Abstract
It is well known that the Cramér–Rao inequality places a lower bound for quantum Fisher information in terms of the variance of any quantum measurement. We establish an upper bound for quantum Fisher information of a parameterized family of density operators in terms of the variance of the generator. These two bounds together yield a generalization of the Heisenberg uncertainty relations from statistical estimation perspective.
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Luo, S. Quantum Fisher Information and Uncertainty Relations. Letters in Mathematical Physics 53, 243–251 (2000). https://doi.org/10.1023/A:1011080128419
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DOI: https://doi.org/10.1023/A:1011080128419