QCQI - Chapter 5: Density Operators & Quantum Operations

Learning Objectives¶

• Density matrices & Bloch representation.

• Channels (CPTP), Kraus, and dilations.

• Choi test; trace distance & fidelity monotonicity.

Density Operators¶

$\rho=\frac12(I+\vec r\cdot\vec \sigma)$; purity $\mathrm{Tr}(\rho^2)=\tfrac{1+\|\vec r\|^2}{2}$.

Eureka! Noise contracts the Bloch vector.

Quantum Operations¶

$\mathcal{E}(\rho)=\sum_i K_i \rho K_i^\dagger$, $\sum_i K_i^\dagger K_i=I$.

Stinespring + Naimark provide physical realizations.

Eureka! Open dynamics = unitary on a bigger space + discard.

Choi Matrix¶

$J(\mathcal{E})=(\mathcal{E}\!\otimes I)(|\Phi\rangle\!\langle \Phi|)$.

CP $\Leftrightarrow J(\mathcal{E})\ge0$, TP $\Leftrightarrow \text{Tr}_{\text{out}}J=I$.

Eureka! One PSD test certifies complete positivity.

Distances & Fidelity¶

$D$ contracts, $F$ increases under CPTP.

Schematics¶

Hands-on Notebook¶

Companion notebook (multi-backend): L06_QCQI_Ch05_Density_Operators_and_Quantum_Operations

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

Mini-Lab¶

• Verify TP/unital; compute $(A,\vec b)$ for common channels.

• Build & test $J(\mathcal{E})$ for amplitude damping.

• Check $D\downarrow$ and $F\uparrow$ under channels.