Quantum Noise & Channel Models

Real quantum systems are never perfectly isolated. They interact with their surroundings---the environment---leading to decoherence and information loss. To model this, we distinguish between closed quantum systems, which evolve unitarily, and open quantum systems, which exchange information with an external environment.

In the environment picture, we consider a system $S$ coupled to an environment $E$. The combined system $SE$ is closed and evolves via a joint unitary $\hat{U}_{\mathrm{SE}}$. If the environment starts in a pure state $|e_0\rangle$, the initial state of the full system is $\rho \otimes |e_0\rangle\langle e_0|$. After evolution, the state of the system alone is obtained by tracing out the environment.

• • A closed system evolves as $\rho \mapsto \hat{U}\rho \hat{U}^{\dagger}$.
• • An open system evolution is described by a completely positive trace-preserving (CPTP) map $\mathcal{E}$.
• • The map $\mathcal{E}$ captures all effects of the environment on the system.

Among the simplest quantum channels are the Pauli channels, which model discrete errors corresponding to bit flips, phase flips, and combined bit-phase flips. Each channel acts probabilistically: with probability $1-p$ the state is unchanged, and with probability $p$ a Pauli operator is applied.

The bit-flip channel applies the $\hat{X}$ operator with probability $p$. The phase-flip channel applies $\hat{Z}$, and the bit-phase-flip channel applies $\hat{Y}$. These channels provide concrete examples of how quantum information degrades and are foundational for understanding quantum error correction.

• • Bit-flip Kraus operators: $E_0 = \sqrt{1-p}\,\hat{I}$, $E_1 = \sqrt{p}\,\hat{X}$.
• • Phase-flip Kraus operators: $E_0 = \sqrt{1-p}\,\hat{I}$, $E_1 = \sqrt{p}\,\hat{Z}$.
• • Bit-phase-flip Kraus operators: $E_0 = \sqrt{1-p}\,\hat{I}$, $E_1 = \sqrt{p}\,\hat{Y}$.
• • On the Bloch sphere, bit-flip shrinks the $x$ component, phase-flip shrinks $z$, and bit-phase-flip shrinks $y$.

Amplitude damping models energy dissipation from a quantum system to its environment, such as spontaneous emission of a two-level atom. It describes the tendency of an excited state $|1\rangle$ to decay to the ground state $|0\rangle$.

The process is characterized by a damping parameter $\gamma$ related to the $T_1$ relaxation time. For evolution time $t$, we have $\gamma = 1 - e^{-t/T_1}$. As $t \rightarrow \infty$, the system relaxes completely to $|0\rangle$ regardless of the initial state.

• • The Kraus operator $E_1$ represents the quantum jump $|1\rangle \rightarrow |0\rangle$.
• • $E_0$ describes the no-jump evolution that renormalizes the amplitude of $|1\rangle$.
• • Amplitude damping is non-unital: it maps the maximally mixed state toward $|0\rangle\langle 0|$.

The depolarizing channel is a symmetric noise process where the qubit is replaced by the maximally mixed state $\hat{I}/2$ with probability $p$, and left unchanged otherwise. It can be written as a uniform mixture of Pauli errors.

Dephasing, or phase damping, describes the loss of off-diagonal elements in the computational basis. It arises from random phase fluctuations and is directly related to the transverse relaxation time $T_2$.

• • Dephasing in the computational basis: $\mathcal{E}(\rho_{ij}) = \rho_{ij}$ for $i=j$ and $\mathcal{E}(\rho_{ij}) = (1-p)\rho_{ij}$ for $i \neq j$.
• • For dephasing, the off-diagonal decay rate is related to $T_2$ by $p = 1 - e^{-t/T_2}$.
• • The depolarizing channel is unital: $\mathcal{E}(\hat{I}) = \hat{I}$.

To quantify how much a quantum state has been affected by noise, we need distance measures between quantum states. Two fundamental quantities are fidelity and trace distance. Fidelity measures the overlap between states, while trace distance measures their distinguishability.

The fidelity between two states $\rho$ and $\sigma$ is defined via the square root of the product. For pure states, fidelity reduces to the squared overlap. Trace distance is half the trace norm of the difference and satisfies a contractivity property under CPTP maps.

• • Fidelity satisfies $0 \leq F(\rho, \sigma) \leq 1$, with $F=1$ if and only if $\rho = \sigma$.
• • Trace distance satisfies the data processing inequality: $T(\mathcal{E}(\rho), \mathcal{E}(\sigma)) \leq T(\rho, \sigma)$ for any CPTP map $\mathcal{E}$.
• • Fidelity is NOT contractive in general: there exist channels for which $F(\mathcal{E}(\rho), \mathcal{E}(\sigma)) < F(\rho, \sigma)$.