into at least one pure entangled state. States that have this property are called distillable. These states are the most useful quantum states since, given enough of them, they can be transformed (with local operations) into any entangled state and hence allow for all possible uses. It came initially as a surprise that not all entangled states are distillable, those that are not are called 'bound entangled'. A different entanglement classification is based on what the quantum correlations present in a state allow A and B to do: one distinguishes three subsets of entangled states: (1) the non-local states, which produce correlations that cannot be explained by a local hidden variable model and thus violate a Bell inequality, (2) the steerable states that contain sufficient correlations for A to modify ("steer") by local measurements the conditional reduced state of B in such a way, that A can prove to B that the state they possess is indeed entangled, and finally (3) those entangled states that are neither non-local nor steerable. All three sets are non-empty. Entropy In this section, the entropy of a mixed state is discussed as well as how it can be viewed as a measure of quantum entanglement. Definition In classical information theory , the Shannon entropy, is associated to a probability distribution, , in the following way: Since a mixed state is a probability distribution over an ensemble, this leads naturally to the definition of the von Neumann entropy: In general, one uses the Borel functional calculus to calculate a non-polynomial function such as . If the nonnegative operator acts on a finite-dimensional Hilbert space and has eigenvalues , turns out to be nothing more than the operator with the same eigenvectors, but the eigenvalues . The Shannon entropy is then: . Since an event of probability 0 should not contribute to the entropy, and given that the convention is adopted. This extends to the infinite-dimensional case as well: if has spectral resolution assume the same convention when calculating As in statistical mechanics, the more uncertainty (number of microstates) the system should possess, the larger the entropy. For example, the entropy of any pure state is zero, which is unsurprising since there is no uncertainty about a system in a pure state. The entropy of any of the two subsystems of the entangled state discussed above is (which can be shown to be the maximum entropy