SciPost Submission Page
Understanding and utilizing the inner bonds of process tensors
by Moritz Cygorek, Erik M. Gauger
Submission summary
| Authors (as registered SciPost users): | Moritz Cygorek |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2404.01287v1 (pdf) |
| Date submitted: | 2024-05-06 14:58 |
| Submitted by: | Cygorek, Moritz |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
Process tensor matrix product operators (PT-MPOs) enable numerically exact simulations for an unprecedentedly broad range of open quantum systems. By representing environment influences in MPO form, they can be efficiently compressed using established algorithms. The dimensions of inner bonds of the compressed PT-MPO may be viewed as an indicator of the complexity of the environment. Here, we show that the inner bonds themselves, not only their dimensions, have a concrete physical meaning: They represent the subspace of the full environment Liouville space which hosts environment excitations that may influence the subsequent open quantum systems dynamics the most. This connection can be expressed in terms of lossy linear transformations, whose pseudoinverses facilitate the extraction of environment observables. We demonstrate this by extracting the environment spin of a central spin problem, the current through a quantum system coupled to two leads, the number of photons emitted from quantum emitters into a structured environment, and the distribution of the total absorbed energy in a driven non-Markovian quantum system into system, environment, and interaction energy terms. Numerical tests further indicate that different PT-MPO algorithms compress environments to similar subspaces. Thus, the physical interpretation of inner bonds of PT-MPOs both provides a conceptional understanding and it enables new practical applications.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Report
The authors show how to extract physically relevant information from the inner bonds obtained from MPO compressions. This is a very interesting, relevant and timely contribution to the blossoming field of PT-MPO methods for open quantum system dynamics. It clearly highlights the meaning and relevance of the algorithms beyond their numerical value. Several examples (some somewhat very trivial, like the easily soluble amplitude damping model eq. (51)) are used to demonstrate the power of the PT-MPO method to reveal information about coupling and bath observables. A final remark: note that the two references [54,55] for the quantum thermodynamic application are not really concerned with coupled heat baths. There are far more suitable references for the scenario considered here.
Recommendation
Ask for minor revision
Strengths
1- Physical interpretation of performant algorithms for simulating open quantum systems.
2- Self-contained and clear explanations of the algorithms
3- Different numerical tests, showing strong evidence for their interpretation
Report
In this work, the authors elucidate the physical meaning of the "virtual" degrees of freedom in the MPO representation of the influence functional. This gives us insight into the success of different algorithms that are used for simulating open (non-Markovian) quantum systems. Moreover, their identification of these virtual degrees of freedom allows to evaluate environment expectation values approximately. The result is tested and confirmed on different benchmark cases. The paper is well-written and self-contained.
This work, therefore, provides a strong motivation for using these algorithms, and is a clear addition to the literature. I therefore recommend publication of this article.
Requested changes
1- For MPS, the singular value decomposition is optimal for truncating the virtual dimension in the MPS, because this optimizes the norm distance between the original and the truncated state. For MPOs, however, the choice for SVD truncation is not always straightforward, because it is not obvious which distance measure should be used. Could the authors argue why SVD truncation is a good choice here?
Recommendation
Publish (meets expectations and criteria for this Journal)
Report
The manuscript suggests a novel numerical method in open quantum systems. By introducing a lossy linear transformation, the authors build a link between the environment and the Process tensor matrix product operators. Most importantly, compare with traditional methods based on integrating out the environments, this linear transformation based on process tensor helps to resolve the environmental observable.
The authors tested carefully the validity of extracting environments from PT-MPO inner bonds in the sense that the compression is lossy. Demonstration of the method in spin system and the light matter coupled case is concrete. Specifically, there is a solid discussion on the stabilization of extracting photon number in the photon emission system. I strongly recommend publication of this manuscript.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Strengths
The manuscript introduces novel matrices \cal T and its pseudoinverse that facilitates insigths into the system and its environment. The formalism is illustrated by several physical examples.
Weaknesses
A possible weakness is the following. Section III B describes a compression of an MPO bond dimension. In the forward sweep subsequent tensors are svd-truncated. The backward sweep seems unnecesary as it only changes the gauge of the virtual indices. According to https://arxiv.org/pdf/1008.3477 sections 4.4.2 and 4.5.1, it would be optimal to use the forward sweep just to bring the MPO to the right-canonical form (without any truncations) and then to perform the truncations during the backwards sweep taking advantage of the mixed canonical form of the MPO. In this form each svd truncation is done in a tensor environment with a Euclidean metric tensor and the svd truncation is an optimal truncation. Would it be possible to implement this procedure in the manuscript?
Report
The manuscript is potentially acceptable. When the weakness is fixed the algorithm may become more powerful/stable.
Requested changes
There is one, see Weaknesses.
Recommendation
Ask for major revision