Quantum Channel

A mathematical description of how quantum states transform, including noise and decoherence effects.

A quantum channel (also called a quantum operation or completely positive trace-preserving map) describes how quantum states evolve, especially when interacting with an environment.

Definition

A quantum channel $\mathcal{E}$ maps density matrices to density matrices:

$$\mathcal{E}:\rho \to {\rho }^{\prime }$$

It must be:

1. Linear: $\mathcal{E}(a{\rho }_1+b{\rho }_2)=a\mathcal{E}({\rho }_1)+b\mathcal{E}({\rho }_2)$
2. Completely positive: Maps positive operators to positive operators (even when extended)
3. Trace-preserving: $\text{Tr}(\mathcal{E}(\rho ))=\text{Tr}(\rho )=1$

Kraus Representation

Every quantum channel can be written as:

$$\mathcal{E}(\rho )=\sum _k{K}_k\rho K_k^{†}$$

where $\{K_k\}$ are Kraus operators satisfying ${\sum }_k{K}_k^{†}{K}_k=I$.

Common Quantum Channels

Unitary Channel

Ideal, noise-free evolution: $$\mathcal{E}(\rho )=U\rho U^{†}$$

Single Kraus operator: $K=U$.

Depolarizing Channel

Random Pauli errors with probability $p$: $$\mathcal{E}(\rho )=(1-p)\rho +\frac{p}{3}(X\rho X+Y\rho Y+Z\rho Z)$$

Turns state toward maximally mixed state.

Amplitude Damping

Models energy loss (e.g., photon emission): $$K_0=\left(\begin{array}{cc}1& 0\\ 0& \sqrt{1-\gamma }\end{array}\right),K_1=\left(\begin{array}{cc}0& \sqrt{\gamma }\\ 0& 0\end{array}\right)$$

Related to T1 decay.

Phase Damping (Dephasing)

Loses phase information without energy loss: $$\mathcal{E}(\rho )=(1-p)\rho +pZ\rho Z$$

Related to T2 decay.

Bit Flip Channel

Random X error: $$\mathcal{E}(\rho )=(1-p)\rho +pX\rho X$$

Phase Flip Channel

Random Z error: $$\mathcal{E}(\rho )=(1-p)\rho +pZ\rho Z$$

Channel Composition

Channels compose naturally: $$({\mathcal{E}}_2\circ {\mathcal{E}}_1)(\rho )={\mathcal{E}}_2({\mathcal{E}}_1(\rho ))$$

Multiple noise sources combine into a single effective channel.

Channel Capacity

The quantum channel capacity quantifies how much quantum information can be reliably transmitted. Different capacities exist for:

• Classical information over quantum channel
• Quantum information (with/without entanglement assistance)

Why It Matters

Quantum channels are essential for:

• Modeling realistic noise in quantum computers
• Designing error correction codes
• Quantum communication protocols
• Understanding decoherence

See also: Density Matrix, Decoherence, Quantum Error Correction