Basic Decoding Theory

This is based on Holevo's Theorem (Holevo, 1973) - Quantum Information Capacity. Given that Alice is trying to send a message to Bob, the decoding error is

Note that

1. If $N\geq d$, then $P_E\geq 0$ – there is no way to reliably distinguish messages

2. lower bound in theorem also holds even if alice sends mixed states $\{\rho_\alpha\}_{\alpha=1}^N$

3. A quantum system with Hilbert space dimension $d$ has information capacity of $\log_2(d)$ bits. This is the maximum number of bits that can be stored, communicated, and read reliably. This is the same amount of infromation from classical d-state system.