No-signalling Principle Theorem

Let
Let Alice have particle and Bob have particle
Then Alice cannot convey any info to Bob just by doing local measurement on .

Proof

Case 1

Bob does a complete measurement on in basis
Let onb of
Write:

Then

Case 2

When alice does a measurement prior to Bob, suppose Alice does measurement on in
Then

If outcome is :

Now Bob does measurement on in

So

Thus

same as in Case 1