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:

${\displaystyle (\Delta \theta )^{2}\geq {\frac {1}{mF_{\rm {Q}}[\varrho ,H]}},}$

where ${\displaystyle m}$ is the number of independent repetitions, and ${\displaystyle F_{\rm {Q}}[\varrho ,H]}$ is the quantum Fisher information.^[1][2]

Here, ${\displaystyle \varrho }$ is the state of the system and ${\displaystyle H}$ is the Hamiltonian of the system. When considering a unitary dynamics of the type

${\displaystyle \varrho (\theta )=\exp(-iH\theta )\varrho _{0}\exp(+iH\theta ),}$

where ${\displaystyle \varrho _{0}}$ is the initial state of the system, ${\displaystyle \theta }$ is the parameter to be estimated based on measurements on ${\displaystyle \varrho (\theta ).}$

Simple derivation from the Heisenberg uncertainty relation

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

${\displaystyle \varrho =\sum _{k}p_{k}\vert \Psi _{k}\rangle \langle \Psi _{k}\vert .}$

The Heisenberg uncertainty relation is valid for all ${\displaystyle \vert \Psi _{k}\rangle }$

${\displaystyle (\Delta A)_{\Psi _{k}}^{2}(\Delta B)_{\Psi _{k}}^{2}\geq {\frac {1}{4}}|\langle i[A,B]\rangle _{\Psi _{k}}|^{2}.}$

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

${\displaystyle (\Delta \theta )_{A}^{2}\geq {\frac {1}{4\min _{\{p_{k},\Psi _{k}\}}[\sum _{k}p_{k}(\Delta B)_{\Psi _{k}}^{2}]}}.}$

Here ^[4]

${\displaystyle (\Delta \theta )_{A}^{2}={\frac {(\Delta A)^{2}}{|\partial _{\theta }\langle A\rangle |^{2}}}={\frac {(\Delta A)^{2}}{|\langle i[A,B]\rangle |^{2}}}}$

is the error propagation formula, which roughly tells us how well ${\displaystyle \theta }$ can be estimated by measuring ${\displaystyle A.}$ Moreover, the convex roof of the variance is given as^[5][6]

${\displaystyle \min _{\{p_{k},\Psi _{k}\}}\left[\sum _{k}p_{k}(\Delta B)_{\Psi _{k}}^{2}\right]={\frac {1}{4}}F_{Q}[\varrho ,B],}$

where ${\displaystyle F_{Q}[\varrho ,B]}$ is the quantum Fisher information.

References

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].