Holevo's theorem

Remarks

Proof idea: encode $X$ in the classical-quantum state $\sigma_{XQ} = \sum_x p_x \lvert x \rangle\langle x \rvert \otimes \rho_x$. Its quantum mutual information is $I(X:Q)_\sigma = \chi$. The measurement is a quantum channel on $Q$ that turns $\sigma_{XQ}$ into the classical state of $(X, Y)$. Quantum mutual information does not increase under local channels, a consequence of strong subadditivity, so $I(X:Y) \le \chi$.

Since $\chi \le S(\rho) \le \log d$, at most $\log_2 d$ bits about $X$ can be read out of a $d$-dimensional system. For $n$ qubits, $d = 2^n$, so $n$ qubits carry at most $n$ bits of accessible classical information. Example: if the $\rho_x$ are pure states, then $\chi = S(\rho)$. For the two non-orthogonal states $\lvert 0 \rangle$ and $\lvert + \rangle = (\lvert 0 \rangle + \lvert 1 \rangle)/\sqrt 2$ sent with probability $1/2$ each, $\chi = S(\rho) < 1$ bit, so they can never be told apart perfectly.

Going deeper

For a student

The Holevo quantity $\chi$ is the main tool for classical communication over quantum channels. The Holevo–Schumacher–Westmoreland theorem shows that $\chi$, maximised over input ensembles, is achievable with many uses of a channel, so $\chi$ is the one-shot building block of the classical capacity. The proof by monotonicity of mutual information is a typical use of strong subadditivity. The bound $I(X:Y) \le \chi$ is usually strict; the maximum of $I(X:Y)$ over all measurements is called the accessible information.