Quantum correlations are inherently statistical. If a measurement is repeated several times under identical conditions, one does not necessarily get the same result at each repetition. Therefore, in general, one cannot predict the result of a single measurement with certainty (see uncertainty).
However, the statistical frequency of each result is univocally determined by the experimental conditions. Hence, the probability of getting a given result can be predicted. The probabilities p(a1)..... p(an), associated with the possible values a1.....an of an observable A, form a probability distribution (provided that p(a1) + p(a2)+...+ p(an) = 1).
Probability distributions are of course employed in classical physics too. There, however, the use of probability is supposed to reflect our contingent ignorance about facts that are relevant to the complete description of what is going on. For example, we lack an infinitely accurate knowledge of the initial conditions, and this prevents us from drawing deterministic predictions (i.e. predictions that assign probability 0 to all the outcomes but one). But even though, in some cases, the structure of the correlations observed in classical physics may not be deterministic, it is nonetheless compatible (to a large extent) with the assumption of ‘underlying’ determinism - i.e. the assumption that the observed phenomena are brought about by a deterministic chain of events. Innocuous as it may seem, this assumption turns out to be untenable in quantum mechanics (see implications and uncertainty for further discussion).
A distinctive feature of quantum probability distributions is that they display wavelike patterns and obey the uncertainty relations.
The formal tool allowing one to deal with quantum probabilities are the so-called quantum states. According to the instrumentalist view, first advocated by Max Born and the Copenhagen School, quantum states do not directly represent any real physical properties. Rather, they merely provide the information necessary to draw all sorts of statistical predictions about results obtained in a well-specified experimental context (see origins).
Quantum states are usually represented by vectors belonging to a special kind of vector space called a Hilbert space. The representation of states in the Hilbert space is closely related to the principle of superposition. This principle, first formulated by P.A.M. Dirac, asserts that if two or more quantum states correspond to situations that are physically possible, so does their superposition. The term ‘superposition’ designates the linear combination of vectors (given two vectors
and 
, a linear combination of them is the vector: 
, where 
are two arbitrary numbers). The principle of superposition is a way to state that, in order to predict the correlations observed in the quantum world, we need all the vectors belonging to the Hilbert space. State vectors provide a very general and powerful predictive structure, which covers both deterministic and indeterministic models (see origins). Quantum particles displaying wavelike patterns, for example, are readily represented in the state space (this is not too surprising, since waves, like vectors, obey a superposition principle).
The fundamental implications of the superposition principle have been intensely debated. If, in contrast to the instrumentalist view, quantum states are supposed to describe a real state of affairs (and not merely to predict possible occurrences), then a number of paradoxes (like the one of Schrödinger's cat) and fundamental puzzles (like the so-called ‘measurement problem’) arise (see implications).
New research fields have been created to study the far-reaching implications of the superposition principle. An important issue that is currently being addressed, both theoretically and experimentally, is how to manipulate the quantum information encoded in the so-called entangled states (see implications) .