This immediately gives the joint-convexity of the squared Bures distance, ds 2 $_{Bures}$( aρ$_{1}$ + (1 − a ) ρ$_{2}$, aσ$_{1}$ + (1 − a ) σ$_{2}$ ) = 2 ( 1 − √ F ( aρ$_{1}$ + (1 − a ) ρ$_{2}$, aσ$_{1}$ + (1 − a ) σ$_{2}$ ) ) ≤ 2 1 − ( a √ F ( ρ$_{1}$, σ$_{1}$ ) + (1 − a ) √ F ( ρ$_{2}$, σ$_{2}$ ) )) ≤ a 2 ( 1 − F ( ρ$_{1}$, σ$_{1}$ ) ) + (1 − a ) 2 ( 1 − √ F ( ρ$_{2}$, σ$_{2}$ ) ) ≤ a d 2 $_{Bures}$( ρ$_{1}$, σ$_{1}$ ) + (1 − a ) d 2 $_{Bures}$( ρ$_{2}$, σ$_{2}$ ) , 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. 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, 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 ρ ( x ) = L $_{x}$[ ρ$_{0}$ ] = ∑ i p$_{i}$ L $_{x}$[ | ψ$_{i}$ 〉〈 ψ$_{i}$ | ] , (8) dρ = dρ dx dx = ∑ i p$_{i}$dx L ′ $_{x}$[ | ψ$_{i}$ 〉〈 ψ$_{i}$ | ] , (9) d 2 $_{Bures}$( ρ ( x ) , ρ ( x ) + dρ ) = d 2 Bures ( ∑ i p$_{i}$ L $_{x}$[ | ψ$_{i}$ 〉〈 ψ$_{i}$ | ] , ∑ i p$_{i}$ ( L $_{x}$[ | ψ$_{i}$ 〉〈 ψ$_{i}$ | ] + dx L ′ $_{x}$[ | ψ$_{i}$ 〉〈 ψ$_{i}$ | ]) ) ≤ ∑ i p$_{i}$d 2 $_{Bures}$( L $_{x}$[ | ψ$_{i}$ 〉〈 ψ$_{i}$ | ] , L $_{x}$[ | ψ$_{i}$ 〉〈 ψ$_{i}$ | ] + dx L ′ $_{x}$[ | ψ$_{i}$ 〉〈 ψ$_{i}$ | ]) ≤ d 2 $_{Bures}$( L $_{x}$[ | ψ$_{0}$ 〉〈 ψ$_{0}$ | ] , L $_{x}$[ | ψ$_{0}$ 〉〈 ψ$_{0}$ | ] + dx L ′ $_{x}$[ | ψ$_{0}$ 〉〈 ψ$_{0}$ | ]) , (10) 4 δx$_{min}$ | ρ ( x ) ≥ δx$_{min}$ | L x [ | ψ 0 〉〈 ψ 0 | ] , (7)