QCQI – Chapter 11: Quantum Shannon Theory & Channel Capacities

Course Slides (MSU)

Learning Objectives¶

• Schumacher: $S(\rho)$ qubits per signal.

• HSW $\chi$ and classical capacity.

• Coherent information & LSD theorem.

• Entanglement-assisted $C_E=\max I(A{:}B)$.

Schumacher Compression¶

Typical subspace of size $\approx 2^{nS(\rho)}$ captures nearly all probability.

Eureka! Quantum entropy is the optimal compression rate.

Classical Capacity (HSW)¶

$C^{(1)}=\max_{\{p_x,\rho_x\}}\chi$; regularize for $C$ in general.

Eureka! Holevo $\chi$ bounds accessible information and is achievable.

Quantum Capacity¶

Coherent info $I_c(\rho,\mathcal{N})=S(\mathcal{N}(\rho))-S((\mathrm{id}\otimes\mathcal{N})(\Phi))$.

LSD: $Q=\lim\limits_{k}\frac{1}{k}\max I_c(\rho^{(k)},\mathcal{N}^{\otimes k})$ (single-letter for degradable channels).

Entanglement-Assisted¶

$C_E(\mathcal{N})=\max_\rho I(A{:}B)$ on Choi state.

Additive and single-letter.

Schematics¶

Hands-on Notebook¶

Companion notebook (multi-backend): L15_QCQI_Ch11_Quantum_Shannon_Theory_and_Channel_Capacities

Set backend in the first cell to cirq, pennylane, braket, pyquil, or qiskit.

Mini-Lab¶

• $\chi$ vs. noise for depolarizing.

• One-letter $Q$ lower bound via $I_c$.

• Entanglement-assisted $C_E$ numeric search.