In these cases, the solution then lies on the boundary of the n-dimensional Bloch sphere. This can be seen as related to linear inversion giving states which lie outside the valid space (the Bloch sphere). MLE in these cases picks a nearby point that is valid, and the nearest points are generally on the boundary. This is not physically a problem, the real state might have zero eigenvalues. However, since no value may be less than 0, an estimate of an eigenvalue being 0 implies that the estimator is certain the value is 0, otherwise they would have estimated some formula_52 greater than 0 with a small degree of uncertainty as the best estimate. This is where the problem arises, in that it is not logical to conclude with absolute certainty after a finite number of measurements that any eigenvalue (that is, the probability of a particular outcome) is 0. For example, if a coin is flipped 5 times and each time heads was observed, it does not mean there is 0 probability of getting tails, despite that being the most "likely" description of the coin. Bayesian mean estimation (BME) is a relatively new approach which addresses the problems of maximum likelihood estimation. It focuses on finding optimal solutions which are also "honest" in that they include error bars in the estimate. The general idea is to start with a likelihood function and a function describing the experimenter's prior knowledge (which might be a constant function), then integrate over all density matrices using the product of the likelihood function and prior knowledge function as a weight. Given a reasonable prior knowledge function, BME will yield a state strictly within the n-dimensional bloch sphere. In the case of a coin flipped N times to get N heads described above, with a constant prior knowledge function, BME would assign formula_53 as the probability for tails. BME provides a high degree of accuracy in that it minimizes the "operational divergences" of the estimate from the actual state. The number of measurements needed for a full quantum state tomography for a multi-particle system scales exponentially with the number of particles, which makes such a procedure impossible even for modest system sizes. Hence, several methods have been developed to  realize quantum tomography with fewer measurements. The concept of matrix completion and compressed sensing have been applied to reconstruct density matrices from an incomplete set of measurements (that is, a set of measurements which is not a quorum). In general, this is impossible, but under assumptions (for example, if the density matrix is a pure state, or a combination of just a few pure states) then the density matrix has fewer degrees of freedom, and it may be possible to reconstruct the state from the incomplete measurements.