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}
is the number of independent repetitions, and
F
Q
ϱ
,
H
{\displaystyle F_{\rm {Q}}[\varrho ,H]}
is the
quantum Fisher information .
[1]
[2]
Here,
ϱ
{\displaystyle \varrho }
is the
state of the system and
H
{\displaystyle H}
is the
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}}
is the initial state of the system,
θ
{\displaystyle \theta }
is the parameter to be estimated based on measurements on
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 }
can be estimated by measuring
A
.
{\displaystyle A.}
Moreover, the convex roof of the variance is given as
[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]}
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 . 72 (22). American Physical Society (APS): 3439–3443.
Bibcode :
1994PhRvL..72.3439B .
doi :
10.1103/physrevlett.72.3439 .
ISSN
0031-9007 .
PMID
10056200 .
^ 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 .
^ 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 .
^ 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 .
^ 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 .
^ Yu, Sixia (2013). "Quantum Fisher Information as the Convex Roof of Variance".
arXiv :
1302.5311 [
quant-ph ].