No way.
But you can detect what coins transferred from one address and what coins transferred from another.
Look for example at the first address in the network:
12c6DSiU4Rq3P4ZxziKxzrL5LmMBrzjrJX
http://blockchain.info/address/12c6DSiU4Rq3P4ZxziKxzrL5LmMBrzjrJX
https://blockexplorer.com/address/12c6DSiU4Rq3P4ZxziKxzrL5LmMBrzjrJX
Nobody can stop people to send bitcoins to Satoshi's address, first bitcoin address in the network.