In our economy, much art is held for its investment value as much as for its aesthetic value. @Richard has pointed out (in another question) that Roddenberry defined money out of the Star Trek economy but that doesn't deal with the issue of objects of unique value.
With replicators, does original art even have a meaning? Let's suppose that Pollock's Lavender Mist has survived, with impeccable provenance, to the 24th century. It is then replicated. Could any test known to the time distinguish the original from the replica? Are there are any minute differences, undetectable in the first replica, that would build up and be detected in the Nth replica, where N is a big number?
Answer
A replicated object can be distinguished from an original "organic" object at a micro-molecular level. It stands to reason that art forgery is next to impossible unless the "original" was a replicated object in the first place.
From the TNG Technical Manual:
As with all transporter-based replication systems, the food replicators operate at molecular resolution. Because of this, there are significant numbers of single-bit errors in the resulting replicated materials. These errors are not nutritionally significant (although some individuals do claim to be able to taste differences in certain dishes), but certain types of Altarian spices have shown a tendency to become mildly toxic when replicated, so their use is avoided in replicated dishes.
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Because of the massive amount of computer memory required to store even the simplest object, it is impossible to record each molecule individually. Instead, extensive data compression and averaging techniques are used. Such techniques reduce memory storage required for molecular patterns by factors approaching 2.7 x 109. The resulting single-bit inaccuracies do not significantly impact the quality of most reproduced objects, but preclude the use of replicator technology to re-create living objects. Single-bit molecular errors could have severely detrimental effects on living DNA molecules and neural activity. Cumulative effects have been shown to closely resemble radiation-induced damage.
and
Another example of replicator limits is the single-bit DNA errors that led Data and Beverly to suspect Romulan trickery in the episode "Data's Day."
In order to replicate at the quantum level (e.g. with no errors), you would need access to the original. You could then move it from one place to another using the replicator (basically by using them as a glorified transporter) but you couldn't create a truly identical duplicate except by the sort of freakish accident that we see in TNG: Second Chances.
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