Physics-inspired forms of the Bayesian Cramér-Rao bound

Abstract

Using differential geometry, I derive a form of the Bayesian Cramér-Rao bound that remains invariant under reparametrization. With the invariant formulation at hand, I find the optimal and naturally invariant bound among the Gill-Levit family of bounds. By assuming that the prior probability density is the square of a wavefunction, I also express the bounds in terms of functionals that are quadratic with respect to the wavefunction and its gradient. The problem of finding an unfavorable prior to tighten the bound for minimax estimation is shown, in a special case, to be equivalent to finding the ground state of a Schrödinger equation, with the Fisher information playing the role of the potential. To illustrate the theory, two quantum estimation problems, namely, optomechanical waveform estimation and subdiffraction incoherent optical imaging, are discussed.

13 pages, 3 figures. v4: added generalization for a vectoral parameter of interest in the Appendix. Published

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Cited by in corpus (4)

• Estimation of Gaussian random displacement using non-Gaussian states
• Quantum limit to subdiffraction incoherent optical imaging. II. A parametric-submodel approach
• Optimizing one-axis twists for variational Bayesian quantum metrology
• Mutual Information Bounded by Fisher Information