4.7 Quantum communication

Now is a good moment to introduce Alice and Bob (not their real names): our two protagonists who always need to communicate with each other. These two play the major role in many communication dramas, though they remain rather lacking in character development.

This time Alice is sending quantum states to Bob, and Bob does his best to identify them correctly by choosing appropriate measurements. Let us start with a simple observation: if a quantum state of the carrier of information is described by a state vector in a 2^n-dimensional Hilbert space, then the carrier can carry at most n bits of information. For example, Alice can choose one of the 2^n states from a pre-agreed orthonormal basis \{|e_k\rangle\}, and Bob will be able to distinguish them reliably by choosing the \{|e_k\rangle\} basis for his measurement.

But can Alice and Bob do better than that? Can Alice send more than n bits of information per carrier by encoding them in states |s_1\rangle,\ldots,|s_N\rangle where N \geqslant 2^n? Can Bob choose a clever measurement and reliably distinguish between all such states?

The answer is no.