# Equivalence of simple formulations of qubit entanglement

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?

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Definition (1) is only correct for pure states, in which case it is indeed equivalent to (2) and (3). – Peter Shor Feb 20 '13 at 18:43
@PeterShor that could be an answer – David Z Feb 20 '13 at 19:51
@David: For an answer, I'd also want to say what happens with mixed states (definition (3) seems to have been formulated for mixed states, because otherwise talking about a "distribution" is unnecessary). This would make the explanation much longer. – Peter Shor Feb 20 '13 at 20:13
@PeterShor: A simple example with two-qubit systems worked out: e.g, showing out why the fact that a state can/cannot be factored implies that measuring the value of one of the bits doesn't/does constrain the value of the other (if that's true). – raxacoricofallapatorius Feb 20 '13 at 20:19