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[6] [7] It is also known as Fréchet-Cramér–Rao or Fréchet-Darmois-Cramér-Rao lower bound. It states that the precision of any unbiased estimator is at most the Fisher information ; or (equivalently) the reciprocal of the Fisher information is a lower bound on its variance .
The Cramér–Rao bound [9] [10] states that the inverse of the Fisher information is a lower bound on the variance of any unbiased estimator of θ. Van Trees (1968) and Frieden (2004) provide the following method of deriving the Cramér–Rao bound, a result which describes use of the Fisher information.
Equivalently, the estimator achieves equality in the Cramér–Rao inequality for all θ. The Cramér–Rao lower bound is a lower bound of the variance of an unbiased estimator, representing the "best" an unbiased estimator can be. An efficient estimator is also the minimum variance unbiased estimator (MVUE). This is because an efficient ...
Hence, a lower bound on the entanglement depth is obtained as F Q [ ϱ , J z ] N ≤ k . {\displaystyle {\frac {F_{\rm {Q}}[\varrho ,J_{z}]}{N}}\leq k.} A related concept is the quantum metrological gain , which for a given Hamiltonian is defined as the ratio of the quantum Fisher information of a state and the maximum of the quantum Fisher ...
The quantum Cramér–Rao bound is the quantum analogue of the classical Cramér–Rao bound. It bounds the achievable precision in parameter estimation with a quantum system: It bounds the achievable precision in parameter estimation with a quantum system:
In information theory and statistics, Kullback's inequality is a lower bound on the Kullback–Leibler divergence expressed in terms of the large deviations rate function. [1]
The Brascamp–Lieb inequality is also related to the Cramér–Rao bound. [9] While Brascamp–Lieb is an upper-bound, the Cramér–Rao bound lower-bounds the variance of (()). The Cramér–Rao bound states
A great many important inequalities in information theory are actually lower bounds for the Kullback–Leibler divergence.Even the Shannon-type inequalities can be considered part of this category, since the interaction information can be expressed as the Kullback–Leibler divergence of the joint distribution with respect to the product of the marginals, and thus these inequalities can be ...