Quantum Cramer - Rao Inequality

The quantum Cramer – Rao inequality is the lower bound inequality for the standard error in quantum estimation theory , similar to the Cramer – Rao inequality in classical estimation theory.

Wording

Consider the quantum estimation of the density operator $\rho (\theta )$ ${\ displaystyle \ rho (\ theta)}$ using a probability operator measure $d\Pi ({\hat {\theta }})$ ${\ displaystyle d \ Pi ({\ hat {\ theta}})}$ giving an assessment ${\hat {\theta }}$ ${\ displaystyle {\ hat {\ theta}}}$ The posterior density probability distribution of a quantum estimate can be calculated as $q({\hat {\theta }}|\theta )=q({\hat {\theta }}|\theta )d^{m}\theta =\operatorname {Tr} [\rho (\theta )d\Pi ({\hat {\theta }})]$ ${\ displaystyle q ({\ hat {\ theta}} | \ theta) = q ({\ hat {\ theta}} | \ theta) d ^ {m} \ theta = \ operatorname {Tr} [\ rho (\ theta) d \ Pi ({\ hat {\ theta}})]}$ . The mathematical expectations of quantum estimates are obtained in the form ${\bar {\theta }}_{j}=E({\hat {\theta }}_{j}|\theta )=\int _{\Theta }{\hat {\theta }}_{j}\,\operatorname {Tr} [\rho (\theta )d\Pi ({\hat {\theta }})]$ ${\ displaystyle {\ bar {\ theta}} _ {j} = E ({\ hat {\ theta}} _ {j} | \ theta) = \ int _ {\ Theta} {\ hat {\ theta}} _ {j} \, \ operatorname {Tr} [\ rho (\ theta) d \ Pi ({\ hat {\ theta}})]}$ . Here $\operatorname {Tr}$ ${\ displaystyle \ operatorname {Tr}}$ Is the trace of an operator in a Hilbert space. Consider unbiased estimates, that is, estimates for which the identity holds: ${\bar {\theta }}_{j}=E({\hat {\theta }}_{j}|\theta )=\theta _{j}$ ${\ displaystyle {\ bar {\ theta}} _ {j} = E ({\ hat {\ theta}} _ {j} | \ theta) = \ theta _ {j}}$ . Covariance of unbiased estimates $B_{ij}$ ${\ displaystyle B_ {ij}}$ are given by: $B_{ij}=E[({\hat {\theta }}_{i}-{\bar {\theta }}_{i})({\hat {\theta }}_{j}-{\bar {\theta }}_{j})|\theta ]=\int _{\Theta }({\hat {\theta }}_{i}-{\bar {\theta }}_{i})({\hat {\theta }}_{j}-{\bar {\theta }}_{j})\operatorname {Tr} \rho (\theta )\,d\Pi ({\hat {\theta }})$ ${\ displaystyle B_ {ij} = E [({\ hat {\ theta}} _ {i} - {\ bar {\ theta}} _ {i}) ({\ hat {\ theta}} _ {j} - {\ bar {\ theta}} _ {j}) | \ theta] = \ int _ {\ Theta} ({\ hat {\ theta}} _ {i} - {\ bar {\ theta}} _ { i}) ({\ hat {\ theta}} _ {j} - {\ bar {\ theta}} _ {j}) \ operatorname {Tr} \ rho (\ theta) \, d \ Pi ({\ hat {\ theta}})}$ . With a quadratic loss function, the average risk is ${\bar {C}}=\sum _{i=1}^{m}\sum _{j=1}^{m}g_{ij}B_{ij}=\operatorname {tr} GB$ ${\ displaystyle {\ bar {C}} = \ sum _ {i = 1} ^ {m} \ sum _ {j = 1} ^ {m} g_ {ij} B_ {ij} = \ operatorname {tr} GB }$ . Here $\operatorname {tr}$ ${\ displaystyle \ operatorname {tr}}$ Is the trace of the matrix ^[1] .

The first form of quantum Cramer-Rao inequality ^[2] :

{\tilde {Y}}BY\geqslant {\tilde {Y}}A^{-1}Y

{\ displaystyle {\ tilde {Y}} BY \ geqslant {\ tilde {Y}} A ^ {- 1} Y}

.

The second form of the Cramer-Rao quantum inequality ^[2] :

{\tilde {Z}}B^{-1}Z\geqslant {\tilde {Z}}AZ

{\ displaystyle {\ tilde {Z}} B ^ {- 1} Z \ geqslant {\ tilde {Z}} AZ}

.

Here $A_{ij}=\operatorname {Tr} \left({\frac {\partial \rho }{\partial \theta _{i}}}L_{j}\right)$ ${\ displaystyle A_ {ij} = \ operatorname {Tr} \ left ({\ frac {\ partial \ rho} {\ partial \ theta _ {i}}} L_ {j} \ right)}$ , $L_{k}$ ${\ displaystyle L_ {k}}$ determined by the formula ${\frac {\partial \rho }{\partial \theta _{k}}}={\frac {1}{2}}(\rho L_{k}+L_{k}\rho )$ ${\ displaystyle {\ frac {\ partial \ rho} {\ partial \ theta _ {k}}} = {\ frac {1} {2}} (\ rho L_ {k} + L_ {k} \ rho)}$ , $Y, Z$ ${\ displaystyle Y, Z}$ we obtain from $\operatorname {Re} \operatorname {Tr} \left(\int _{\Theta }T_{\theta }\rho T_{L}\,d\Pi ({\hat {\theta }})\right)={\tilde {Y}}Z$ ${\ displaystyle \ operatorname {Re} \ operatorname {Tr} \ left (\ int _ {\ Theta} T _ {\ theta} \ rho T_ {L} \, d \ Pi ({\ hat {\ theta}}) \ right) = {\ tilde {Y}} Z}$ where $T_{\theta }=1\sum _{j=1}^{m}y_{j}({\hat {\theta }}_{j}-\theta _{j})$ ${\ displaystyle T _ {\ theta} = 1 \ sum _ {j = 1} ^ {m} y_ {j} ({\ hat {\ theta}} _ {j} - \ theta _ {j})}$ , $T_{L}=\sum _{k=1}^{L}z_{k}L{k}$ ${\ displaystyle T_ {L} = \ sum _ {k = 1} ^ {L} z_ {k} L {k}}$ .

Notes

↑ Helstrom, 1979 , p. 295.
↑ ¹ ² Helstrom, 1979 , p. 297.

Literature

Helstrom K. Quantum theory of hypothesis testing and evaluation. - M .: Mir, 1979.- 344 p.

[_59f41d8d018e4854-1] Helstrom, 1979 , p. 295.

[_59f41d8d018e4856-2] ¹ ² Helstrom, 1979 , p. 297.

Quantum Cramer - Rao Inequality

Wording

See also

Notes

Literature

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