the entropy of the subsystems is greater than zero. In this sense, the systems are "entangled". This has specific empirical ramifications for interferometry. The above example is one of four Bell states, which are (maximally) entangled pure states (pure states of the space, but which cannot be separated into pure states of each and ). Now suppose Alice is an observer for system , and Bob is an observer for system . If in the entangled state given above Alice makes a measurement in the eigenbasis of , there are two possible outcomes, occurring with equal probability: Alice measures 0, and the state of the system collapses to . Alice measures 1, and the state of the system collapses to . If the former occurs, then any subsequent measurement performed by Bob, in the same basis, will always return 1. If the latter occurs, (Alice measures 1) then Bob's measurement will return 0 with certainty. Thus, system has been altered by Alice performing a local measurement on system . This remains true even if the systems and are spatially separated. This is the foundation of the EPR paradox. The outcome of Alice's measurement is random. Alice cannot decide which state to collapse the composite system into, and therefore cannot transmit information to Bob by acting on her system. Causality is thus preserved, in this particular scheme. For the general argument, see no-communication theorem. Ensembles As mentioned above, a state of a quantum system is given by a unit vector in a Hilbert space. More generally, if one has less information about the system, then one calls it an 'ensemble' and describes it by a density matrix, which is a positive-semidefinite matrix, or a trace class when the state space is infinite-dimensional, and has trace 1. Again, by the spectral theorem, such a matrix takes the general form: where the wi are positive-valued probabilities (they sum up to 1), the vectors are unit vectors, and in the infinite-dimensional case, we would take the closure of such states in the trace norm. We can interpret as representing an ensemble where is the proportion of the ensemble whose states are . When a mixed state has rank 1, it therefore describes a 'pure ensemble'. When there is less than total information about the state of a quantum system we need density matrices to represent the state. Experimentally, a mixed ensemble might be realized