  Quantum states are the key mathematical objects in quantum mechanics, and
entanglement lies at the heart of the nascent fields of quantum information
processing and computation. What determines whether an arbitrary quantum state
is entangled or separable is therefore very important for investigating both
fundamental physics and practical applications. Here we show that an arbitrary
bipartite state can be divided into a unique purely entangled structure and a
unique purely separable structure. We show that whether a quantum state is
entangled or not is determined by the ratio of its weight of the purely
entangled structure and its weight of the purely separable structure. We
provide a general algorithm for the purely entangled structure and the purely
separable structure, and a general algorithm for the best separable
approximation (BSA) decomposition, that has been a long-standing open problem.
Our result implies that quantum states exist as families in theory, and that
the entanglement (separability) of family members can be determined by
referring to a crucial member of the family.
