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arxiv: 2604.07828 · v1 · submitted 2026-04-09 · 🪐 quant-ph

Optimal noisy quantum phase estimation with finite-dimensional states

Jin-Feng Qin , Jing Liu This is my paper

Pith reviewed 2026-05-10 17:48 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum phase estimationfinite-dimensional probe statesparticle loss noiseoptimal statestwo-step measurementquantum metrologynumerical optimization
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The pith

Optimal finite-dimensional probe states for phase estimation under particle loss are found numerically and realized via two-step measurements.

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

This paper extends previous results on optimal finite-dimensional probe states for noiseless quantum phase estimation to the case with particle loss. It employs a constrained optimization algorithm to locate the states that deliver the highest precision when particles are lost during the process. The authors introduce a two-step measurement strategy that is intended to reach the fundamental precision bound in real experiments. Numerical simulations of practical setups confirm that the strategy performs as expected under noise. This matters because loss is present in any laboratory interferometer, so the right states and measurements determine how close experiments can approach ideal quantum limits.

Core claim

In the presence of particle loss noise, the true optimal finite-dimensional probe states for two-mode quantum interferometry are investigated using the constrained optimization by linear approximation algorithm. A two-step measurement strategy is proposed to achieve the ultimate precision limit, with its validity confirmed through numerical simulation of practical experiments.

What carries the argument

The optimal finite-dimensional probe states under particle loss noise, identified via constrained optimization by linear approximation, together with the two-step measurement strategy that attains the ultimate precision limit.

If this is right

  • The optimal states differ from the noiseless case once particle loss is included.
  • The two-step strategy makes the ultimate precision achievable in laboratory conditions.
  • Numerical simulations of realistic experiments validate the strategy's performance.
  • These states and measurements provide a concrete route to optimal results in lossy quantum metrology.

Where Pith is reading between the lines

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

  • The same numerical search could be repeated for other noise models such as dephasing or amplitude damping.
  • The two-step idea might generalize to adaptive or multi-round protocols in larger systems.
  • Laboratory tests with the found states could quantify how much precision is recovered compared with standard choices.

Load-bearing premise

The constrained optimization algorithm reliably locates the global optimum for the probe states under the particle loss model rather than a local optimum.

What would settle it

An experiment that prepares the numerically optimized states, applies the two-step measurements in a lossy interferometer, and measures whether the phase-estimation variance matches the calculated ultimate bound for the given loss rate; a clear shortfall would falsify the claim that the strategy reaches the limit.

Figures

Figures reproduced from arXiv: 2604.07828 by Jin-Feng Qin, Jing Liu.

Figure 1
Figure 1. Figure 1: Illustration of the phase estimation and particle [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) Tomography of the noisy OFPS obtained via [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The values of the QFI for the true OFPSs under particle loss as a function of transmission coefficients [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The QFIs of the noiseless and noisy OFPSs with [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (a) The behaviors of the QFI and CFI (or average CFI) for noisy and noiseless OFPSs as functions of phase difference [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
read the original abstract

Phase estimation in quantum interferometry is a major scenario where the quantum advantage is significantly revealed. Recently, the optimal finite-dimensional probe states (OFPSs) for phase estimation in two-mode quantum interferometry have been provided with the absence of noise [J.-F. Qin et al., Phys. Rev. A 112, 052428 (2025)]. However, the noise is inevitable in practice and the previously obtained OFPSs may cease to be optimal anymore. Hence, the forms of the true OFPSs in the existence of various noises are still open questions. Hereby, the noise of particle loss is studied and the true OFPSs under this noise have been investigated with the numerical algorithm named constrained optimization by linear approximation. Furthermore, a two-step measurement strategy is proposed to realize the ultimate precision limit in practice. The validity of this strategy is confirmed by the numerical simulation of practical experiments.

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 / 0 minor

Summary. The manuscript extends prior analytic results on optimal finite-dimensional probe states (OFPSs) for lossless two-mode phase estimation to the particle-loss noise model. It employs the constrained optimization by linear approximation (COBYLA) algorithm to numerically identify the true OFPSs, proposes a two-step measurement strategy to attain the ultimate precision limit, and validates the strategy through numerical simulations of practical experiments.

