Weak convexity of Fisher information matrix and superresolved localization of blinking sources of light

Abstract

A group of techniques known by the general name of single-molecule localization microscopy reaches a nanometer-scale spatial resolution of point light emitters, well below the diffraction limit of the traditional microscopy. The key feature of these techniques is blinking, alternation of bright and dark states, of each emitter so that no more than one emitter is bright within the width of the point-spread function of the microscope during a time sufficient for its localization. We give a formulation of the optical part of these techniques in terms of quantum metrology, where the limit of precision is determined by the Fisher information on the emitters positions contained in the measurement data. We show that the advantage in resolution provided by making the emitters blink is a consequence of the fundamental property of Fisher information, its convexity. In particular, we prove the weak matrix convexity and the trace convexity of the Fisher information matrix -- two fundamental results in the multiparameter estimation theory. We show also that the advantage in information is inherent to the quantum state of light itself before the measurement and is related to the convexity of the quantum Fisher information matrix.

arXiv comment18 pages, 7 figures, final accepted version

References in corpus (20)

• Quantum metrology
• Advances in Photonic Quantum Sensing
• Quantum Theory of Superresolution for Two Incoherent Optical Point Sources
• A perspective on multiparameter quantum metrology: from theoretical tools to applications in quantum imaging
• Multiparameter Quantum Metrology of Incoherent Point Sources: Towards Realistic Superresolution
• Extremal properties of the variance and the quantum Fisher information
• Advances in quantum imaging
• Tutorial: Optical quantum metrology
• Subdiffraction incoherent optical imaging via spatial-mode demultiplexing
• Single spin stochastic optical reconstruction microscopy
• Quantum limits of localisation microscopy
• Uncertainty relations with the variance and the quantum Fisher information based on convex decompositions of density matrices
• Ultra-sensitive separation estimation of optical sources
• Quantum Teleportation Using Quantum Non-Demolition Technique
• Super-resolution Optical Fluctuation Imaging -- fundamental estimation theory perspective
• Correlated noise in Brownian motion allows for super resolution
• Limits on Spectral Resolution Measurements by Quantum Probes
• Optimal correlation order in super-resolution optical fluctuation microscopy
• Lost photon enhances superresolution
• Optimizing measurement tradeoffs in multiparameter spatial superresolution