I'm reading some very elementary treatments of quantum computation and am unsure about the correspondence among "definitions" of qubit entanglement.
One definition states that (1) the bits of a two-qubit system are entangled if the state cannot be expressed as the (tensor) product of two one-qubit states. Another "definition" states that (2) a two-qubit system is entangled if "we cannot determine the state of each qubit separately" or if (3) "measuring one qubit determines the distribution of the other".
Perhaps my confusion is a simple matter of not fully understanding what the "independent" means and how it relates to the tensor product; but it's not clear to me how these statements are related. Does, for example (1) imply (2)? Are (2) and (3) equivalent?