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Enhancing Quantum Metrology with High-order Fisher Information and Experiments
by Xin-Zhu Liu, Jun-Li Jiang, Yan-Han Yang, Li-Ming Zhao, Xue Yang, Shao-Ming Fei and Ming-Xing Luo
Submission summary
| Authors (as registered SciPost users): | Ming-Xing Luo |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202509_00041v1 (pdf) |
| Date submitted: | Sept. 23, 2025, 4:19 p.m. |
| Submitted by: | Ming-Xing Luo |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Experimental |
Abstract
Fisher information plays a central role in statistics and quantum metrology, providing the basis for the celebrated Cramér-Rao bound. In this work, we introduce a new information measure based on higher-order Fisher information and show that it naturally leads to a generalized uncertainty relation for parameter estimation, which can be regarded as an extension of the Cramér-Rao bound. As an application, we analyze the case of quantum phase estimation with a single qubit and compare our theoretical bounds with the well-known established hierarchical bounds. Finally, we experimentally validate the proposed framework using a photonic platform.
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:
Awaiting resubmission
Reports on this Submission
Report
In this work, the authors propose an extension of the Cramér–Rao bound based on what they refer to as high-order Fisher information. In the first four chapters, they present their theoretical extension, which consists of incorporating higher-order derivatives of the likelihood function into the parameter-estimation framework. They then extend this approach to quantum systems and provide examples of its application in thermodynamic settings. In the final section, they apply their findings to a one-qubit phase-estimation problem, demonstrating how their high-order bound behaves in experimental scenarios.
I believe that the authors’ findings merit publication in some form, but the manuscript in its current state should not be accepted by SciPost Physics.
The manuscript should be thoroughly revised to improve its presentation, as in its current form the authors’ work is not easily readable. They should present their results more clearly, and I suggest adding, independently of the experimental application, a simple example (if one exists) in which the standard Cramér–Rao bound is outperformed by their proposed bound.
I also have some questions regarding the authors’ work:
1)
In their proofs of the bounds in Eqs. (2) and (5), the authors use the condition of an unbiased estimator. I believe that, similarly to the standard Cramér–Rao bound, this condition can be relaxed to that of a locally unbiased estimator. If so, the authors should comment on this point in their work.
2)
I think the authors should comment more explicitly on the distance between their bound and the Cramér–Rao bound, both in classical and quantum scenarios. This would be particularly interesting in view of potential extensions to multi-parameter estimation, where the Quantum Cramér–Rao bound is no longer tight. Do the authors have any insights on this?
3)
The high-order information used in this work is not additive. If this framework is applied to a many-body system in an interferometric scheme, will the scaling of their new bound always lie between the shot-noise and Heisenberg scalings? Is Heisenberg scaling achievable?
4)
In their derivation of the bound in Eq. (5), the authors should comment further on its attainability. I am interested in the measurement operators that maximize the bound. Are these projective measurements onto the eigenstates of their modified SLD in Eq. (C.1)?
More generally, might this new class of measurements be significantly more difficult to implement in practice—especially in many-body systems, where such measurements could be highly non-local? If so, the practical advantage suggested by the new bound may be mitigated by the increased complexity of the required experimental protocol.
5)
In Figure 4, the authors present some of their experimental results. Do they have an explanation for why the experimental data and the theoretical prediction for the purple curve do not match?
I believe that the authors’ findings merit publication in some form, but the manuscript in its current state should not be accepted by SciPost Physics.
The manuscript should be thoroughly revised to improve its presentation, as in its current form the authors’ work is not easily readable. They should present their results more clearly, and I suggest adding, independently of the experimental application, a simple example (if one exists) in which the standard Cramér–Rao bound is outperformed by their proposed bound.
I also have some questions regarding the authors’ work:
1)
In their proofs of the bounds in Eqs. (2) and (5), the authors use the condition of an unbiased estimator. I believe that, similarly to the standard Cramér–Rao bound, this condition can be relaxed to that of a locally unbiased estimator. If so, the authors should comment on this point in their work.
2)
I think the authors should comment more explicitly on the distance between their bound and the Cramér–Rao bound, both in classical and quantum scenarios. This would be particularly interesting in view of potential extensions to multi-parameter estimation, where the Quantum Cramér–Rao bound is no longer tight. Do the authors have any insights on this?
3)
The high-order information used in this work is not additive. If this framework is applied to a many-body system in an interferometric scheme, will the scaling of their new bound always lie between the shot-noise and Heisenberg scalings? Is Heisenberg scaling achievable?
4)
In their derivation of the bound in Eq. (5), the authors should comment further on its attainability. I am interested in the measurement operators that maximize the bound. Are these projective measurements onto the eigenstates of their modified SLD in Eq. (C.1)?
More generally, might this new class of measurements be significantly more difficult to implement in practice—especially in many-body systems, where such measurements could be highly non-local? If so, the practical advantage suggested by the new bound may be mitigated by the increased complexity of the required experimental protocol.
5)
In Figure 4, the authors present some of their experimental results. Do they have an explanation for why the experimental data and the theoretical prediction for the purple curve do not match?
Requested changes
- See report.
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