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arxiv: 2606.01445 · v2 · pith:R5HM2QEAnew · submitted 2026-05-31 · ⚛️ physics.optics

Multiparameter Maximum Information States for Coherent Diffraction Measurements

Bram Verreussel , Jacob Seifert , Allard Pieter Mosk This is my paper

Pith reviewed 2026-06-28 16:14 UTC · model grok-4.3

classification ⚛️ physics.optics
keywords Fisher informationmultiparameter estimationcoherent diffractionscattering matrixmaximum information statesnuisance parametersoptical metrology
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The pith

The scattering matrix allows maximization of the Fisher information matrix over input modes to achieve optimal multiparameter precision in coherent light measurements.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends the concept of maximum information states from single-parameter to multiparameter estimation in optical metrology. Fisher information, which limits the precision of estimates under photon noise, becomes a matrix when multiple parameters are involved. Several scalar functions of this matrix are considered to find input modes that optimize precision across parameters simultaneously. Strategies are also developed to handle nuisance parameters that affect measurements but are not of interest. These ideas are tested numerically on a system of 2D coupled dipoles.

Core claim

Fisher information for coherent diffraction can be written in terms of the scattering matrix, and for multiple parameters this matrix can be optimized over the choice of input modes by maximizing scalar functions such as its trace or determinant, yielding input states that simultaneously improve precision for all parameters of interest while accounting for nuisance effects.

What carries the argument

The Fisher information matrix constructed from the scattering matrix of the optical system, which is maximized over input modes to define multiparameter maximum information states.

If this is right

  • Multiple parameters can be estimated with higher joint precision by choosing input light patterns that maximize functions of the Fisher matrix.
  • Nuisance parameters can be suppressed in the optimization without sacrificing precision on the parameters of interest.
  • The approach applies to any linear scattering system where the scattering matrix is known.
  • Photon-noise-limited measurements achieve the Cramér-Rao bound more closely with these optimized states.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the scattering matrix can be measured or modeled accurately, this method could guide experimental design in complex media like biological tissue.
  • Extending to dynamic systems might allow real-time adaptation of input modes for ongoing multiparameter tracking.

Load-bearing premise

The scattering matrix of the system is known and fixed, allowing computation of the Fisher information matrix for different inputs.

What would settle it

A numerical test in the 2D coupled dipole system where the variance of parameter estimates using the optimized input modes is compared to that using random modes, expecting lower variance for the optimized case.

Figures

Figures reproduced from arXiv: 2606.01445 by Allard Pieter Mosk, Bram Verreussel, Jacob Seifert.

Figure 1
Figure 1. Figure 1: Schematic of the setup. Wavefront synthesis represents a device that can [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Intensity plots of maximum information states. (a) is optimized for the [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Cramér-Rao Lower Bounds (CRLB) of our 3 different parameters expressed [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Cramér-Rao Lower Bounds (CRLB) for the two parameters of interest [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Bias as a function of nuisance parameter deviation. Bias is in units of [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
read the original abstract

In metrology, Fisher information is an important metric that quantifies the precision that can be achieved in a measurement. For optical measurements using coherent light it has been shown that Fisher information can be expressed simply using the scattering matrix of the system. Fisher information can be maximized over the input modes to achieve maximum information states, which produce optimally precise estimates for a parameter when the system is limited by photon noise. Here, we extend this approach to multiparameter estimation, in which case Fisher information takes the form of a matrix. We consider several scalar functions of the Fisher matrix to optimize the precision in multiple parameters at the same time. We also consider strategies for dealing with nuisance parameters, which can degrade the achievable precision of other parameters but are not of interest to measure. We corroborate our findings numerically using a scattering system of 2D coupled dipoles.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript extends single-parameter maximum information states to multiparameter estimation for coherent diffraction measurements. Fisher information is expressed as a matrix derived from the scattering matrix S; several scalar functions of this matrix are optimized over input modes to maximize simultaneous precision across multiple parameters, with additional strategies proposed for nuisance parameters. Numerical corroboration is provided via a 2D coupled-dipole scattering system.

Significance. If the algebraic extension holds, the work supplies a systematic route to input-mode optimization for multiparameter precision under photon-noise limits when S is known exactly. The numerical example verifies the matrix construction and scalar-function optimization under ideal conditions, which is a modest but useful check on the formalism.

major comments (2)
  1. [Abstract / numerical section] Abstract and numerical-results section: the stated numerical corroboration on 2D coupled dipoles supplies no error bars, exclusion criteria, or direct comparison showing that the chosen scalar functions of the Fisher matrix improve joint precision beyond single-parameter baselines; the central multiparameter claim therefore rests on unshown quantitative detail.
  2. [Numerical section] Numerical section: S is computed exactly from the model parameters with no added estimation noise, so the example confirms the algebra under perfect knowledge but does not test whether the reported precision gains survive when S itself must be recovered from finite noisy measurements—the assumption underlying practical use of the method.
minor comments (2)
  1. Define the scalar functions (trace, det, etc.) of the Fisher matrix with explicit equations in the main text rather than referring only to the single-parameter precursor.
  2. Clarify the treatment of nuisance parameters: state whether they are marginalized, projected out, or optimized jointly, and indicate which scalar function is used in each case.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments and positive recommendation. We respond to each major comment below.

read point-by-point responses

  1. Referee: [Abstract / numerical section] Abstract and numerical-results section: the stated numerical corroboration on 2D coupled dipoles supplies no error bars, exclusion criteria, or direct comparison showing that the chosen scalar functions of the Fisher matrix improve joint precision beyond single-parameter baselines; the central multiparameter claim therefore rests on unshown quantitative detail.

    Authors: We agree that the numerical section can be strengthened by the addition of direct comparisons to single-parameter baselines and error bars from repeated realizations. In the revised manuscript we will include these quantitative comparisons to demonstrate the joint-precision improvements. revision: yes

  2. Referee: [Numerical section] Numerical section: S is computed exactly from the model parameters with no added estimation noise, so the example confirms the algebra under perfect knowledge but does not test whether the reported precision gains survive when S itself must be recovered from finite noisy measurements—the assumption underlying practical use of the method.

    Authors: The numerical example is presented to verify the algebraic construction and scalar-function optimization assuming exact knowledge of S, consistent with the scope of the manuscript. Extending the test to noisy recovery of S would require separate estimation procedures that lie outside the present work; we will add an explicit statement of this scope limitation in the revised text. revision: partial

Circularity Check

0 steps flagged

Minor self-citation to single-parameter result; multiparameter extension algebraically independent with no reduction by construction

full rationale

The paper states that 'it has been shown' Fisher information can be expressed using the scattering matrix for the single-parameter case, then extends this to the multiparameter Fisher matrix by considering scalar functions (trace, det, etc.) and nuisance-parameter strategies. This prior result is referenced but not load-bearing for the new claims, which derive directly from the matrix extension without fitting, self-definition, or ansatz smuggling. Numerical checks with 2D dipoles use the exact model S and verify the algebra under known S, without circular reduction. No steps match the enumerated circularity patterns; the derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard definition of Fisher information for photon-noise-limited coherent measurements and on the availability of the scattering matrix; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption Fisher information quantifies the precision achievable in a parameter estimate from noisy measurements
    Invoked at the opening of the abstract as the starting metric for optical measurements.
  • domain assumption For coherent light the Fisher information can be expressed using the scattering matrix
    Stated as already shown for the single-parameter case and used as the basis for the multiparameter extension.

pith-pipeline@v0.9.1-grok · 5675 in / 1385 out tokens · 26871 ms · 2026-06-28T16:14:04.067667+00:00 · methodology

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