# How do I calculate Holevo information when information is knowable in principle, but unknown?

I am trying to better understand the meaning of Holevo information $\chi$. Suppose Alice starts with data encoded on qubits in the $\{0,1\}$ basis. She takes one of these qubits, originally in pure state $\lvert0\rangle$; randomly chooses from a uniform distribution of SU(2) transformations with which to encrypt the qubit; sends the qubit to Bob.

I have the formula $\chi=S(\sum_j p_j \rho_j)-\sum_j p_j S(\rho_j)$, where $\rho_j$ are all the encrypted states Alice chooses from, with corresponding probabilities $p_j$. The first term is clearly equal to $1$ (the entropy of a completely mixed qubit state); the second term is zero, since Alice always sends pure states (albeit unknown ones); so $\chi=1$.

Have I done this correctly? It makes sense to me, but it's not the way I see others handling the calculation.

edit: note that Bob does not know Alice's key. See comment for clarification.

• The part 'randomly encrypts..' is a bit unclear. Do you mean encode? If you use tge Holevo quantity like this, then badically you are enconding a continous alphabet. So this beeing one makes sense. But if you say you only have 0 as the alphabet, then it the information is 0. Can you be more explicit whar information is encoded by alice? – lalala Mar 17 '18 at 23:00
• Without getting too much into my application, let me restate: Alice sends 0 or 1 to Bob, but in a random basis. Let's say Eve intercepts the qubit, but we're guaranteed that she doesn't know the basis. What is the Holevo bound on Eve's information about Alice's plaintext? So I do mean in "encrypt" in the sense that the key is meant to be secret. – ostrichCamel Mar 17 '18 at 23:33
• In that case it is not true. The alphabet she receives is actually for each 0 or 1 the fully mixed state. So the Holevo quantity is 0 for this setup. – lalala Mar 18 '18 at 8:25