Past exam of the introduction to quantum information course of the University of Oxford / 2021 / 2 / a / Solution

Each local reduced density operator of the Bell state is $I/2$. Alice's unrecorded local measurement leaves Bob's reduced state unchanged: he still sees a fair random bit. Only after learning her outcome through a classical channel can he identify the corresponding conditional state. Alice cannot choose the random outcome to encode a bit, so the correlations obey the no-signalling theorem.

If the shared state is the trusted pure Bell state, measuring both sides in the $Z$ basis gives identical, uniformly random raw bits. A pure joint state of Alice and Bob has no correlations with an external system, so these bits can form a shared secret. In an actual quantum key distribution protocol, the parties must verify the necessary state/device assumptions and authenticate their classical communication; correlation alone, without a security test, does not certify secrecy against an untrusted source.