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: [math]\displaystyle{ (\Delta \theta)^2 \ge \frac 1 {m F_{\rm Q}[\varrho,H]}, }[/math]
where [math]\displaystyle{ m }[/math] is the number of independent repetitions, and [math]\displaystyle{ F_{\rm Q}[\varrho,H] }[/math] is the quantum Fisher information.[1][2]
Here, [math]\displaystyle{ \varrho }[/math] is the state of the system and [math]\displaystyle{ H }[/math] is the Hamiltonian of the system. When considering a unitary dynamics of the type
[math]\displaystyle{ \varrho(\theta)=\exp(-iH\theta)\varrho_0\exp(+iH\theta), }[/math]
where [math]\displaystyle{ \varrho_0 }[/math] is the initial state of the system, [math]\displaystyle{ \theta }[/math] is the parameter to be estimated based on measurements on [math]\displaystyle{ \varrho(\theta). }[/math]
Simple derivation from the Heisenberg uncertainty relation
Let us consider the decomposition of the density matrix to pure components as
[math]\displaystyle{ \varrho=\sum_k p_k \vert\Psi_k\rangle\langle\Psi_k\vert. }[/math]
The Heisenberg uncertainty relation is valid for all [math]\displaystyle{ \vert\Psi_k\rangle }[/math]
[math]\displaystyle{ (\Delta A)^2_{\Psi_k}(\Delta B)^2_{\Psi_k}\ge \frac 1 4 |\langle i[A,B] \rangle_{\Psi_k}|^2. }[/math]
From these, employing the Cauchy-Schwarz inequality we arrive at [3]
[math]\displaystyle{ (\Delta\theta)^2_A \ge \frac{1}{4\min_{\{p_k,\Psi_k\}}[\sum_k p_k (\Delta B)_{\Psi_k}^2]}. }[/math]
Here [4]
[math]\displaystyle{ (\Delta\theta)^2_A= \frac{(\Delta A)^2}{|\partial_{\theta}\langle A \rangle|^2}=\frac{(\Delta A)^2}{|\langle i[A,B] \rangle|^2} }[/math]
is the error propagation formula, which roughly tells us how well [math]\displaystyle{ \theta }[/math] can be estimated by measuring [math]\displaystyle{ A. }[/math] Moreover, the convex roof of the variance is given as[5][6]
[math]\displaystyle{ \min_{\{p_k,\Psi_k\}}\left[\sum_k p_k (\Delta B)_{\Psi_k}^2\right]=\frac1 4 F_Q[\varrho, B], }[/math]
where [math]\displaystyle{ F_Q[\varrho, B] }[/math] is the quantum Fisher information.
References
- ↑ Braunstein, Samuel L.; Caves, Carlton M. (1994-05-30). "Statistical distance and the geometry of quantum states". Physical Review Letters (American Physical Society (APS)) 72 (22): 3439–3443. doi:10.1103/physrevlett.72.3439. ISSN 0031-9007. PMID 10056200. Bibcode: 1994PhRvL..72.3439B.
- ↑ 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. doi:10.1006/aphy.1996.0040. Bibcode: 1996AnPhy.247..135B.
- ↑ 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. doi:10.1103/PhysRevResearch.4.013075. Bibcode: 2022PhRvR...4a3075T.
- ↑ 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. doi:10.1103/RevModPhys.90.035005. Bibcode: 2018RvMP...90c5005P.
- ↑ Tóth, Géza; Petz, Dénes (20 March 2013). "Extremal properties of the variance and the quantum Fisher information". Physical Review A 87 (3): 032324. doi:10.1103/PhysRevA.87.032324. Bibcode: 2013PhRvA..87c2324T.
- ↑ Yu, Sixia (2013). "Quantum Fisher Information as the Convex Roof of Variance". arXiv:1302.5311 [quant-ph].
 |  |
