Quantum Cramér–Rao bound explained

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:

(\Delta \theta) ^{2} \ge (1)/(m F _{\rm} [\varrho,H]) ,

where

m is the number of independent repetitions, and F _{\rm} [\varrho,H] is the quantum Fisher information.^{[1] [2]}

Here,

\varrho is the state of the system and H is the Hamiltonian of the system. When considering a unitary dynamics of the type\varrho(\theta) = \exp( - iH\theta)\varrho _{0\exp(+iH\theta),}

where

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

Simple derivation from the Heisenberg uncertainty relation

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

\varrho = \sum _{k} p _{k} \vert\Psi _{k\rangle\langle\Psi} _{k\vert.}

The Heisenberg uncertainty relation is valid for all

\vert\Psi _{k\rangle} (\Delta A) ^{2} _{\Psi _{k}} (\Delta B) ^{2} _{\Psi _{k}} \ge (1)/(4) |\langle i[A,B] \rangle _{\Psi _{k}} | ^{2.}

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

(\Delta\theta) ^{2} _{A} \ge (1)/(4min _{\{p _{k,\Psi} _{k\}}) [\sum _{k} p _{k} (\Delta B) ^{2 ]}.} _{\Psi _{k}}

Here ^[4]

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

is the error propagation formula, which roughly tells us how well

\theta can be estimated by measuring A. Moreover, the convex roof of the variance is given as^{[5] [6]} min _{\{p _{k,\Psi} _{k\}} }\left[\sum_k p_k (\Delta B)_{\Psi_k}^2\right]=\frac1 4 F_Q[\varrho, B],

where

F _{Q[\varrho,} B] is the quantum Fisher information.

Notes and References

1. Braunstein . Samuel L. . Caves . Carlton M. . Carlton Caves . Statistical distance and the geometry of quantum states . Physical Review Letters . American Physical Society (APS) . 72 . 22 . 1994-05-30 . 0031-9007 . 10.1103/physrevlett.72.3439 . 10056200 . 3439–3443. 1994PhRvL..72.3439B .
2. Braunstein . Samuel L. . Caves . Carlton M. . Milburn . G.J. . Carlton Caves . Generalized Uncertainty Relations: Theory, Examples, and Lorentz Invariance . Annals of Physics . April 1996 . 247 . 1 . 135–173 . 10.1006/aphy.1996.0040. quant-ph/9507004 . 1996AnPhy.247..135B . 358923 .
3. Tóth . Géza . Fröwis . Florian . Uncertainty relations with the variance and the quantum Fisher information based on convex decompositions of density matrices . Physical Review Research . 31 January 2022 . 4 . 1 . 013075 . 10.1103/PhysRevResearch.4.013075. 2109.06893 . 2022PhRvR...4a3075T . 237513549 .
4. Pezzè . Luca . Smerzi . Augusto . Oberthaler . Markus K. . Schmied . Roman . Treutlein . Philipp . Quantum metrology with nonclassical states of atomic ensembles . Reviews of Modern Physics . 5 September 2018 . 90 . 3 . 035005 . 10.1103/RevModPhys.90.035005. 1609.01609 . 2018RvMP...90c5005P . 119250709 .
5. Tóth . Géza . Petz . Dénes . Extremal properties of the variance and the quantum Fisher information . Physical Review A . 20 March 2013 . 87 . 3 . 032324 . 10.1103/PhysRevA.87.032324. 2013PhRvA..87c2324T . 1109.2831 . 55088553 .
6. Yu . Sixia . Quantum Fisher Information as the Convex Roof of Variance . 2013 . 1302.5311. quant-ph .