content_image/372020.jpg content_image/372033.jpg content_image/372029.jpg content_image/372030.jpg content_image/372031.jpg
This immediately gives the joint-convexity of the squared Bures distance, 

<img src='content_image/372020.jpg'>

which proves the Lemma. 

With this Lemma, it is straightforward to show the following theorem: 

Theorem. Let ρ$_{0}$ = ∑ i p$_{i}$ | ψ$_{i}$ 〉〈 ψ$_{i}$ | be an arbitrary mixed state ( 0 ≤ p$_{i}$ ≤ 1 , ∑ i p$_{i}$ = 1 ), and ρ ( x ) = L $_{x}$[ ρ$_{0}$ ] the state generated from ρ$_{0}$ by an arbitrary linear map L x which depends smoothly on the parameter x . Then the sensitivity for measuring x in the state ρ ( x ) is bounded by the best achievable sensitivity in the state obtained by propagating any of the pure states | ψ$_{i}$ 〉 with the same map, i.e. 

<img src='content_image/372033.jpg'>

where | ψ$_{0}$ 〉 is the pure state in the set of all {| ψ$_{i}$ 〉} which leads to the best sensitivity, i.e. min$_{{}$$_{|}$$_{ψ}$$_{i}$$_{〉}}$ δx$_{min}$ | L x [ | ψ i 〉〈 ψ i | ] = δx$_{min}$ | L x [ | ψ 0 〉〈 ψ 0 | $_{]}$. 

Proof. Due to the linearity of L $_{x}$[ ρ ] , we have 

and thus, for smooth x -dependence of L $_{x}$, 

where L ′ x denotes the derivative of L x with respect to x . Therefore, 

<img src='content_image/372029.jpg'>

<img src='content_image/372030.jpg'>

<img src='content_image/372031.jpg'>

where we have designated | ψ$_{0}$ 〉 to be the state which maximizes d 2 $_{Bures}$( L $_{x}$[ | ψ$_{i}$ 〉〈 ψ$_{i}$ | ] , L $_{x}$[ | ψ$_{i}$ 〉〈 ψ$_{i}$ | ] + dx L ′ $_{x}$[ | ψ$_{i}$ 〉〈 ψ$_{i}$ | ]) over all | ψ$_{i}$ 〉 , and thus minimizes 

