Reply to Commentary on "Unified Framework for Open Quantum Dynamics with Memory" Addressing Critiques and Highlighting Contributions

Kernekoncepter

This response addresses critiques raised by Makri et al. on the authors' recent paper concerning the relationship between memory kernels and influence functions in open quantum system dynamics, clarifying misunderstandings and emphasizing the novel contributions of their work.

Resumé

• Bibliographic Information: Ivander, F., Lindoy, L. P., & Lee, J. (2024). Reply to “Comment on “Unified Framework for Open Quantum Dynamics with Memory””. arXiv preprint arXiv:2411.02409v1.
• Research Objective: This response paper aims to address the critiques raised by Makri et al. regarding the authors' original work on establishing an explicit relationship between the Nakajima-Zwanzig memory kernel (K) and influence functions (I) in open quantum dynamics.
• Methodology: The authors directly address each of the four main critiques presented by Makri et al. by referencing specific equations and analyses from both their original paper and the works cited by Makri et al.
• Key Findings: The authors refute the claims made by Makri et al. by demonstrating that (1) Makri's 2020 paper does not explicitly connect I and K, (2) the chosen GQME discretization, while specific, does not present significant numerical concerns, (3) Makri's driven SMatPI work was acknowledged and differentiated in the Supplementary Notes of the original paper, and (4) Wang and Cai's work was appropriately cited for the number of Dyck paths.
• Main Conclusions: The authors successfully clarify the misunderstandings surrounding their original work and reiterate its novel contributions, including the explicit connection between I and K for various open quantum dynamics setups, the construction of K without projection-free dynamics inputs, and the characterization of the bath spectral function from reduced system dynamics.
• Significance: This response paper contributes to the ongoing discussion within the field of open quantum system dynamics, particularly concerning the relationship between memory kernels and influence functions. It clarifies the authors' contributions and highlights the importance of precise analysis and interpretation in scientific discourse.
• Limitations and Future Research: The response paper primarily focuses on addressing the critiques raised by Makri et al. and does not delve into further research avenues. However, it implicitly encourages continued exploration of open quantum system dynamics, particularly for systems beyond the simplified Class 1 problems.

Statistik

Citater

"After a close inspection of the cited paper again, we could not find a set of equations in Ref. 3 that write K in terms of I along the discretized time axis."
"The cited paper only mentions an observation of a close resemblance between SMatPI and TTM and does not explicitly construct K in terms of I like our work did."
"As we show in our paper (see, for instance, Fig. 3)1 and also evidenced by TTM,6 this does not raise significant concerns in numerical examples."
"We cited Makri’s driven SMatPI approach4 in the Supplementary Notes H of our paper to note differences if one were to develop new methods based on our formulation."
"We only cited their preprint,5 and not the final version published in the journal.11 This was an oversight on our part."

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Reply to "Comment on "Unified Framework for Open Quantum Dynamics with Memory""

by Felix Ivande... kl. arxiv.org 11-06-2024

https://arxiv.org/pdf/2411.02409.pdf

Dybere Forespørgsler

How might the explicit relationship between memory kernels and influence functions be applied to develop more efficient numerical methods for simulating open quantum systems, particularly for complex systems beyond the simplified cases discussed?

The explicit relationship between memory kernels and influence functions uncovered by Ivander et al. offers a promising avenue for developing more efficient numerical methods for simulating open quantum systems, especially for complex systems. Here's how:

Circumventing Projection-Free Techniques: Traditional methods for calculating memory kernels often rely on computationally expensive projection-free dynamics simulations. By establishing a direct link to influence functions, which are directly computable from the system-bath Hamiltonian, one can potentially construct memory kernels without resorting to these expensive techniques. This becomes particularly advantageous for complex systems where projection-free methods become intractable.

Exploiting Time-Translational Invariance: The authors' choice of Trotter ordering leads to a time-translationally invariant memory kernel. This property can be exploited to reduce the computational cost. Instead of calculating the memory kernel at every time step, one could potentially pre-compute and store a smaller set of time-independent tensors, significantly reducing the computational burden for long-time simulations.

Tensor Network Approaches: The relationship between memory kernels and influence functions can be leveraged in conjunction with advanced tensor network techniques. These techniques, already showing promise in simulating complex quantum systems, could be adapted to efficiently represent and manipulate the high-dimensional tensors arising in this context. This could pave the way for simulating open quantum systems with significantly larger system and bath sizes.

