig. 8 is taken from Ref [2] or the discussion on MIPT).
I would encourage the Author to modify the manuscript to make the angle into the field more unique. One possibility could be to modify Section 6 and discuss other types of Many-body applications, for example taken from the research experience of the Author himself. I can for example notice that dissipative impurities, such as Ref 33, 96 from the Author, are not properly covered in Ref. 2.
Similarly it is the case of boundary driven open quantum systems.
Once the final section is improved and made stronger I think the Lecture notes could provide a nice resource to enter the field.
Requested changes
1)Improve and expand the final Section 6 to make the work more unique and interesting
Recommendation
Ask for major revision
Strengths
- Concise introduction to open systems
- Good practice problems
- Pedagogical and accessible
Report
I think this is a very nice and useful introduction to the field of open systems for the beginning graduate student. I particularly like the inclusion of a good number of practice problems. I have a few small points where I think the clarity could be improved:
1. On page 12, there is a discussion of the eigenvalues of the Lindbladian having nonpositive real part: “If the dynamics is well behaved at long times—as is true in most physical situations—all nonzero eigenvalues in the summation must have nonpositive real parts.” It could perhaps be useful to point out that this is always true for finite dimensional system (i.e. from the contraction-mapping theorem). What constitutes well-behaved dynamics is also a bit vague – one could reasonably define this as the Lindbladian being negative semi-definite, in which case this is a tautology. It could also be useful to point out there are very simple, physically motivated models in infinite dimensional Hilbert spaces (bosonic modes) where this fails, such as a simple parametric amplifier.
2. Also on page 12, when discussing the spectrum of a spin model vs a damped Harmonic oscillator, it is written: “The wedge shape in Fig. 1(b) is characteristic of a classical fixed point, whereas the parabolic shapes in Fig. 1(c) indicate damped oscillations”. I believe either of these could reasonably be called decaying oscillations. The wedge shape in 1(b) is characteristic of a non-interacting system, where there is a mode structure.
3. In section 4.1, and especially at the top of page 14 when discussing the delta function at the boundary of the integral, the text plays a bit fast and loose with the stochastic calculus. I appreciate the point of these lecture notes are not suitable for a rigorous derivation, but perhaps a reference to a proper derivation using the Ito calculus would help point the curious student in the right direction for how to put this on a more rigorous footing.
4. On page 14, colored noise is introduced without defining it. Should mention that this is any noise source with a non-flat spectral density (and perhaps some experimentally motivated examples, like 1/f noise).
5. On page 29, the text reads: “In many-body systems, generic states, including eigenstates of an interacting Hamiltonian, have correlations between far-off sites.” It is important to qualify that this is only eigenstates in the bulk of the spectrum. The reader might infer this to mean many-body ground states, which of course can only have long range correlations if the Hamiltonian is gapless.
6. On page 43, the text reads: “One can formulate a broader class of continuous phase transitions in terms of a spontaneous weak-symmetry breaking, where the symmetry-broken phase is gapless”. It is important to be careful with language here. From a quantum optics perspective, having two degenerate zero modes might be defined as a gapless spectrum, but from a more traditional condensed matter perspective, a spectrum is only gapless if there is a continuum of modes around the ground state energy, whereas just having multiple (but a finite number of) steady states separated by a finite gap would just be a ground state degeneracy. This is especially important when discussing phases of matter, which are typically only well defined for gapped Liouvillians/Hamiltonians (in the condensed matter sense), and the phase transition point is gapless. Some clarifying remarks might be useful to avoid confusion.
7. In the discussion of XXZ subject to boundary dissipation, one should also add the reference: Prosen, PRL 107, 137201 (2011). This gives exact, analytic results in agreement with Ref. [108] in the text.
1. On page 12, there is a discussion of the eigenvalues of the Lindbladian having nonpositive real part: “If the dynamics is well behaved at long times—as is true in most physical situations—all nonzero eigenvalues in the summation must have nonpositive real parts.” It could perhaps be useful to point out that this is always true for finite dimensional system (i.e. from the contraction-mapping theorem). What constitutes well-behaved dynamics is also a bit vague – one could reasonably define this as the Lindbladian being negative semi-definite, in which case this is a tautology. It could also be useful to point out there are very simple, physically motivated models in infinite dimensional Hilbert spaces (bosonic modes) where this fails, such as a simple parametric amplifier.
2. Also on page 12, when discussing the spectrum of a spin model vs a damped Harmonic oscillator, it is written: “The wedge shape in Fig. 1(b) is characteristic of a classical fixed point, whereas the parabolic shapes in Fig. 1(c) indicate damped oscillations”. I believe either of these could reasonably be called decaying oscillations. The wedge shape in 1(b) is characteristic of a non-interacting system, where there is a mode structure.
3. In section 4.1, and especially at the top of page 14 when discussing the delta function at the boundary of the integral, the text plays a bit fast and loose with the stochastic calculus. I appreciate the point of these lecture notes are not suitable for a rigorous derivation, but perhaps a reference to a proper derivation using the Ito calculus would help point the curious student in the right direction for how to put this on a more rigorous footing.
4. On page 14, colored noise is introduced without defining it. Should mention that this is any noise source with a non-flat spectral density (and perhaps some experimentally motivated examples, like 1/f noise).
5. On page 29, the text reads: “In many-body systems, generic states, including eigenstates of an interacting Hamiltonian, have correlations between far-off sites.” It is important to qualify that this is only eigenstates in the bulk of the spectrum. The reader might infer this to mean many-body ground states, which of course can only have long range correlations if the Hamiltonian is gapless.
6. On page 43, the text reads: “One can formulate a broader class of continuous phase transitions in terms of a spontaneous weak-symmetry breaking, where the symmetry-broken phase is gapless”. It is important to be careful with language here. From a quantum optics perspective, having two degenerate zero modes might be defined as a gapless spectrum, but from a more traditional condensed matter perspective, a spectrum is only gapless if there is a continuum of modes around the ground state energy, whereas just having multiple (but a finite number of) steady states separated by a finite gap would just be a ground state degeneracy. This is especially important when discussing phases of matter, which are typically only well defined for gapped Liouvillians/Hamiltonians (in the condensed matter sense), and the phase transition point is gapless. Some clarifying remarks might be useful to avoid confusion.
7. In the discussion of XXZ subject to boundary dissipation, one should also add the reference: Prosen, PRL 107, 137201 (2011). This gives exact, analytic results in agreement with Ref. [108] in the text.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Report
See attached document.
Recommendation
Ask for minor revision
Strengths
1- a very useful and pedagogical introduction to Markovian master equations 2 - acceptance criteria match those of SciPost Physics Lecture Notes 3 - covers the stochastic unraveling, weak, strong, and dynamical symmetries in just 46 pages 4- There are also exercises
Report
The paper meets all the criteria of SciPost Physics Lecture Notes and therefore I recommend publishing the paper.
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)