Quantum Cramér–Rao bound

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:

where is the number of independent repetitions, and is the quantum Fisher information.[1][2]

Here, is the state of the system and is the Hamiltonian of the system. When considering a unitary dynamics of the type

where is the initial state of the system, is the parameter to be estimated based on measurements on

Simple derivation from the Heisenberg uncertainty relation

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Let us consider the decomposition of the density matrix to pure components as

 

The Heisenberg uncertainty relation is valid for all  

 

From these, employing the Cauchy-Schwarz inequality we arrive at [3]

 

Here [4]

 

is the error propagation formula, which roughly tells us how well   can be estimated by measuring   Moreover, the convex roof of the variance is given as[5][6]

 

where   is the quantum Fisher information.

References

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  1. ^ Braunstein, 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.
  2. ^ 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.
  3. ^ Tóth, Géza; Fröwis, Florian (31 January 2022). "Uncertainty relations with the variance and the quantum Fisher information based on convex decompositions of density matrices". Physical Review Research. 4 (1): 013075. arXiv:2109.06893. Bibcode:2022PhRvR...4a3075T. doi:10.1103/PhysRevResearch.4.013075. S2CID 237513549.
  4. ^ Pezzè, Luca; Smerzi, Augusto; Oberthaler, Markus K.; Schmied, Roman; Treutlein, Philipp (5 September 2018). "Quantum metrology with nonclassical states of atomic ensembles". Reviews of Modern Physics. 90 (3): 035005. arXiv:1609.01609. Bibcode:2018RvMP...90c5005P. doi:10.1103/RevModPhys.90.035005. S2CID 119250709.
  5. ^ Tóth, Géza; Petz, Dénes (20 March 2013). "Extremal properties of the variance and the quantum Fisher information". Physical Review A. 87 (3): 032324. arXiv:1109.2831. Bibcode:2013PhRvA..87c2324T. doi:10.1103/PhysRevA.87.032324. S2CID 55088553.
  6. ^ Yu, Sixia (2013). "Quantum Fisher Information as the Convex Roof of Variance". arXiv:1302.5311 [quant-ph].