A study points to a possible information advantage from repeatedly using small entangled photon states in a sensing arm when some light is lost. In its modeled comparisons, a state called rotated NOON outperformed the conventional NOON state and BAT, and marginally outperformed flat, after the number of passes and beam-splitter reflectivity were optimized. The result is a model comparison, not a completed laboratory demonstration.
Here, phase metrology means estimating an optical phase. The study asks whether small, loss-robust entangled probes inside a multipass interferometer can perform better on that task under loss than conventional NOON-state interferometry and a single-photon multipass analogue. It also compares BAT, flat, rotated NOON and an optimized general state.
The score counted each particle pass
To compare different resources on the same footing, the study used quantum Fisher information per particle-pass, written . Quantum Fisher information, or QFI, is the upper benchmark for the phase information available from the quantum state; N is the particle number and p is the number of passes. The denominator treats particle-passes as the query budget, so the score asks how much information is obtained for each particle-sensing interaction.
Loss was modeled through imperfect cavity mirrors with reflectivity below one, and it was placed only in the sensing arm; the reference arm was treated as effectively lossless. The calculations assumed asymptotic estimation theory, meaning a sufficiently large number of independent measurements rather than a finite-data setting. These assumptions define the conditions behind the reported comparison.
The state design changed the result
In the lossy NOON calculation, particle number and pass number could be traded because they appear together as a product. At the displayed loss setting of 0.95, modeled NOON states with 1, 2 and 5 particles reached the same maximum QFI per particle-pass, although the smaller states needed more passes to reach it. In this comparison, the number of passes was therefore part of the resource calculation.
That pattern did not make all the state families equivalent. Flat and BAT achieved higher maximum QFI per particle-pass than NOON. After optimization over passes and beam-splitter reflectivity, rotated NOON outperformed NOON and BAT and marginally outperformed flat. The paper also draws a practical distinction: unlike flat, rotated NOON is described as creatable in a laboratory with current technology. The comparison remains model-based and depends on the stated loss and optimization settings.
An optimized general state was included as a benchmark, with coefficients selected for the largest peak QFI per particle-pass. At N = 4, it was presented as useful for comparison but generally impractical to prepare.
The measurement mattered too
The researchers then checked the ordering with a practical readout. The modeled setup sends light through a beam splitter and counts particles at the outputs, with the beam-splitter reflectivity adjustable. The optimization covered the local phase and output reflectivity and, for rotated NOON, the input reflectivity as well, with the calculation repeated for each pass number.
With that readout, the classical Fisher information ranked rotated NOON above BAT above NOON across the displayed loss values. Classical Fisher information describes the information associated with a chosen measurement, while QFI is the upper benchmark. The reported ordering therefore belongs to the specified photon-counting arrangement and displayed parameter range, rather than applying automatically to every possible measurement.
The proposed test is still ahead
For N = 4 and a loss setting of 0.95, the reported rotated-NOON operating point was 17 passes with beam-splitter reflectivity 0.83. It is a modeled optimum: the analysis reports no empirical validation, uncertainty interval or optimization-convergence information. The authors' conclusion is that the hybrid protocol is compatible with experimentally accessible four-photon states and simple photon counting, provided the number of passes can be controlled.
That conclusion leaves the central practical test open. Because the analysis assumes asymptotic estimation and places loss only in the sensing arm, it does not establish finite-data performance or show that the same ranking would hold under a different loss arrangement. An implementation with controlled pass number would be needed to test the predicted hierarchy.
The work is a preprint. Its front matter identifies an arXiv version 2 document dated 1 September 2026.
Paper data and sources
Original title: Enhanced quantum metrology with robust multipass interferometry
Authors: Sayak Mukherjee, José Afonso Oliveira, Sean William Moore, Jacob A. Dunningham
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text