Certifying the quantum Fisher information from a given set of mean values: a semidefinite programming approach

Guillem Müller-Rigat1, Anubhav Kumar Srivastava1, Stanisław Kurdziałek2, Grzegorz Rajchel-Mieldzioć1, Maciej Lewenstein1,3, and Irénée Frérot4,5

1ICFO - Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
2Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warszawa, Poland
3ICREA, Pg. Lluís Companys 23, 08010 Barcelona, Spain
4Univ Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38000 Grenoble, France
5Laboratoire Kastler Brossel, Sorbonne Université, CNRS, ENS-PSL Research University, Collège de France, 4 Place Jussieu, 75005 Paris, France

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We introduce a semidefinite programming algorithm to find the minimal quantum Fisher information compatible with an arbitrary dataset of mean values. This certification task allows one to quantify the resource content of a quantum system for metrology applications without complete knowledge of the quantum state. We implement the algorithm to study quantum spin ensembles. We first focus on Dicke states, where our findings challenge and complement previous results in the literature. We then investigate states generated during the one-axis twisting dynamics, where in particular we find that the metrological power of the so-called multi-headed cat states can be certified using simple collective spin observables, such as fourth-order moments for small systems, and parity measurements for arbitrary system sizes.

Quantum systems may be investigated from the perspective of the resource they represent in quantum metrology applications. This resource is quantified by the so-called quantum Fisher information (QFI). In this work, we introduce a mathematical technique to quantify the minimal QFI in a given metrology scenario, compatible with some given measured mean values. We show that some popular experiments on spin ensembles allow one to prepare very useful states for metrology, beyond what was previously envisioned.

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