6.3.3 Quantum Channels#

Prompts

What is a quantum channel? Why must physical evolution of density matrices be completely positive and trace-preserving (CPTP)?
State the Kraus representation theorem. How does the completeness relation $\sum_k K_k^\dagger K_k = I$ ensure trace preservation?
Compare the depolarizing, amplitude damping, and dephasing channels: what physical noise does each model, and what are their Kraus operators?
How are unitary evolution and projective measurement special cases of quantum channels?

Lecture Notes#

Overview#

A quantum channel is a linear map that describes the most general physical evolution of a quantum system. Unlike unitary evolution (which is reversible), quantum channels include irreversible processes: measurement, decoherence, noise. The key insight is that any physical process transforming density matrices must satisfy complete positivity (CPTP) and trace preservation. The Kraus representation provides the universal form: every CPTP channel can be written as a sum of “jump operators” $\mathcal{E}(\hat{\rho}) = \sum_k K_k \hat{\rho} K_k^\dagger$. In this section, we define CPTP maps rigorously, derive the Kraus theorem, and examine canonical examples (depolarizing, dephasing, amplitude damping) that model real quantum noise.

Definition: Quantum Channels#

A quantum channel (or quantum operation) is a linear map:

\[ \mathcal{E}: \text{Density matrices on } \mathcal{H}_\text{in} \to \text{Density matrices on } \mathcal{H}_\text{out} \]

representing any physical process: unitary evolution, measurement, dissipation, entanglement with environment, etc.

CPTP Properties#

A valid quantum channel must satisfy four conditions:

1. Linearity:

(141)#\[ \mathcal{E}(p\hat{\rho}_1 + q\hat{\rho}_2) = p\mathcal{E}(\hat{\rho}_1) + q\mathcal{E}(\hat{\rho}_2) \]

2. Trace preservation:

(142)#\[ \text{Tr}(\mathcal{E}(\hat{\rho})) = \text{Tr}(\hat{\rho}) = 1 \]

Probabilities are conserved; output is a valid density matrix.

3. Positivity:

(143)#\[ \hat{\rho} \geq 0 \quad \Rightarrow \quad \mathcal{E}(\hat{\rho}) \geq 0 \]

Positive semidefinite inputs map to positive semidefinite outputs.

4. Complete positivity:

(144)#\[ \mathcal{E} \otimes \mathcal{I}_n \text{ is positive for all } n \]

Even when the system is entangled with auxiliary systems, the map preserves positivity.

Physical Motivation for Complete Positivity

Consider two qubits A and B, entangled. You apply a channel $\mathcal{E}$ to A only. If $\mathcal{E}$ is merely positive but not completely positive, the map $\mathcal{E} \otimes \mathcal{I}_B$ could produce negative eigenvalues in the joint system (negative probability!). This is unphysical. Therefore, all physical channels must be completely positive.

Definition: CPTP Channel

A completely positive, trace-preserving (CPTP) map $\mathcal{E}$ is a linear map on density matrices satisfying linearity, trace preservation, positivity, and complete positivity. CPTP maps are the mathematical characterization of all physical quantum processes.

Kraus Representation Theorem#

Theorem: A linear map $\mathcal{E}$ on density matrices is CPTP if and only if it can be written in Kraus form:

(145)#\[ \mathcal{E}(\hat{\rho}) = \sum_{k=1}^r K_k \hat{\rho} K_k^\dagger \]

where the Kraus operators $\{K_k\}$ satisfy:

(146)#\[ \sum_k K_k^\dagger K_k = I \]

The Kraus representation ensures complete positivity and traces preservation:

Proof: Kraus Representation Satisfies CPTP

Trace-preserving: $$\text{Tr}(\sum_k K_k \hat{\rho} K_k^\dagger) = \sum_k \text{Tr}(\hat{\rho} K_k^\dagger K_k) = \text{Tr}(\hat{\rho} \sum_k K_k^\dagger K_k) = \text{Tr}(\hat{\rho} \cdot I) = \text{Tr}(\hat{\rho})$$

Positive semidefinite output: If $\hat{\rho} \geq 0$, then $\mathcal{E}(\hat{\rho}) = \sum_k K_k \hat{\rho} K_k^\dagger \geq 0$ (sum of positive operators).

Completely positive: The Kraus structure ensures positivity even under tensor product with identity (standard proof involves expanding in eigenbasis).

Non-Uniqueness of Kraus Operators#

Given one set $\{K_k\}$, other valid sets exist related by unitary rotation:

\[ K_k' = \sum_j U_{kj} K_j \]

where $\{U_{kj}\}$ is unitary. Different Kraus representations encode the same physical channel.

Discussion

The Kraus representation is non-unique, but the channel map $\mathcal{E}(\hat{\rho})$ is always the same. What does this tell you about the relationship between Kraus operators and physical observables? Could two very different sets of Kraus operators describe the same noise process?

