Bitcoin and preceding alternative cryptocurrencies have attempted to solve this problem through the use of transaction mixers or ring signatures. But they score very poorly on the metric called the traceability set. The traceability set is a key metric to understanding how private a cryptocurrency is. The traceability set in formerly proposed solutions is limited by the size of the mixing cycle or ring signature. Each mixing cycle or ring signature is limited by the number of transactions per cycle, which is transitively limited by the the block size of the cryptocurrency. Thus, the traceability set in previous attempts at privacy tends to only be a few hundred.
With Zcoin, the traceability set is on a dramatically higher magnitude. Instead of having a traceability set limited to the few hundreds, Zcoin has a traceability set that encompasses all minted coins in an accumulator. Thus, the magnitude of the traceability could be in the order of many thousands rather than hundreds. So its privacy level is magnitudes higher than cryptocurrencies that rely on mixing or ring signatures.
The other problem is that tumbling methods are only secure under the assumption of a lack of topological analysis and pre-existing network data, which is an incorrect threat model. There have been multiple research papers demonstrating that taking a separate network topology like Facebook can be used to de-anonymize a cryptocurrency as long as a long chain of transaction history exists. With all previous cryptocurrencies, a long chain of transactions is publicly viewable on the blockchain and prone to topological analysis.
With Zcoin, this long chain of transaction history simply does not exist, and there is zero information leakage about the sender and receiver of a transaction, so it is not prone to topological analysis and so the link between the sender and receiver disappears.