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if L (ˆ ρ$_{H}$ ) is almost as large as L (ˆ ρ$_{MLE}$ ), then ˆ ρ$_{H}$ is al- most as plausible as ˆ ρ$_{MLE}$ . If they have identical proper- ties, we may as well pick the more plausible one – but if ˆ ρ$_{H}$ is substantially different in some way, then it should be considered on its merits unless ˆ ρ$_{MLE}$ has significantly higher likelihood. We have already seen that the HMLE estimate has substantially different properties, so let’s confirm that it is not significantly less likely. 

Consider the classical case. If n$_{k}$ = 0, then p$_{MLE}$ assigns ˆ p$_{k}$ = 0. But it’s equally plausible that p$_{k}$ > 0, since if p$_{k}$ < 1 N then “ k ” probably won’t appear in the first N samples. The likelihood function bears this out: the most likely state sets ˆ p$_{k}$ = 0, but nearby states with nonzero p$_{k}$ have almost the same likelihood. If p$_{H}$ assigns 

p ′ k = β N and p ′ j = $^{(}$1 − β N ) n$_{j}$ N for j ̸ = k , then 

Likelihood ratios between e − 1 and e are “barely worth mentioning” [15], so if β < 1, then p$_{H}$ is essentially as plausible as p$_{MLE}$ . Actually, p$_{MLE}$ comprises K − 1 independent parameters, and in this case likelihood ra- tios between e − K and e K are insignificant. [Typically, L ( p$_{true}$ ) ≈ e − $^{K}$L ( p$_{MLE}$ ), so tighter significance criteria would reject the true state.] If p$_{MLE}$ assigns zero prob- ability to M < K different events, and p$_{H}$ hedges all M of them, then the argument leading to Eq. 11 gives a likelihood ratio of e − $^{Mβ}$, which is not significant. 

For quantum HMLE, it’s possible to show the same result for the HMLE estimate ˆ ρ$_{H}$ : 

The proof is a bit long, and can be found in Appendix A of the arxiv.org version. 

Conclusions: Hedging is a simple, well-motivated so- lution to the zero eigenvalue problem. It is also easy to implement – unlike, for instance, Bayesian techniques (which tend to be an order of magnitude harder to cal- culate). HMLE can be implemented by a near-trivial change to any MLE routine. Because the hedged likeli- hood goes smoothly to zero near the boundary, no ex- plicit positivity constraint is needed. So in fact it may be easier than MLE, as simple gradient-crawling meth- ods should work (though care is necessary for small β , where the boundary roll-off becomes sharper). 

Because ˆ ρ$_{H}$ is always full-rank, it can safely be used for predictive tasks like gambling and data compression. HMLE works well for qubits, providing improved accu- racy by almost all metrics. Further studies will reveal how well it works for larger systems. Pure states are best estimated using very small values of β , and in gen- eral the optimal value of β is not clear. This contrast with the classical case, where β ≈ 1 2 is known to be asymptoti- cally optimal, suggests that alternative hedging functions (which do not correspond to “add β ”) may work better for quantum estimation. 

Acknowledgments: This paper owes much to dis- cussions with Alexei Gilchrist, Daniel James, Jordan La- Pointe, and Rob Spekkens. I am supported by the Gov- ernment of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research & Innovation. 

The point of this section is to demonstrate that the hedged maximum likelihood estimate ˆ ρ$_{H}$ is never signifi- 

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## Appendix A 