Common Quantum Channels#

1. Unitary Channel#

(147)#\[ \mathcal{E}_U(\hat{\rho}) = U \hat{\rho} U^\dagger \]

Kraus form: Single operator $K = U$ with $U^\dagger U = I$.

Properties: Reversible (inverse is $\mathcal{E}_{U^\dagger}$); entropy conserved; no information loss.

2. Depolarizing Channel#

With probability $p$, the state becomes completely random; with probability $1-p$, unchanged:

(148)#\[ \mathcal{E}_\text{depol}(\hat{\rho}) = (1-p)\hat{\rho} + p\frac{I}{d} \]

where $d = \dim(\mathcal{H})$.

Kraus operators (qubit, $d=2$):

(149)#\[ K_0 = \sqrt{1-p}\, I, \quad K_1 = \sqrt{p/3}\, \hat{\sigma}^x, \quad K_2 = \sqrt{p/3}\, \hat{\sigma}^y, \quad K_3 = \sqrt{p/3}\, \hat{\sigma}^z \]

Physical interpretation: Errors on all three Pauli axes with equal probability; models generic noise.

3. Amplitude Damping ($T_1$ Process)#

Excited state decays to ground state at rate $\Gamma = 1/T_1$:

(150)#\[ \mathcal{E}_\text{amp}(\hat{\rho}) = K_0 \hat{\rho} K_0^\dagger + K_1 \hat{\rho} K_1^\dagger \]

with:

(151)#\[\begin{split} K_0 = \begin{pmatrix} 1 & 0 \\ 0 & \sqrt{1-\gamma} \end{pmatrix}, \quad K_1 = \begin{pmatrix} 0 & \sqrt{\gamma} \\ 0 & 0 \end{pmatrix} \end{split}\]

where $\gamma = 1 - \mathrm{e}^{-t/T_1}$ is the decay probability.

Physical interpretation: Energy dissipation; ground state is absorbing (fixed point).

4. Phase Damping ($T_2$ Process, Dephasing)#

Coherences decay without energy loss at rate $\Gamma_\phi = 1/T_2$:

(152)#\[ \mathcal{E}_\text{phase}(\hat{\rho}) = K_0 \hat{\rho} K_0^\dagger + K_1 \hat{\rho} K_1^\dagger \]

with:

(153)#\[\begin{split} K_0 = \begin{pmatrix} 1 & 0 \\ 0 & \sqrt{1-\gamma} \end{pmatrix}, \quad K_1 = \begin{pmatrix} 0 & 0 \\ 0 & \sqrt{\gamma} \end{pmatrix} \end{split}\]

where $\gamma = 1 - \mathrm{e}^{-t/T_2}$.

Physical interpretation: Random phase fluctuations; off-diagonal elements (coherences) decay; diagonal (populations) preserved.

Reversibility and Irreversibility#

Reversible (Unitary) Channel:

Single Kraus operator $K = U$ (unitary).
Entropy conserved: $S(\mathcal{E}(\hat{\rho})) = S(\hat{\rho})$.
Inverse channel exists: $\mathcal{E}^{-1}(\sigma) = U^\dagger \sigma U$.
State can be recovered perfectly.

Irreversible (Non-Unitary) Channel:

Multiple Kraus operators ($r > 1$) or degenerate structure.
Entropy generally increases: $S(\mathcal{E}(\hat{\rho})) > S(\hat{\rho})$ (output more mixed).
No inverse: original state cannot be recovered (information lost).
Examples: Depolarizing, amplitude damping.

Summary#

Quantum channel: CPTP linear map from density matrices to density matrices; describes all physical quantum evolution.
CPTP: Completely positive and trace-preserving; ensures valid quantum states and conserved probabilities.
Kraus representation $\mathcal{E}(\hat{\rho}) = \sum_k K_k \hat{\rho} K_k^\dagger$ with $\sum_k K_k^\dagger K_k = I$: universal form for channels; non-unique.
Common channels: Unitary (reversible), depolarizing (generic noise), amplitude damping ($T_1$), phase damping ($T_2$).
Reversibility: Unitary channels conserve entropy; non-unitary channels increase entropy (information loss).
Unification: Unitary evolution is a special case ($r=1$); projective measurement is a special case (see 6.3.2 POVM).

Homework#

1. Kraus Completeness Relation. Consider a quantum channel with Kraus operators $\{K_1, K_2, K_3\}$. Show that the completeness relation $\sum_k K_k^\dagger K_k = I$ is necessary and sufficient for the channel to be trace-preserving. Start with the trace of the output $\text{Tr}(\mathcal{E}(\hat{\rho})) = \sum_k \text{Tr}(K_k \hat{\rho} K_k^\dagger)$ and use the cyclic property of trace.

2. Depolarizing Channel on Bloch Sphere. A qubit depolarizing channel with parameter $p$ is given by:

\[ \mathcal{E}_\text{depol}(\hat{\rho}) = (1-p)\hat{\rho} + \frac{p}{2}I \]

Express this in Kraus form using Pauli operators $\{I, \hat{\sigma}^x, \hat{\sigma}^y, \hat{\sigma}^z\}$.

