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"One entanglement measure E in d dimensions is asymptotic continuous if for two arbitrary states ρ and σ we've in the asymptotic regime n---->∞ that

||ρ - σ|| ---> 0

with

[E(ρ) - E( σ)]/log(d) ---> 0 "

I can understand that for two states which are close to each other, the value for the entanglement measure is expected to be similar, but what I don't understand is why we have to use the asymptotic regime. I've searched in various papers about entanglement measures but I haven't found a clear answer for the question.

Thank you everyone for reading me and I will be very grateful if you can help me with any contribution.