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, in scenarios of varying complexity and danger. These two play the major role in many communication dramas, though they remain rather lacking in character development. In this episode of their story, Alice is sending quantum states, called carriers, to Bob, and Bob is trying his best to correctly identify them by choosing appropriate measurements.

Let us start with a simple observation: if the carriers are described by state vectors in a 2^n-dimensional Hilbert space, then they can encode at most n bits of information.⁹⁵ For example, Alice can choose one of the 2^n states from a pre-agreed orthonormal basis \{|e_k\rangle\}_{k=1,\ldots,2^n}, and Bob will be able to distinguish them reliably by choosing the same 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.

---

95. This is just like the classical scenario: the space of binary strings of length n (which encode exactly n bits of information, by definition) is of dimension 2^n, since we describe any such string by picking between 0 and 1 for each digit, and we have n-many digits.↩︎