(a) Write down all four Kraus operators explicitly.

(b) Verify the completeness relation $\sum_k K_k^\dagger K_k = I$.

(c) Interpret the action on the Bloch vector: if $\hat{\rho} = \frac{1}{2}(I + \boldsymbol{r} \cdot \hat{\boldsymbol{\sigma}})$ with Bloch vector $\boldsymbol{r}$, show that the output has Bloch vector $\boldsymbol{r}' = (1-p)\boldsymbol{r}$.

3. Amplitude Damping and Spontaneous Emission. A two-level atom decays from excited state $\vert 1\rangle$ to ground state $\vert 0\rangle$ with lifetime $T_1$. After time $t$, the amplitude damping channel is given by Kraus operators:

\[\begin{split} K_0 = \begin{pmatrix} 1 & 0 \\ 0 & \sqrt{1-\gamma} \end{pmatrix}, \quad K_1 = \begin{pmatrix} 0 & \sqrt{\gamma} \\ 0 & 0 \end{pmatrix} \end{split}\]

where $\gamma = 1 - \mathrm{e}^{-t/T_1}$.

(a) Verify the completeness relation $K_0^\dagger K_0 + K_1^\dagger K_1 = I$.

(b) Apply the channel to a pure state $\vert\psi\rangle = \alpha\vert 0\rangle + \beta\vert 1\rangle$. Write the output density matrix $\mathcal{E}_\text{amp}(\vert\psi\rangle\langle\psi\vert)$ and interpret physically.

(c) Show that the ground state $\vert 0\rangle$ is a fixed point of the channel: $\mathcal{E}_\text{amp}(\vert 0\rangle\langle 0\vert) = \vert 0\rangle\langle 0\vert$.

4. Completely Positive Trace-Preserving (CPTP) Maps. Consider the map $\mathcal{E}$ defined by:

\[ \mathcal{E}(\hat{\rho}) = \frac{1}{2}\hat{\rho} + \frac{1}{2}\sigma^z \hat{\rho} \sigma^z \]

where $\sigma^z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$.

(a) Show that this map is linear and trace-preserving.

(b) Express this map in Kraus form with operators $K_0$ and $K_1$.

(c) Verify complete positivity by checking that the map remains positive when tensored with identity: test on the maximally entangled state $\vert\Phi^+\rangle = \frac{1}{\sqrt{2}}(\vert 00\rangle + \vert 11\rangle)$ and confirm the resulting density matrix is positive semidefinite.

5. Amplitude Damping as Kraus Map. For the amplitude damping channel (Problem 3), consider a pure state $\hat{\rho}_\text{in} = \vert+\rangle\langle+\vert$ where $\vert+\rangle = \frac{1}{\sqrt{2}}(\vert 0\rangle + \vert 1\rangle)$.

(a) Compute $\mathcal{E}_\text{amp}(\hat{\rho}_\text{in})$ for $\gamma = 0.5$.

(b) What is the purity of the output state? Is information lost?

6. Composing Channels. Consider two channels:

Phase flip: $\mathcal{E}_\text{phase}(\hat{\rho}) = (1-q)\hat{\rho} + q\sigma^z\hat{\rho}\sigma^z$
Amplitude damping: $\mathcal{E}_\text{amp}$ from Problem 3 with $\gamma = 0.1$

(a) Express the phase-flip channel in Kraus form.

(b) The composition $\mathcal{E}_\text{comp} = \mathcal{E}_\text{amp} \circ \mathcal{E}_\text{phase}$ applies phase flip first, then amplitude damping. Write the Kraus operators of the composed channel.

(c) Show that the composed channel is still CPTP. Is the composition commutative? Does $\mathcal{E}_\text{amp} \circ \mathcal{E}_\text{phase} = \mathcal{E}_\text{phase} \circ \mathcal{E}_\text{amp}$? Explain physically why or why not.

7. Distinguishing Quantum Noise Channels. Three noise channels (all with parameter $p = 0.5$) act on a qubit:

Bit-flip (X noise): $\mathcal{E}_X(\hat{\rho}) = (1-p)\hat{\rho} + p\sigma^x\hat{\rho}\sigma^x$
Phase-flip (Z noise): $\mathcal{E}_Z(\hat{\rho}) = (1-p)\hat{\rho} + p\sigma^z\hat{\rho}\sigma^z$
Depolarizing: $\mathcal{E}_\text{depol}(\hat{\rho}) = (1-p)\hat{\rho} + \frac{p}{2}I$ (random Pauli, normalized)

(a) Apply each channel to the input state $\hat{\rho}_\text{in} = \frac{1}{2}(I + \hat{\sigma}^x)$ (eigenstate of $\sigma^x$). Compare the outputs: which channel preserves the most information?

(b) Conceptual: The depolarizing channel is called “universal noise” because it cannot distinguish the computational basis. Explain why bit-flip and phase-flip channels can be distinguished by an appropriate measurement on the output, but depolarizing cannot.

6.3.3 Quantum Channels

Contents