2 The usual fix for this problem is MLE [2]. A likelihood function is derived from the data, a relative entropy cost function and a Dirichlet- β prior Common examples of Dirichlet priors include the “flat” Lebesgue measure ( β = 1), and Jeffreys’ prior ( β = 1 $_{2}$). Given any prior, we can minimize expected relative en- tropy by: (1) updating the prior to a posterior via Bayes’ Rule; and (2) reporting its mean value. For the Dirichlet- β prior, this gives the “add β ” rule. The “add β ” rule is not intrinsically Bayesian, how- ever. A n¨ aive estimator following Eq. 2 can simulate it by adding β dummy observations of each letter k . This yields new frequencies { n$_{k}$ + β } and a total of N + Kβ observations. To generalize to non-integer β , we observe that the likelihood function is L ( p ) = Pr( { n$_{k}$ }| p ) = ∏ k p n$_{k}$ k , and adding β dummy observations of each letter yields a hedged likelihood function and we assign the ˆ ρ that maximizes it. Maximization over all trace-1 Hermitian matrices yields ˆ ρ$_{tomo}$ (just as in the classical case), but restricting to ρ ≥ 0 yields a non-negative ˆ ρ$_{MLE}$ . However, ˆ ρ$_{MLE}$ can still assign zero probabilities – just like its classical counterpart (Eq. 2). If ˆ ρ$_{tomo}$ is not strictly positive, ˆ ρ$_{MLE}$ will have at least one zero eigenvalue [6], so this is rather common. Moreover, the zero probabilities in ˆ ρ$_{MLE}$ are less justified than those in p$_{MLE}$ , because they generally correspond to a mea- surement outcome | ψ 〉〈 ψ | that is not an element of the measured POVM, and could never have appeared. In contrast, Eq. 2 assigns p$_{k}$ = 0 only when “ k ” has been given N chances to appear and (so far) has not. So al- though ˆ ρ$_{MLE}$ may be the right estimator for some task, its zero eigenvalues represent a level of confidence that is implausible and (for predictive tasks like gambling and compression) catastrophic. Prediction demands a hedged estimator. P$_{0}$ ( p )d p ∝ ∏ k p β − 1 k d p$_{k}$. (5) L ( ρ ) = Pr( { n$_{i}$ }| ρ ) = ∏ i Tr[ ρE$_{i}$ ] $^{n$_{i}$}$, (9) L $^{′}$( p ) = ∏ k p n$_{k}$ + β k = ( ∏ k n β k ) L ( p ) , (6) Pr( i ) = Tr[ ρE$_{i}$ ] . (7) whose maximum value is achieved by Eq. 4. When β is not an integer, the hedged likelihood (Eq. 6) remains well-defined, and the “add β ” rule still maximizes it. Quantum Hedging: The quantum analogue of a distribution p is a d × d density matrix ρ . It cannot be observed directly; observing a sample of ρ requires choosing a particular measurement M . Experimental- ists often divide the samples into groups and measure M j on the N$_{j}$ samples in group j , but L ( ρ ) depends only on observed events, not the unobserved alternatives, so we may pretend that all N samples were measured by M = ⋃ j w$_{j}$ M $_{j}$, where w$_{j}$ = N j N . M corresponds to a POVM , a set of positive operators { E$_{i}$ } summing to 1l, which determine the probability of outcome “ i ” as Bayesian mean estimators are hedged, and with suit- able priors they have extremely good predictive behavior [6]. However, for quantum estimation there are no closed- form solutions, and numerical integration is hard. This is unfortunate, because Bayes estimators for classical prob- abilities work very well. They yield “add β ” rules when applied to Dirichlet- β priors, and Dirichlet priors are well motivated. Jeffreys’ prior ( β = 1 $_{2}$) yields asymptotically minimax-optimal estimators for data compression [12], Krichevskiy showed that “add 0 . 50922 . . . ” outperforms all other rules for predicting the next letter [13], and Braess et al [14] pointed out that β ≈ 1 generally works well because large- N behavior depends only weakly on β . This suggests adapting “add β ” to quantum state esti- mation (independent of Bayesian arguments). However, obvious methods like adding dummy counts don’t work. Suppose we estimate a qubit source by measuring σ$_{x}$ , σ$_{y}$ , and σ$_{z}$ ten times each, and – by unlikely chance – all the outcomes are +1. ˆ ρ$_{tomo}$ lies well outside the Bloch sphere, and ˆ ρ$_{MLE}$ is the projector onto its largest eigen- vector. Now, if we add β = 1 dummy counts, ˆ ρ$_{tomo}$ is still outside the Bloch sphere, and ˆ ρ ′ MLE is unchanged! The frequencies { n$_{i}$ } thus provide information about ρ . Interpreting this information is the central problem of quantum state estimation. The oldest and simplest procedure, linear inversion to- mography [11], is based on Eq. 2. Inverting Born’s Rule (Eq. 7) yields an estimate ˆ ρ$_{tomo}$ satisfying If these equations are overcomplete, ˆ ρ$_{tomo}$ is chosen by least-squares fitting. Frequently, some of ˆ ρ$_{tomo}$ ’s eigen- values are negative – a serious problem, for they repre- sent probabilities. This occurs because linear inversion is blind to the shape of the space of quantum states (which assign probabilities to all measurements). It tries to fit data from a single POVM M , and happily assigns neg- ative probabilities for measurements that weren’t per- formed. The underlying problem is that MLE tries to fit the observed data, with no consideration of unobserved mea- surements – but the resulting quantum state makes pre- dictions about those unobserved measurements. Adding dummy data works in the classical case because there are only K different events that can be observed or pre- dicted, so by adding a dummy observation of each one, we rule out the possibility of assigning p$_{k}$ = 0 to any event. A quantum state assigns probabilities to infinitely Tr[ˆ ρ$_{tomo}$E$_{i}$ ] = n$_{i}$ N for i = 1 . . . m. (8)