SciPost Submission Page
Knowledge-dependent optimal Gaussian strategies for phase estimation
by Ricard Ravell Rodríguez, Simon Morelli
Submission summary
| Authors (as registered SciPost users): | Simon Morelli |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2412.16023v2 (pdf) |
| Date submitted: | 2025-01-24 10:26 |
| Submitted by: | Morelli, Simon |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
When estimating an unknown phase rotation of a continuous-variable system with homodyne detection, the optimal probe state strongly depends on the value of the estimated parameter. In this article, we identify the optimal pure single-mode Gaussian probe states depending on the knowledge of the estimated phase parameter before the measurement. We find that for a large prior uncertainty, the optimal probe states are close to coherent states, a result in line with findings from noisy parameter estimation. But with increasingly precise estimates of the parameter it becomes beneficial to put more of the available energy into the squeezing of the probe state. Surprisingly, there is a clear jump, where the optimal probe state changes abruptly to a squeezed vacuum state, which maximizes the Fisher information for this estimation task. We use our results to study repeated measurements and compare different methods to adapt the probe state based on the changing knowledge of the parameter according to the previous findings.
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 #1 by Robert Trenyi (Referee 1) on 2025-3-27 (Invited Report)
Report
The manuscript is about searching for the optimal probe states for metrology in the continuous variable setting, namely for single-mode Gaussian states with given energy. The authors consider this task from two different perspectives, the frequentist (quantum Fisher information based) and the Bayesian one. The main finding is that the optimal state strongly depends on the approach used for the estimation, that is, on the prior knowledge of the parameter to be estimated. Interestingly, they find that when we have good knowledge of the phase parameter, then the best strategy is to use all the energy for squeezing. On the other hand, when the prior knowledge is poor then it is better to prioritize the displacement of the probe state.
The authors also investigate the case of repeated measurements, where the best strategy is to update probe states after each measurement round. For practicality, they also consider simplified versions of the fully adaptive case, which can be almost as good as the fully adaptive strategy.
A noisy scheme with phase diffusive noise as an example is also considered. Their method can be utilized as this type of noise can be thought of as having a higher prior uncertainty in the parameter, therefore it is better to put more energy in displacing the probe state.
As far as I know these results are new and can be of interest to the quantum information community, especially in designing metrological experiments with continuous variable systems.
The manuscript is sufficiently self-contained, whenever it cannot be, then the proper references are provided. The manuscript is well written and logically structured, so it can be followed easily. The figures are easy to grasp and informative. Moreover, the manuscript has a clear message, calculations that seem correct, even though I did not check all of them. I only checked till Eq. (14) and the derivation of (A3) in Appendix A. All of the analytical calculations seem reproducible.
Based on the above, I strongly recommend the publication of the manuscript in SciPost Physics.
I have the following suggestions/questions, and also found some typos:
• At the end of the introduction there should be a brief sentence about the content of the appendices.
• Regarding the previous point I did not find direct references to Appendix B, C in the main text (apart from referring to Fig. 6 that is in Appendix C), even though it contains useful information.
• 6th line of the 2nd paragraph of III.A.: The vector r should contain (<q>,<p>) instead of (<q>,<q>)?
• It would be useful to define the anticommutator {,} when defining the covariance matrix in the second paragraph of III.A.
• „In this section, we describe the estimation scenario that we are looking at.” sounds too informal to me (Section III. beginning)
• Can the authors give a simple intuitive argument already at the beginning of IV.C. why is it better to allocate all the energy in the squeezing of the state for the frequentist approach?
• III.B: typo in title homodye -> homodyne
• In Eqs. (14) and (16) the entries of the matrices should be better spaced for readability, if possible, in this two-column format
• What is the intuitive reason that the parameter \varphi changes significantly between the two approaches (observation of IV.C)?
• Is there a particular reason why \varphi=2\pi-2\theta is used instead of the previously used \varphi=-2\theta in the penultimate paragraph of Section IV.C?
• For Eq. (16), it should be stated which equation is evaluated from II.A for clarity.
• Do I understand correctly that the authors claim that there cannot be other optimal strategies apart from HUS and LUS based on numerical evidence?
• What is exactly the difference between the “simplified” and the “pushed further” strategies in Section V? The authors should clarify this a bit more in my opinion.
• Would it be possible to obtain meaningful results for the case when the noise does not commute with the dynamics? Maybe the authors can comment on that.
• 1st reference: Americal Journal ->American Journal
• Just out of curiosity: Are there results for general non-Gaussian states? What can they achieve in each metrological setting?
• Also out of curiosity: Do similar results also hold for other types of measurements (like heterodyne)?
Recommendation
Publish (meets expectations and criteria for this Journal)