( Quantum) FISHER INFORMATION FOCUS
regarding a parameter of interest, such as particle location, in a given measurement scheme( see info box). The quantum Fisher information( QFI), defined as the maximum FI over all possible measurement schemes, then sets the more fundamental Quantum Cramér- Rao bound( QCRB) on estimation precision. It depends only on the properties of the probe light and the specimen. The FI and the QFI can thus guide the development of even more precise localization and tracking schemes.
Here, we discuss recent advances in the estimation-theoretical analysis of localization techn < iques. We start with the textbook example of localizing an elastically scattering particle in the near and the far field [ 3 ]. The much simpler far-field treatment gives rise to the intuitive picture of Fisher information flux [ 4 ]. Based on this framework, we assess the localization precision in interferometric scattering
( iScat), dark-field [ 5 ], and fluorescence microscopy [ 2,6,7 ]. Finally, we discuss MINFLUX microscopy [ 8,9 ], which increases the information content per detected photon using adaptive measurements, prior knowledge, and tailored probe beams [ 10 ].
RESULTS When illuminating a small non-absorptive particle with a plane wave, the light field induces an oscillating dipole that elastically scatters photons, which carry information about the particle ' s location( Fig. 1A). A quantum description of this dynamic interaction predicts transient peaks of the QFI in the near field [ 3 ]. The QFI oscillates at the optical frequency, indicating that information flows back and forth between the field and the internal degrees of freedom of the particle. The peaks can surpass the QFI accessible in the far field by orders of magnitude( Fig. 1B). It remains to be explored
Microscopy can be understood as a parameter estimation process [ 1,2 ]: First, a probe state of light ρ 0 is generated. This state could be as simple as a plane wave, have structured amplitude and phase, or involve entanglement or squeezing. Second, the probe state interacts with the sample, which encodes information about a parameter of interest θ into the probe state ρ 0 → ρ( θ). Third, a measurement is performed on ρ( θ). Finally, one estimates the parameter from the outcome. As measurements are noisy, the estimate varies randomly and deviates from the true value θ. In the absence of bias, the standard deviation σ quantifies the error of the estimate and obeys the CRB σ 2 ≥ FI – 1, where the FI quantifies the sensitivity of the measurement outcome distribution to deviations of θ. An estimator that saturates the CRB is called efficient. In the limit of many measurements, maximum-likelihood estimation is both unbiased and efficient. The QFI quantifies the sensitivity of ρ( θ) to θ-deviations and upper-bounds the FI for any measurement, which implies the QCRB σ 2 ≥ QFI – 1. Both the FI and the QFI grow linearly with the number of independent measurements, which for classical light equals the number of detected photons N d – hence the shot noise limit σ 2 ∙ N d
– 1 in efficient and optimal measurements.
Photoniques 131 I www. photoniques. com 59