I have seen estimates that, for example, the human brain can store 2.5 petabytes of information, which presumably prevents the use of a replicator to create a human brain. The implication is that simpler objects require far less memory to store. How much memory would it take to store, for example, a glass of water? Or an isolinear chip?
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A single isolinear chip can hold 2.15 kiloquads. While that amount is never completely defined, this page: homepage.usask.ca/~lowey/kiloquad.html has a rather interesting breakdown, which suggest that a single isolinear chip could hold an entire brains worth of information. Also, you're confusing memory with capacity. The "memory" needed to store water, for example, would be three bytes (H2O), where you need the memory/computing is in the creation of the substance from the formula. Those routines are what would take up storage.– JohnPCommented Dec 9, 2013 at 18:06
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@JohnP and paracaudex: This is too long for a comment and only tangential to the question here, but see the second half of my answer on what cannot be replicated (below the "EDIT") for the probable reason it takes almost an entire station to store 5 people's minds.– IzkataCommented Dec 10, 2013 at 0:25
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@JohnP Uhh, I come from StackOverflow and I can tell you that storing water in three bytes is beyond ridiculous. You need the quantum numbers of every electron (possibly every quark), how the electrons are interacting, the isotopes present (is it oxygen 16? 18?) and who knows how much else. Now I;m not Jon Skeet, but I couldn't even represent a single electron in three bytes.– ApproachingDarknessFishCommented Dec 10, 2013 at 7:37
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3@ValekHalfHeart: it definitely wouldn’t take more than 640 KB though.– Paul D. WaiteCommented Dec 10, 2013 at 10:08
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@ValekHalfHeart - You're talking about the processing needed to PRODUCE water. Simply storing the formula OF water would take...three bytes. Same with the information of a brain. It can hold 2.5 petabytes, so you could store the equivalent information amount on a single isolinear (Theoretically). The replication OF the brain is what would take the massive amount over and above the simple information storage.– JohnPCommented Dec 10, 2013 at 16:08
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2 Answers
This is less of an answer than sort of a mental meander...
For the Star Trek-style replicator that we're all familiar with, you need to not only know the atoms involved and their position, but you also need to know the state of each atom (velocity, kinetic energy, quantum state, etc). For some items, like a glass of water, or a hot cup of Earl Grey, you don't need to store the state of individual atoms; it doesn't matter which water molecule is at a particular position with a particular energy, so the replicator can fill a specified volume with N atoms/molecules of the same type with an average kinetic energy. That's not very much information at all. Even an isolinear chip could be stored in a compressed manner, depending on whether you need to replicate not just the chip but what's on it.
Compare that with what a transporter has to process to send a live person from point A to point B. For you, it matters what state the individual atoms of your body are in. Your blood is flowing through your veins at a particular pressure, your heart is contracting, your diaphragm is moving, your brain and nervous system are sending electrical signals throughout your body, your digestive system is dealing with breakfast, etc; all of this has to be replicated accurately at the destination for you to still be "you" instead of a lifeless lump of flesh and bone. That means the transporter pattern buffer has to know not just the element and position of each atom, but its velocity, kinetic energy, and quantum state. Multiply that by the number of atoms in the body (one estimate I've seen is on the order of 7 * 1027 for a 70 kg person), and the storage requirements get daunting.
Presumably one could replicate a brain of some sort and all that would be initially required is the organism's genome along with whatever algorithms you wish to use to determine the final state.
For example in the Enterprise episode "Dead Stop" Tucker was able to order pan-fried catfish from an alien replicator. T'pol surmised that the alien computer scanned the Enterprise data banks and took the information it needed to make the meal from scratch, specifically the catfish genome and the recipe for pan-fried catfish.
So, even though the Enterprise did not have a replicator, it nonetheless was able to store the information required to replicate catfish and rather trivially. The alien computer had to crunch a lot of numbers and make a lot of guesses about the environment where a catfish develops to edible health to produce the final result, though. It would have also had to have its own standard computations for applying heat through a pan and the physical consequences.
Say Tucker ordered "catfish brains, raw," it would potentially have been a much simpler computation than to produce the whole fish and fry it, especially if the alien computer is able to determine which parts of the genome are necessary to make a brain and what is necessary from the fish's environment to develop a healthy one. Then again if Tucker is ordering catfish brains for edible consumption, fully functioning brains may not be required, so long as they are made according to taste (no doubt an acquired one).
The restrictions for making brains from replicators in the Federation are more legal than technical. Doctor Bashir objects to the trade of biomimetic gel in the DS9 episode "In the Pale Moonlight" because "it could be used for illegal replication experiments." Along with many prohibited items that cannot be replicated, like deadly poisons, which Tuvok notes in the Voyager episode "Death Wish," there are whole fields of replicator research that no one pursues because they are just so morally repugnant. Even materials that could potentially be used to pursue them are outlawed.