5 ∇ log h evaluated at ˆ ρ$_{H}$ into the integral yields an upper bound. Defining the eigenvalues of ˆ ρ$_{MLE}$ as λ 0 k and those of ˆ ρ$_{H}$ as λ f $_{k}$, cantly less plausible than the MLE estimate, i.e. The MLE estimate maximizes the log-likelihood ( l ( ρ ) ≡ log L ( ρ )), while the HMLE estimate maximizes the hedged log-likelihood ( l $^{′}$( ρ ) = l ( ρ ) + log h ( ρ ), where h ( ρ ) is given in Eq. 1). Both are convex functions on a convex subset of R d $^{2}$− $^{1}$. It is convenient to think of − l ( ρ ) and − l $^{′}$( ρ ) as potential energy functions, and of ˆ ρ$_{MLE}$ and ˆ ρ$_{H}$ as the corresponding equilibrium states. In this picture, the gradients ∇ l ( ρ ) and ∇ log h ( ρ ) are force fields (which balance perfectly at ˆ ρ$_{H}$ ), and the logarithm of the likelihood ratio This means that L (ˆ ρ$_{H}$ ) is at least e − $^{dβ}$L (ˆ ρ$_{MLE}$ ), so for β < 1 it is not significantly less plausible. This does not necessarily mean that ˆ ρ$_{H}$ is close to ˆ ρ$_{MLE}$ . When L ( ρ ) is nearly flat, hedging can cause sub- stantial deflection – precisely because there is no gradi- ent in L to oppose it. When L is sharply peaked around ˆ ρ$_{MLE}$ , hedging has comparatively little effect. Δ l ≤ ∫ ˆ ρ$_{H}$ ρ =ˆ ρ$_{MLE}$ β ∑ k d λ$_{k}$ λ$_{k}$ = β d ∑ k =1 λ f k − λ 0 k λ f k ≤ βd. L (ˆ ρ$_{H}$ ) (ˆ ρ$_{MLE}$ ) ≥ e − $^{dβ}$. log$^{(}$ L (ˆ ρ$_{MLE}$ ) (ˆ ρ$_{H}$ ) ) = l (ˆ ρ$_{MLE}$ ) − l (ˆ ρ$_{H}$ ) = Δ l ∇ log h ( ρ ) = βρ − 1 h ( ρ ) = det( ρ ) $^{β}$, ˆ ρ$_{H}$ = ∑ k n$_{k}$ + β N + dβ | k 〉〈 k | . (13) d l = −∇ l · dˆ ρ = ∇ log h | β ′ · dˆ ρ. is the amount of work done by ∇ log h ( ρ ) by adiabatically changing the equilibrium from ˆ ρ$_{MLE}$ → ˆ ρ$_{H}$ . Because h ( ρ ) depends only on ρ ’s eigenvalues, the cor- responding “force” The point of this section is to show that the hedging function given in Eq. 1, is orthogonal to unitary rotations, and acts only on the spectrum of ρ . Furthermore, while it diverges at the boundary, it becomes rapidly and monotonically weaker away from the boundary. So, although it inexorably forces ˆ ρ off the boundary, it does not necessarily push it very far. Let us imagine that the hedging parameter (denoted β $^{′}$) is adiabatically increased from zero to β . For each β $^{′}$, there is an equilibrium ˆ ρ$_{β}$ ′ . Increasing β ′ by d β shifts it a distance dˆ ρ and does work is the only smooth hedging function that reproduces the “add β ” rule for measurements in any single orthonor- mal basis. That is, when N i.i.d. d -dimensional quan- tum systems all have been measured in a single ba- sis denoted {| 0 〉 . . . | d − 1 〉} , and outcome | k 〉〈 k | has ap- peared n$_{k}$ times, the maximum of the hedged likelihood L ′ = h ( ρ ) L ( ρ ) should be Appendix B First, consider hedging according to Eq. 1. The hedged likelihood is Integrating ∇ log h | β ′ · dˆ ρ along the entire path yields Δ l . Since ∇ log h is orthogonal to unitary changes in ˆ ρ , the integral is only sensitive to motion within the eigenvalue simplex, so It’s equally valid (and more convenient) to maximize its logarithm, It’s tempting to evaluate this directly as This function’s gradient thus has two components, one from the likelihood and one from the hedging function. The likelihood depends only on the diagonal elements of ρ , so its gradient is orthogonal to off-diagonal variations. The hedging function is unitarily invariant, so its gradi- ent is orthogonal to unitary rotations. If we vary only over diagonal matrices ρ = ∑ k p$_{k}$ | k 〉〈 k | , then this prob- lem reduces to the classical one and it’s easy to show that but β ′ changes with ρ . Instead, we upper-bound the in- tegral by observing that − l ( ρ ) and − log h ( ρ ) are both concave, so their second derivatives are strictly positive. As ˆ ρ$_{β}$ ′ moves away from the maximum of l ( ρ ) and to- ward that of log h ( ρ ), the components of −∇ l and ∇ log h parallel to dˆ ρ are strictly increasing. Thus, substituting L $^{′}$( ρ ) = det( ρ ) β ∏ k 〈 k | ρ | k 〉 $^{n$_{k}$}$. log L $^{′}$( ρ ) = β log det( ρ ) + ∑ k n$_{k}$ log 〈 k | ρ | k 〉 . ∇ log h | β ′ · dˆ ρ = ∑ k ∂ log h ∂λ$_{k}$ d λ$_{k}$. Δ l = ∫ ˆ ρ$_{H}$ ρ =ˆ ρ$_{MLE}$ β ′ ∑ k d λ$_{k}$ λ$_{k}$ ,