Quantum Channel Capacity

Quantum channel capacity is the maximum rate at which quantum information — not classical bits, but quantum states themselves — can be transmitted reliably through a noisy quantum channel. Unlike classical channel capacity, which Shannon proved is governed by a single formula, quantum channel capacity is fragmented into multiple distinct capacities depending on what resources the sender and receiver share: the quantum capacity \(Q\) for transmitting quantum states, the classical capacity \(C\) for transmitting classical bits, and the entanglement-assisted capacity when prior entanglement is available.

The most striking feature of quantum channel capacity is that the quantum capacity can be zero even when the classical capacity is positive. A channel may faithfully transmit classical information while destroying quantum coherence entirely — a phenomenon with no classical analog, rooted in the no-cloning theorem and the fragility of entanglement under decoherence. The quantum capacity is given by the regularized coherent information, a quantity that is notoriously difficult to compute and is not known to be additive across channel uses.

This non-additivity means that the capacity of a quantum channel may only be achieved by encoding information across entangled inputs to multiple uses of the channel — a feature that makes quantum channel theory substantially more complex than its classical counterpart. The quantum error correction threshold theorem can be understood as the statement that below a critical noise level, the quantum capacity of a physical channel is positive.