Meaning of entropy of a bitstring in NIST SP 800 - 90A From NIST SP 800 - 90A from January 2012 (see http://csrc.nist.gov/publications/nistpubs/800-90A/SP800-90A.pdf), page 6:


  For the purposes of this Recommendation, an n-bit string is said to have 
  full entropy if that bit string is estimated to contain at least 
  $(1-\epsilon) n$ bits of entropy, where $0 &lt;= \epsilon &lt;= 2^{-64}$.


My understanding of entropy is that entropy is a property of the probability distribution of a source of bit strings, and not a property of a single bit string.

So what is NIST specifying here? Is there a way to estimate the entropy of the underlying source, given a bit string produced by that source? Maybe statistical tests, like Maurer's universal test? This could only be probabilistic since in principle, a true source of randomness can create any bit string...
"Entropy" is more accurately defined, in cryptography, as "that which the attacker does not know". For instance, suppose that every day you take all rates at the closure of the New York stock exchange, and hash them with SHA-256. The resulting value is very unpredictable (otherwise you could become very rich), so, from a "physics" point of view, there is a lot of entropy here. However, once the rates are published, everybody knows them: thus, from a cryptographic point of view, the entropy is nil.

To estimate the entropy, you have to analyze the process that generates the purportedly random value, and imagine an attacker trying to guess that value: the attacker has access to a validation box that returns "true" if the attacker found exactly the value, and "false" otherwise. You may say that the process yields "$n$ bits of entropy" if, on average, the attacker must invoke the validation box $2^{n-1}$ times before hitting the right value.

The important point here is that the estimate can come only from an analysis of the generation process. It cannot be tested with statistics, because statistics do not capture the level of knowledge of the attacker.