The quantum Fisher information is a central quantity in quantum metrology and is the quantum analogue of the classical Fisher information.[1][2][3][4][5] It is one of the central quantities used to qualify the utility of an input state, especially in Mach–Zehnder (or, equivalently, Ramsey) interferometer-based phase or parameter estimation.[1][3][6] It is shown that the quantum Fisher information can also be a sensitive probe of a quantum phase transition (e.g. recognizing the superradiant quantum phase transition in the Dicke model[6]). The quantum Fisher information of a state with respect to the observable is defined as
where and are the eigenvalues and eigenvectors of the density matrix respectively, and the summation goes over all and such that .
When the observable generates a unitary transformation of the system with a parameter from initial state ,
the quantum Fisher information constrains the achievable precision in statistical estimation of the parameter via the quantum Cramér–Rao bound as
where is the number of independent repetitions.
It is often desirable to estimate the magnitude of an unknown parameter that controls the strength of a system's Hamiltonian with respect to a known observable during a known dynamical time . In this case, defining , so that , means estimates of can be directly translated into estimates of .
^ abHelstrom, C (1976). Quantum detection and estimation theory. Academic Press. ISBN 0123400503.
^Holevo, Alexander S (1982). Probabilistic and statistical aspects of quantum theory (2nd English ed.). Scuola Normale Superiore. ISBN 978-88-7642-378-9.
^ abBraunstein, Samuel L.; Caves, Carlton M. (1994-05-30). "Statistical distance and the geometry of quantum states". Physical Review Letters. 72 (22). American Physical Society (APS): 3439–3443. Bibcode:1994PhRvL..72.3439B. doi:10.1103/physrevlett.72.3439. ISSN 0031-9007. PMID 10056200.
^Braunstein, Samuel L.; Caves, Carlton M.; Milburn, G.J. (April 1996). "Generalized Uncertainty Relations: Theory, Examples, and Lorentz Invariance". Annals of Physics. 247 (1): 135–173. arXiv:quant-ph/9507004. Bibcode:1996AnPhy.247..135B. doi:10.1006/aphy.1996.0040. S2CID 358923.
^Paris, Matteo G. A. (21 November 2011). "Quantum Estimation for Quantum Technology". International Journal of Quantum Information. 07 (supp01): 125–137. arXiv:0804.2981. doi:10.1142/S0219749909004839. S2CID 2365312.
^ abWang, Teng-Long; Wu, Ling-Na; Yang, Wen; Jin, Guang-Ri; Lambert, Neill; Nori, Franco (2014-06-17). "Quantum Fisher information as a signature of the superradiant quantum phase transition". New Journal of Physics. 16 (6): 063039. arXiv:1312.1426. Bibcode:2014NJPh...16f3039W. doi:10.1088/1367-2630/16/6/063039. ISSN 1367-2630.
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Global Information![{\displaystyle F_{\rm {Q}}[\varrho ,A]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b69f7312fc239501ff1042cedcacc8fdc1fb4cc6)
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