No-cloning theorem
An unknown quantum state cannot be copied. Proved in 1982, and the consequences run through the entire field.
Wootters and Zurek showed that no operation can take an arbitrary state and produce two copies of it. The proof is short and follows directly from the linearity of quantum operations. This is not a matter of technical limitation but of what the theory permits.
It creates both problems and opportunities. Classical error correction works by copying data three times and voting — impossible here, which is why quantum error correction had to be invented from scratch. At the same time it is exactly what makes quantum key distribution secure: an eavesdropper cannot take a copy and let the original pass unnoticed.