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I'm not even sure the carrier has the information. At least in countries where the iPhone is sold unlocked.

The UDID has nothing to do with the IMEI/IMSI which are both known by the carriers.

I'd be worried if the carrier has this info. I bought the iPhone 4 from Apple, the micro SIM from vodafone, two separate unconnected transactions.

So what stops the carrier to tie your billing information (which I presume has your name and address on it) to the SIM purchase ?

Every time you switch that phone on that phone ID is tied to the SIM on the network.