Квантовое неравенство Крамера — Рао — неравенство для нижней границы для среднеквадратической ошибки в квантовой теории оценивания , аналогичное неравенству Крамера — Рао в классической теории оценивания.
Рассмотрим квантовое оценивание оператора плотности
ρ
(
θ
)
{\displaystyle \rho (\theta )}
d
Π
(
θ
^
)
{\displaystyle d\Pi ({\hat {\theta }})}
θ
^
{\displaystyle {\hat {\theta }}}
q
(
θ
^
|
θ
)
=
q
(
θ
^
|
θ
)
d
m
θ
=
Tr
[
ρ
(
θ
)
d
Π
(
θ
^
)
]
{\displaystyle q({\hat {\theta }}|\theta )=q({\hat {\theta }}|\theta )d^{m}\theta =\operatorname {Tr} [\rho (\theta )d\Pi ({\hat {\theta }})]}
θ
¯
j
=
E
(
θ
^
j
|
θ
)
=
∫
Θ
θ
^
j
Tr
[
ρ
(
θ
)
d
Π
(
θ
^
)
]
{\displaystyle {\bar {\theta }}_{j}=E({\hat {\theta }}_{j}|\theta )=\int _{\Theta }{\hat {\theta }}_{j}\,\operatorname {Tr} [\rho (\theta )d\Pi ({\hat {\theta }})]}
Tr
{\displaystyle \operatorname {Tr} }
θ
¯
j
=
E
(
θ
^
j
|
θ
)
=
θ
j
{\displaystyle {\bar {\theta }}_{j}=E({\hat {\theta }}_{j}|\theta )=\theta _{j}}
B
i
j
{\displaystyle B_{ij}}
B
i
j
=
E
[
(
θ
^
i
−
θ
¯
i
)
(
θ
^
j
−
θ
¯
j
)
|
θ
]
=
∫
Θ
(
θ
^
i
−
θ
¯
i
)
(
θ
^
j
−
θ
¯
j
)
Tr
ρ
(
θ
)
d
Π
(
θ
^
)
{\displaystyle B_{ij}=E[({\hat {\theta }}_{i}-{\bar {\theta }}_{i})({\hat {\theta }}_{j}-{\bar {\theta }}_{j})|\theta ]=\int _{\Theta }({\hat {\theta }}_{i}-{\bar {\theta }}_{i})({\hat {\theta }}_{j}-{\bar {\theta }}_{j})\operatorname {Tr} \rho (\theta )\,d\Pi ({\hat {\theta }})}
C
¯
=
∑
i
=
1
m
∑
j
=
1
m
g
i
j
B
i
j
=
tr
G
B
{\displaystyle {\bar {C}}=\sum _{i=1}^{m}\sum _{j=1}^{m}g_{ij}B_{ij}=\operatorname {tr} GB}
tr
{\displaystyle \operatorname {tr} }
[ 1]
Первая форма квантового неравенства Крамера-Рао[ 2]
Y
~
B
Y
⩾
Y
~
A
−
1
Y
{\displaystyle {\tilde {Y}}BY\geqslant {\tilde {Y}}A^{-1}Y}
Вторая форма квантового неравенства Крамера-Рао[ 2]
Z
~
B
−
1
Z
⩾
Z
~
A
Z
{\displaystyle {\tilde {Z}}B^{-1}Z\geqslant {\tilde {Z}}AZ}
Здесь
A
i
j
=
Tr
(
∂
ρ
∂
θ
i
L
j
)
{\displaystyle A_{ij}=\operatorname {Tr} \left({\frac {\partial \rho }{\partial \theta _{i}}}L_{j}\right)}
L
k
{\displaystyle L_{k}}
∂
ρ
∂
θ
k
=
1
2
(
ρ
L
k
+
L
k
ρ
)
{\displaystyle {\frac {\partial \rho }{\partial \theta _{k}}}={\frac {1}{2}}(\rho L_{k}+L_{k}\rho )}
Y
,
Z
{\displaystyle Y,Z}
Re
Tr
(
∫
Θ
T
θ
ρ
T
L
d
Π
(
θ
^
)
)
=
Y
~
Z
{\displaystyle \operatorname {Re} \operatorname {Tr} \left(\int _{\Theta }T_{\theta }\rho T_{L}\,d\Pi ({\hat {\theta }})\right)={\tilde {Y}}Z}
T
θ
=
1
∑
j
=
1
m
y
j
(
θ
^
j
−
θ
j
)
{\displaystyle T_{\theta }=1\sum _{j=1}^{m}y_{j}({\hat {\theta }}_{j}-\theta _{j})}
T
L
=
∑
k
=
1
L
z
k
L
k
{\displaystyle T_{L}=\sum _{k=1}^{L}z_{k}L{k}}
Хелстром К. Квантовая теория проверки гипотез и оценивания. — М. : Мир, 1979. — 344 с.
![{\displaystyle q({\hat {\theta }}|\theta )=q({\hat {\theta }}|\theta )d^{m}\theta =\operatorname {Tr} [\rho (\theta )d\Pi ({\hat {\theta }})]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7032fa786934a2da1b3baeeef41d91b4350fee1)
![{\displaystyle {\bar {\theta }}_{j}=E({\hat {\theta }}_{j}|\theta )=\int _{\Theta }{\hat {\theta }}_{j}\,\operatorname {Tr} [\rho (\theta )d\Pi ({\hat {\theta }})]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a02492bff19c9107907b991e00d530df1e7ef7e)
![{\displaystyle B_{ij}=E[({\hat {\theta }}_{i}-{\bar {\theta }}_{i})({\hat {\theta }}_{j}-{\bar {\theta }}_{j})|\theta ]=\int _{\Theta }({\hat {\theta }}_{i}-{\bar {\theta }}_{i})({\hat {\theta }}_{j}-{\bar {\theta }}_{j})\operatorname {Tr} \rho (\theta )\,d\Pi ({\hat {\theta }})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c75777dba867292cf33c95452936bfce35a1d7ca)