Extending to More General Cases: While the authors primarily focus on specific cases (e.g., Class 1), their work lays the groundwork for extensions to more general and complex scenarios. Future research could explore generalizing the relationship between memory kernels and influence functions to encompass:

Non-Gaussian Baths: Moving beyond Gaussian baths to consider more realistic, structured environments.
Strong Coupling Regimes: Developing methods applicable to systems strongly coupled to their environments, where perturbative approaches break down.
Non-Markovian Dynamics: Accurately capturing memory effects that play a crucial role in many open quantum systems.

By exploring these directions, the explicit connection between memory kernels and influence functions holds the potential to significantly advance our ability to simulate and understand the behavior of complex open quantum systems.

Could there be alternative interpretations or mathematical frameworks that reconcile the differing viewpoints on the relationship between memory kernels and influence functions presented by the authors and Makri et al.?

While the authors and Makri et al. seem to have differing viewpoints on the explicitness of the relationship between memory kernels and influence functions, there might be alternative interpretations or mathematical frameworks that could reconcile these perspectives. Here are some possibilities:

Continuous vs. Discrete Time: A key difference lies in the treatment of time. Makri et al.'s work primarily operates in the continuous-time domain, while Ivander et al. focus on a specific discretization scheme. It's possible that the explicit relationship between K and I becomes more apparent or easier to formulate in the discrete-time setting due to the chosen discretization. Exploring different discretization schemes or carefully analyzing the continuous-time limit could bridge the gap.

Choice of Representation: The choice of mathematical representation for the memory kernel and influence functions could influence the perceived explicitness of their relationship. Different representations might highlight different aspects of the connection. For instance, exploring representations based on operator bases or path integral formulations could offer new insights.

Implicit vs. Explicit Connections: While Makri et al. acknowledge a "close resemblance" between their formulation and the GQME, they might not consider it an explicit connection in the same way Ivander et al. do. It's possible that the relationship is implicit in Makri et al.'s framework but requires further mathematical manipulation or interpretation to be expressed in the explicit form presented by Ivander et al.

Focus on Different Aspects: The two groups might be emphasizing different aspects of the relationship. Makri et al.'s focus on numerical efficiency and practical implementation might lead them to prioritize different aspects of the connection compared to Ivander et al.'s focus on establishing a formal, explicit relationship.

Further investigation into these possibilities could lead to a more unified understanding of the connection between memory kernels and influence functions, potentially revealing a deeper underlying mathematical structure.

If the bath spectral function can be characterized solely from the reduced system dynamics, what new possibilities does this open up for experimental characterization and control of open quantum systems in realistic environments?

The ability to characterize the bath spectral function, J(ω), directly from the reduced system dynamics, ρ(t), presents exciting new possibilities for experimental characterization and control of open quantum systems in realistic environments:

Direct Probing of the Environment: Traditionally, characterizing the environment required separate, often challenging, measurements. This new capability allows experimentalists to directly probe the crucial features of the environment (encoded in J(ω)) solely by observing how the system itself evolves over time. This is particularly valuable for complex environments where direct characterization is difficult or impossible.

Simplifying Quantum Device Characterization: In areas like quantum computing and sensing, precisely characterizing the environment surrounding qubits is crucial for optimizing performance and understanding decoherence sources. This technique could offer a streamlined approach to characterize the noise spectrum affecting qubits by analyzing their readily accessible dynamics.

Tailoring System-Environment Interactions: A precise knowledge of J(ω) opens doors for tailoring the system-environment interaction. By engineering the system or its surroundings to modify J(ω), experimentalists could potentially enhance desired couplings, suppress unwanted noise sources, or even induce novel quantum effects.

Validating Theoretical Models: Directly extracting J(ω) from experimental data provides a powerful tool for validating theoretical models of open quantum systems. By comparing the experimentally determined J(ω) with theoretical predictions, researchers can refine their models and gain a deeper understanding of the system-environment interaction.

Quantum Control in Realistic Environments: Precise knowledge of the environment is essential for implementing sophisticated quantum control protocols. By characterizing J(ω) directly from system dynamics, one could develop more robust control strategies that account for the specific features of the environment, paving the way for quantum technologies operating reliably in realistic, non-ideal settings.

Overall, the ability to characterize the bath spectral function solely from reduced system dynamics represents a significant step towards understanding and harnessing open quantum systems in practical applications. It promises to bridge the gap between theoretical models and experimental reality, enabling the development of more robust and controllable quantum devices for various quantum technologies.