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:
( Δ θ ) 2 ≥ 1 m F Q [ ϱ , H ] , {\displaystyle (\Delta \theta )^{2}\geq {\frac {1}{mF_{\rm {Q}}[\varrho ,H]}},}
where m {\displaystyle m} F Q [ ϱ , H ] {\displaystyle F_{\rm {Q}}[\varrho ,H]} quantum Fisher information .[1] [2]
Here, ϱ {\displaystyle \varrho } state of the system and H {\displaystyle H} Hamiltonian of the system. When considering a unitary dynamics of the type
ϱ ( θ ) = exp ( − i H θ ) ϱ 0 exp ( + i H θ ) , {\displaystyle \varrho (\theta )=\exp(-iH\theta )\varrho _{0}\exp(+iH\theta ),}
where ϱ 0 {\displaystyle \varrho _{0}} θ {\displaystyle \theta } ϱ ( θ ) . {\displaystyle \varrho (\theta ).}
Simple derivation from the Heisenberg uncertainty relation Let us consider the decomposition of the density matrix to pure components as
ϱ = ∑ k p k | Ψ k ⟩ ⟨ Ψ k | . {\displaystyle \varrho =\sum _{k}p_{k}\vert \Psi _{k}\rangle \langle \Psi _{k}\vert .}
The Heisenberg uncertainty relation is valid for all | Ψ k ⟩ {\displaystyle \vert \Psi _{k}\rangle }
( Δ A ) Ψ k 2 ( Δ B ) Ψ k 2 ≥ 1 4 | ⟨ i [ A , B ] ⟩ Ψ k | 2 . {\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]
( Δ θ ) A 2 ≥ 1 4 min { p k , Ψ k } [ ∑ k p k ( Δ B ) Ψ k 2 ] . {\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]
( Δ θ ) A 2 = ( Δ A ) 2 | ∂ θ ⟨ A ⟩ | 2 = ( Δ A ) 2 | ⟨ i [ A , B ] ⟩ | 2 {\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 θ {\displaystyle \theta } A . {\displaystyle A.} [5] [6]
min { p k , Ψ k } [ ∑ k p k ( Δ B ) Ψ k 2 ] = 1 4 F Q [ ϱ , B ] , {\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 F Q [ ϱ , B ] {\displaystyle F_{Q}[\varrho ,B]} quantum Fisher information .
References ![{\displaystyle (\Delta \theta )^{2}\geq {\frac {1}{mF_{\rm {Q}}[\varrho ,H]}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38bf713de6686a5333ef08df3a367bdca334ed09)
![{\displaystyle F_{\rm {Q}}[\varrho ,H]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c54f77b6fc319c1ca22ab43ef12c9eee14766ed)
