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Suppose Alice sends $m$ qubits to Bob, and I calculate the Holevo bound $\chi$ on the mutual information between Alice's plaintext message and the bit string obtained by Bob. My understanding is that Bob's best strategy can reveal no more than $\chi$ bits of Alice's string; for the remaining $m-\chi$ bits, he can do no better than to randomly guess. Therefore, if Bob tries to guess Alice's string, $\chi$ gives an upper bound $P_\chi=(\frac{1}{2})^{m-\chi}$ on his probability of success.

To my surprise, this seems to be false. I am working on a problem where I can calculate $\chi(m)$; I can also calculate the success probability $P_\text{B}(m)$ of a specific strategy Bob could adopt. Based on numerical examples for specific values of $m$, $P_B>P_\chi$.

What is wrong with my formula for $P_\chi$? What's the correct way to do it?

(I'm happy to give more details about my calculations if that would help.)

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