  When a pair of physical systems interact, they form a composite system, and, in quantum mechanics as in classical mechanics, there is a rule for constructing the state-space of a composite system from those of its components, a rule that tells us how to obtain, from the state-spaces, \(H_A\) and \(H_B\) for \(A\) and \(B\), respectively, the state-space — called the ‘tensor product’ of \(H_A\) and \(H_B\), and written \(H_A \otimes H_B\) — of the pair. There are two important things about the rule; first, so long as \(H_A\) and \(H_B\) are Hilbert spaces, \(H_A \otimes H_B\) will be as well, and second, there are some facts about the way \(H_A \otimes H_B\) relates to \(H_A\) and \(H_B\), that have surprising consequences for the relations between the complex system and its parts. In particular, it turns out that the state of a composite system is not uniquely defined by those of its components. What this means, or at least what it appears to mean, is that there are, according to quantum mechanics, facts about composite systems (and not just facts about their spatial configuration) that don’t supervene on facts about their components; it means that there are facts about systems as wholes that don’t supervene on facts about their parts and the way those parts are arranged in space. The significance of this feature of the theory cannot be overplayed; it is, in one way or another, implicated in most of its most difficult problems.