"The ink can be TRACED back 400 years" is a complete sentence "The ink can be TRACED back 400 years BEFORE" is an incomplete sentence. You would need to explain "Before what?" to make it a complete sentence.

"The ink can be traced back to 400 years before" This is a passive construction, implying the an unnamed agent can tracce the ink. I think someone is confusing before with earlier. Other than that it is correct and understandable. "The ink traced back to 400 years before." This sentence uses trace in active voice, and it not correct. Someone has to trace the ink back 400 years. The word, before, is redundant and not needed because the word, back, already means before.

Ryan is correct. I think the word you're looking for is "ago." You can say "The ink can be traced back to 400 years ago." This means "400 years in the past." In order to use the word "before," you would need to give more information. For example, you could say, "John found the book in his closet on Friday. His mother had given it to him ten years before." We can then understand that his mother gave him the book ten years *before he found it in his closet.* But "ten years before" doesn't mean "ten years ago." It means "ten years earlier than [something else that happened]." So as Ryan says, you would need to explain "400 years before [what?]"