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I think “edge” is qualitative “alpha” is quantitative. Eg you can have an edge because you have a better view of flow, positioning, you are faster, less constrained, have systems + knowledge to put on trades others can’t. Alpha is what you expect to make from your edge.
Asking for a *precise* or *quantitative* definition is missing the point. Like asking “what do you mean *precisely* by love”?
Agree that you can fake alpha though, eg add some duration risk to an equity portfolio, this looks like alpha when regressed against equities 
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