`Lazy' quantum ensembles

We compare different strategies aimed to prepare an ensemble with a given density matrix $\rho$. Preparing the ensemble of eigenstates of $\rho$ with appropriate probabilities can be treated as `generous' strategy: it provides maximal accessible information about the state. Another extremity is the so-called `Scrooge' ensemble, which is mostly stingy to share the information. We introduce `lazy' ensembles which require minimal efforts to prepare the density matrix by selecting pure states with respect to completely random choice. We consider two parties, Alice and Bob, playing a kind of game. Bob wishes to guess which pure state is prepared by Alice. His null hypothesis, based on the lack of any information about Alice's intention, is that Alice prepares any pure state with equal probability. Then, the average quantum state measured by Bob turns out to be $\rho$, and he has to make a new hypothesis about Alice's intention solely based on the information that the observed density matrix is $\rho$. The arising `lazy' ensemble is shown to be the alternative hypothesis which minimizes the Type I error.