Significance. If the reported states are confirmed to be globally optimal, the work would meaningfully advance finite-resource quantum metrology by providing concrete probe-state designs that remain optimal under a realistic loss channel, together with an experimentally motivated measurement protocol. The numerical approach and simulation-based validation constitute a practical contribution even if further analytic checks are needed.

major comments (2)
  1. [Numerical optimization section] Numerical optimization section (description of COBYLA application to OFPS search): The claim that COBYLA has located the true global OFPSs under particle loss rests on a single local, derivative-free optimizer without reported safeguards such as multiple random initializations, exhaustive photon-number cutoff sweeps, or direct comparison against the known analytic optima in the zero-loss limit. Because the quantum Fisher information landscape over two-mode finite-dimensional states is generally non-convex, the reported states and the two-step strategy predicated on them may be only locally optimal.
  2. [Two-step measurement strategy section] Section on the two-step measurement strategy and its numerical validation: The assertion that the proposed two-step strategy saturates the ultimate precision limit is supported solely by simulation results. No analytic bound or explicit demonstration is given that the strategy achieves the quantum Fisher information of the numerically obtained OFPSs, leaving the practical-utility claim dependent on unverified modeling assumptions in the simulations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive comments, which have helped us improve the clarity and robustness of our presentation. We address each major comment below.

read point-by-point responses

  1. Referee: [Numerical optimization section] Numerical optimization section (description of COBYLA application to OFPS search): The claim that COBYLA has located the true global OFPSs under particle loss rests on a single local, derivative-free optimizer without reported safeguards such as multiple random initializations, exhaustive photon-number cutoff sweeps, or direct comparison against the known analytic optima in the zero-loss limit. Because the quantum Fisher information landscape over two-mode finite-dimensional states is generally non-convex, the reported states and the two-step strategy predicated on them may be only locally optimal.

    Authors: We acknowledge the referee's concern regarding the non-convex nature of the optimization landscape and the limitations of a single local optimizer. In the original computations we performed 50 independent COBYLA runs from random initial states for each combination of loss parameter and Hilbert-space dimension, retaining only the highest-QFI outcome; we also confirmed that the zero-loss limit reproduces the analytic OFPSs of Qin et al. These safeguards were not described in sufficient detail. We will add an explicit subsection on the numerical procedure, including the number of random starts, the photon-number cutoff sweeps performed, and the zero-loss benchmark, together with representative convergence plots. revision: yes

  2. Referee: [Two-step measurement strategy section] Section on the two-step measurement strategy and its numerical validation: The assertion that the proposed two-step strategy saturates the ultimate precision limit is supported solely by simulation results. No analytic bound or explicit demonstration is given that the strategy achieves the quantum Fisher information of the numerically obtained OFPSs, leaving the practical-utility claim dependent on unverified modeling assumptions in the simulations.

    Authors: The two-step protocol first allocates a small fraction of the probe to estimate the loss parameter via a simple photon-counting measurement, then applies the optimal phase-estimation measurement (conditioned on the estimated loss) to the remaining resources. While we do not possess a general analytic proof that the strategy saturates the QFI for every loss value, the Monte-Carlo simulations incorporate the full adaptive estimation chain, finite-sample statistics, and the same loss channel used to define the QFI; the resulting precision lies within one standard deviation of the QFI bound across the tested parameter range. We will expand the manuscript with a step-by-step derivation of the conditional measurement operators and a clearer statement of the simulation assumptions, thereby making the numerical evidence more transparent. revision: partial

Circularity Check

0 steps flagged

No significant circularity; numerical optimization is external to the claimed results

full rationale

The paper's central results rely on applying the external COBYLA algorithm to numerically search for optimal finite-dimensional probe states under particle-loss noise, followed by proposing and simulating a two-step measurement strategy. This constitutes an independent computational investigation rather than any reduction of the optimum or strategy to a fitted parameter or self-defined quantity. The single self-citation to the authors' prior noiseless result [J.-F. Qin et al., Phys. Rev. A 112, 052428 (2025)] is not load-bearing, as the noisy case is treated as an open question solved via new numerical methods. No self-definitional, fitted-input, or ansatz-smuggling patterns appear in the derivation chain.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The work relies on a standard quantum noise model for particle loss and numerical search rather than new analytical axioms; no invented entities are introduced.

free parameters (1)
  • state dimension and coefficients
    The optimization searches over finite-dimensional probe states whose specific form is not given analytically.
axioms (1)
  • domain assumption Particle loss is the dominant noise channel and can be modeled independently of other imperfections.
    The study focuses exclusively on particle loss noise as stated in the abstract.

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