content_image/441848.jpg content_image/441849.jpg content_image/441850.jpg
Eq. 13 is the maximum. Furthermore, this is a local max- imum (with respect to all variations), because the gradi- ent of h ( ρ ) is orthogonal to unitary rotations and there- fore locally orthogonal to off-diagonal variations. This is also a global maximum, because log L $^{′}$( ρ ) is a convex function. Thus, Eq. 13 maximizes the hedged likelihood. 

Now, consider some other hedging function h $^{′}$( ρ ). If h $^{′}$( ρ ) is not unitarily invariant, then there exists some point ρ such that, in the neighborhood of ρ , the gradi- ent of h $^{′}$( ρ ) is not orthogonal to unitary rotations. Sup- pose that the measured basis {| k 〉} is the diagonal basis of ρ , and the measured frequencies n$_{k}$ are such that ˆ ρ$_{H}$ (given by Eq. 13) is in the neighborhood of ρ . Then, at the point ˆ ρ$_{H}$ , the gradient of L ( ρ ) is orthogonal to off-diagonal variations, but the gradient of h $^{′}$( ρ ) is not. This means that the gradient of the hedged likelihood does not vanish, and thus its maximum cannot coincide with Eq. 13. 

If h $^{′}$( ρ ) is unitarily invariant, then for every measured basis, the maximization can safely be restricted to diago- nal matrices ρ = ∑ k p$_{k}$ | k 〉〈 k | , and it reduces to a classical problem, maximizing 

To reproduce the “add β ” rule, the gradient of log L ′ must vanish – for all { n$_{k}$ } – at p$_{H}$ = { n$_{k}$ + β N $^{}}$. This implies 

and since this condition is automatically satisfied by Eq. 1, 

at every point p$_{H}$ = { n$_{k}$ + β N } for any { n$_{k}$ } . These points are dense in the simplex on which h $^{′}$( p ) is defined. This 

<img src='content_image/441848.jpg'>

<img src='content_image/441849.jpg'>

<img src='content_image/441850.jpg'>

means that h ( ρ ) = h $^{′}$( ρ ), since smooth functions whose derivatives agree at a dense set of points are identical. 

∗ Electronic address: robin@blumekohout.com 

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