Lost Bitcoins purchased in 2011
If I understand correctly, you not only do not know your private key anymore (and probably never knew), you also do not even know the BTC address where your Bitcoins are?

I'm afraid I have to say that under these circumstances it is as if you never had any Bitcoins. I lost some exactly this way as you describe and actually also had purchased them in 2011.

No chance. Best thing you can do is to forget about them and not to burden your mind with the loss.