QCQI – Chapter 11 Detailed Summary: Quantum Shannon Theory & Channel Capacities

Course Notes (MSU)

Learning Objectives¶

• Understand Schumacher compression and the role of von Neumann entropy as the quantum source coding rate.

• Use the Holevo information $\chi$ and the HSW theorem for classical capacity of quantum channels.

• Define coherent information $I_c$ and state the Lloyd–Shor–Devetak (LSD) theorem for quantum capacity.

• Explain entanglement-assisted communication and compute $C_E(\mathcal{N})=\max_\rho I(A{:}B)$.

• Work hands-on with qubit channels (depolarizing, amplitude damping): simulate, estimate entropies/capacities numerically.

Quantum Source Coding (Schumacher)¶

An i.i.d. source emits pure states $\{|\psi_x\rangle,p_x\}$ with average density $\rho=\sum_x p_x|\psi_x\rangle\langle \psi_x|$. For $n$ signals, the typical subspace $\mathcal{T}_\delta^{(n)}$ has dimension $\approx 2^{n S(\rho)}$ and captures almost all probability.

Theorem (Schumacher). For any $\epsilon,\delta>0$ and large $n$, there exist an isometry (encoder) $V:\mathcal{H}^{\otimes n}\to (\mathbb{C}^2)^{\otimes m}$ and a decoder such that $m/n \le S(\rho)+\delta$ and the average fidelity is at least $1-\epsilon$.

Eureka! Entropy is the qubit compression rate: $S(\rho)$ qubits per signal, just like Shannon’s $H(p)$ bits.

Classical Communication over Quantum Channels¶

A quantum channel $\mathcal{N}$ is CPTP. To send classical messages $x$, choose codewords (quantum states) $\{\rho_x\}$ and a POVM at the receiver.

Holevo Information & HSW¶

For ensemble $\mathcal{E}=\{p_x,\rho_x\}$ and outputs $\sigma_x=\mathcal{N}(\rho_x)$,

HSW theorem: The product-state classical capacity is $C^{(1)}(\mathcal{N}) = \max_{\mathcal{E}} \chi(\mathcal{E},\mathcal{N}).$ The (regularized) classical capacity $C(\mathcal{N}) = \lim_{k\to\infty}\frac{1}{k} C^{(1)}(\mathcal{N}^{\otimes k})$; additivity need not hold in general.

Eureka! Accessible information is upper-bounded by Holevo $\chi$; optimal codes approach it (HSW).

Quantum Capacity & Coherent Information¶

Coherent information for input state $\rho$ is $I_c(\rho,\mathcal{N}) = S(\mathcal{N}(\rho)) - S((\mathrm{id}\otimes\mathcal{N})(|\Phi\rangle\!\langle \Phi|))$ where $|\Phi\rangle$ purifies $\rho$. LSD theorem: The quantum capacity $Q(\mathcal{N}) = \lim_{k\to\infty}\frac{1}{k}\max_{\rho^{(k)}} I_c(\rho^{(k)},\mathcal{N}^{\otimes k})$. For degradable channels, the limit is single-letter.

Eureka! Coherent information plays Shannon’s role for quantum data: it is the rate of reliably transmitted qubits.

Entanglement-Assisted Capacities¶

With unlimited prior EPR pairs, the entanglement-assisted classical capacity is $C_E(\mathcal{N}) = \max_\rho I(A{:}B)_{(\mathrm{id}\otimes\mathcal{N})(\Phi_\rho)},$ the quantum mutual information of the channel’s Choi state constructed from the purification $\Phi_\rho$. This is single-letter and additive.

Eureka! Shared entanglement upgrades a noisy quantum channel to a clean classical pipe of size mutual information.

Canonical Qubit Channels¶

Depolarizing:¶

$\mathcal{D}_p(\rho)=(1-p)\rho + \frac{p}{3}(X\rho X+Y\rho Y+Z\rho Z)$. Symmetric and unital; $C_E$ is maximized by $\rho=I/2$.

Amplitude Damping:¶

with Kraus $E_0=\begin{pmatrix}1&0\\0&\sqrt{1-\gamma}\end{pmatrix}$, $E_1=\begin{pmatrix}0&\sqrt{\gamma}\\0&0\end{pmatrix}$. Degradable for $\gamma\le 1/2$ (single-letter $Q$), anti-degradable for large $\gamma$ ($Q=0$).

Pseudo-code Aids¶

Pseudo-code: Schumacher compression (typical-subspace sketch)

Input: source average $\rho$, blocklength $n$, thresholds $(\delta,\epsilon)$ Compute eigenpairs $\{\lambda_i,|i\rangle\}$ of $\rho$ Define projector $P$ onto sequences $i^n$ with empirical entropy $\in [S(\rho)\pm\delta]$ Encoder: isometry $V: P\mathcal{H}^{\otimes n}\to (\mathbb{C}^2)^{\otimes m}$ with $m\approx n S(\rho)$ Decoder: inverse isometry on typical subspace; declare failure otherwise

Pseudo-code: Holevo $\chi$ for an ensemble through a channel

Input: $\{p_x,\rho_x\}$, channel $\mathcal{N}$ Compute $\sigma_x=\mathcal{N}(\rho_x)$ Return $\chi = S(\sum_x p_x \sigma_x) - \sum_x p_x S(\sigma_x)$

Pseudo-code: Coherent information (one-letter lower bound)

Input: state $\rho$, channel $\mathcal{N}$ Let $|\Phi\rangle_{RA}$ be a purification of $\rho$ Compute $\sigma_{RB}=(\mathrm{id}_R\otimes \mathcal{N}_A)(|\Phi\rangle\!\langle \Phi|)$ Return $I_c = S(\sigma_B)-S(\sigma_{RB})$

Pseudo-code: Entanglement-assisted capacity

Input: channel $\mathcal{N}$ Optimize over $\rho$: $C_E(\mathcal{N})=\max_\rho I(R{:}B)$ of $(\mathrm{id}\otimes\mathcal{N})(\Phi_\rho)$

Schematics¶

Schumacher Compression¶

Channel Capacities (concept)¶

Hands-on Notebook (Multi-Backend)¶

Run: L15_QCQI_Ch11_Quantum_Shannon_Theory_and_Channel_Capacities

In the first cell set backend to one of: cirq, pennylane, braket, pyquil, or qiskit.

Mini-Lab Ideas¶

1. Simulate depolarizing/amplitude-damping channels; compute $\chi$ for simple ensembles and plot vs. noise.

2. Numerically maximize $I(R{:}B)$ for $C_E$ on qubit depolarizing; compare with maximally mixed input.

3. Estimate a one-letter lower bound on $Q$ by maximizing $I_c$ over single-qubit states.

4. Emulate Schumacher compression on small $n$ with a typical-subspace projector; measure average fidelity.