pure state, the density matrix is idempotent, , the entropy S(ρ) for it vanishes. Thus, if the system is finite (finite-dimensional matrix representation), the entropy S(ρ) quantifies the departure of the system from a pure state. In other words, it codifies the degree of mixing of the state describing a given finite system. Measurement decoheres a quantum system into something noninterfering and ostensibly classical; so, e.g., the vanishing entropy of a pure state , corresponding to a density matrix increases to for the measurement outcome mixture as the quantum interference information is erased. Properties Some properties of the von Neumann entropy: is zero if and only if represents a pure state. is maximal and equal to for a maximally mixed state, being the dimension of the Hilbert space. is invariant under changes in the basis of , that is, , with a unitary transformation. is concave, that is, given a collection of positive numbers which sum to unity () and density operators , we have satisfies the bound where equality is achieved if the have orthogonal support, and as before are density operators and is a collection of positive numbers which sum to unity () is additive for independent systems. Given two density matrices describing independent systems A and B, we have . is strongly subadditive for any three systems A, B, and C: This automatically means that is subadditive: Below, the concept of subadditivity is discussed, followed by its generalization to strong subadditivity. Subadditivity If are the reduced density matrices of the general state , then This right hand inequality is known as subadditivity. The two inequalities together are sometimes known as the triangle inequality. They were proved in 1970 by Huzihiro Araki and Elliott H. Lieb. While in Shannon's theory the entropy of a composite system can never be lower than the entropy of any of its parts, in quantum theory this is not the case, i.e., it is possible that , while . Intuitively, this can be understood as follows: In quantum mechanics, the entropy of the joint system can be less than the sum of the entropy of its components because the components may be entangled. For instance, as seen explicitly, the Bell state of two spin-½s, is a pure state with zero entropy, but each spin has maximum entropy when considered individually in its reduced density matrix. The entropy in one spin can be "cancelled"