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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

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