QCQI – Chapter 5 Detailed Summary: Density Operators & Quantum Operations

Learning Objectives¶

Work with density operators: mixtures, partial trace, Bloch form.
Define and use quantum operations (CPTP maps) with Kraus operators.
Interpret channels via Stinespring dilation and Choi–Jamiołkowski isomorphism.
Compute trace distance and fidelity; understand data-processing monotonicity.
Model common noise (depolarizing, dephasing, amplitude damping) and its Bloch-affine action.

Density Operators¶

A state is a positive semidefinite operator with unit trace: $\rho\ge 0$ , $\operatorname{Tr}\rho=1$ . For a qubit,

\begin{aligned} \rho = \frac{1}{2}\!\left(I + r_x X + r_y Y + r_z Z\right),\qquad \vec r\in\mathbb{R}^3,\ \|\vec r\|\le 1. \end{aligned}

(1)

Purity: $\operatorname{Tr}(\rho^2)=\tfrac{1+\|\vec r\|^2}{2}$ .

Eureka! Pure $\Leftrightarrow \|\vec r\|=1$ . Noise shrinks $\|\vec r\|$ inward.

Quantum Operations (CPTP)¶

A physical operation $\mathcal{E}$ is linear, completely positive, and trace-preserving (CPTP). Kraus form:

\begin{aligned} \mathcal{E}(\rho)=\sum_i K_i \rho K_i^\dagger,\qquad \sum_i K_i^\dagger K_i = I. \end{aligned}

(2)

Kraus representations are not unique (related by isometries).

Eureka! “Many realizations, one map”: different $\{K_i\}$ sets describe the same channel.

Dilation Pictures¶

Stinespring. There exists an ancilla $E$ , unitary $U$ on $S\!E$ , and fixed $|0\rangle_E$ such that

\begin{aligned} \mathcal{E}(\rho)=\operatorname{Tr}_E\left[U\,(\rho\otimes |0\rangle\!\langle 0|)\,U^\dagger\right]. \end{aligned}

(3)

Naimark. Any POVM arises from a projective measurement on a larger Hilbert space.

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

Choi–Jamiołkowski (CJ) Isomorphism¶

Define the (unnormalized) maximally entangled state $|\Phi\rangle=\sum_j |j\rangle|j\rangle$ and the Choi matrix

\begin{aligned} J(\mathcal{E}) = (\mathcal{E}\otimes I)\big(|\Phi\rangle\!\langle \Phi|\big). \end{aligned}

(4)

$\mathcal{E}$ is CP iff $J(\mathcal{E})\ge 0$ and TP iff $\operatorname{Tr}_{\text{out}} J(\mathcal{E})=I_{\text{in}}$ .

Eureka! Checking CP reduces to checking positivity of a single matrix: the Choi matrix.

Distances & Fidelity¶

Trace distance $D(\rho,\sigma)=\tfrac12\|\rho-\sigma\|_1$ contracts under CPTP:

\begin{aligned} D(\mathcal{E}(\rho),\mathcal{E}(\sigma)) \le D(\rho,\sigma). \end{aligned}

(5)

Uhlmann fidelity $F(\rho,\sigma)=\left(\operatorname{Tr}\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}\right)^2$ is non-decreasing under CPTP:

\begin{aligned} F(\mathcal{E}(\rho),\mathcal{E}(\sigma)) \ge F(\rho,\sigma). \end{aligned}

(6)

Eureka! Processing information can only make states harder to tell apart (distance down, fidelity up).

Canonical Qubit Channels (Bloch-Affine Forms)¶

Depolarizing (prob. $p$ ): $\rho \mapsto (1-p)\rho + \frac{p}{2}I$ ; $\vec r\mapsto (1-p)\vec r$ .
Dephasing / Phase-damping ( $p$ ): $r_x,r_y\mapsto (1-2p)\,r_{x,y}$ ; $r_z$ unchanged.
Amplitude damping ( $\gamma$ ): non-unital; roughly $r_{x,y}\mapsto \sqrt{1-\gamma}\,r_{x,y}$ , $r_z\mapsto (1-\gamma)r_z+\gamma$ .

Eureka! Unital channels map $I\!\!I/2 \mapsto I\!\!I/2$ (Bloch shift $\vec b=0$ ). Amplitude damping is non-unital ( $\vec b\neq 0$ ).

POVMs¶

POVM elements $\{E_m\}$ satisfy $E_m\ge 0$ , $\sum_m E_m=I$ . Probabilities: $p(m)=\operatorname{Tr}(E_m\rho)$ .

Realization via Naimark: embed in larger space, apply unitary, measure projectively, and post-process.

Worked Examples¶

Amplitude damping Kraus set. $K_0=\begin{pmatrix}1&0\\0&\sqrt{1-\gamma}\end{pmatrix}$ , $K_1=\begin{pmatrix}0&\sqrt{\gamma}\\0&0\end{pmatrix}$ .

Choi test. Build $J(\mathcal{E})$ ; check PSD and TP by partial trace.

Monotonicity demo. Pick $\rho,\sigma$ ; compute $D,F$ before/after $\mathcal{E}$ .

Pseudo-code Aids¶

Pseudocode — Apply a channel in Kraus form

Input: density matrix $\rho$ , list of Kraus ops $\{K_i\}$ Check: $\sum_i K_i^\dagger K_i \approx I$ (TP) Return: $\sum_i K_i \rho K_i^\dagger$

Pseudocode — Build the Choi matrix

Input: channel $\mathcal{E}$ via $\{K_i\}$ on $d$ -dim space Construct $|\Phi\rangle=\sum_{j=0}^{d-1}|j\rangle|j\rangle$ and $\Phi=|\Phi\rangle\!\langle \Phi|$ Return $J(\mathcal{E})=\sum_i (K_i\otimes I)\,\Phi\,(K_i^\dagger\otimes I)$

Pseudocode — Stinespring dilation (constructive sketch)

Input: Kraus ops $\{K_i\}$ Form isometry $V=\sum_i K_i\otimes |i\rangle_E$ ; extend $V$ to a unitary $U$ on $S\!E$ Realize $\mathcal{E}(\rho)=\operatorname{Tr}_E\big[U(\rho\otimes|0\rangle\!\langle 0|)U^\dagger\big]$

Schematics¶

Stinespring Dilation¶

Choi Test¶

Hands-on Notebook (Multi-Backend)¶

Run: L06_QCQI_Ch05_Density_Operators_and_Quantum_Operations

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

Mini-Lab Ideas¶

Verify TP/unital conditions for given Kraus sets; compute Bloch-affine $(A,\vec b)$ .
Build $J(\mathcal{E})$ for amplitude damping; check PSD and TP by partial trace.
Numerically confirm $D$ decreases and $F$ increases under (i) dephasing, (ii) amplitude damping.