Adding a backdoor that is secure is very easy to implement. The government just needs to publish a public key. You then encrypt your private key using that public key and include it with whatever you encrypted. This would be much like the lock box on a house that holds the front-door key that only real estate agents showing the house are supposed to be able to access. And there's no reason it would be limited to just one. Opening a connection to a server in Turkey? Better include lock boxes for both your own government and the Turkish authorities.
The only big hole is the security of the government's private key (or more likely, keys).
The tricky part is that this government lock box has to be added to the common protocols. And how many different protocols would have to be updated? TLS, ssh, PGP, etc. What are the odds of introducing flaws that allow for new attack vectors when introducing the back door? For example, could you trick a victim's computer into thinking it needed to include the lock box for some jurisdiction that you control?
Now while the technical side of this could be made to work, as a public policy, it's a horrible idea. Let's not just say it can't be done and forget about it. We need to fight this as bad policy.