Decoherence Channels
Quantum channels describe noise as completely positive trace-preserving maps; common ones model bit-flip, phase damping, and energy loss.
Noise as a map on states
Real qubits undergo non-unitary evolution when they interact with an environment. Such open-system dynamics are described by quantum channels: maps that take a density matrix to another valid density matrix. Mathematically they are completely positive and trace-preserving (CPTP) maps.
Operator-sum form
Every channel can be written as E(rho) = sum_k K_k rho K_k-dagger, where the Kraus operators K_k satisfy sum_k K_k-dagger K_k = I. This form encodes the effect of tracing out an environment the qubit has become entangled with. Unitary evolution is the special case of a single Kraus operator.
Standard qubit channels
- Bit-flip: applies X with some probability p, leaves the qubit alone otherwise
- Phase-flip: applies Z with probability p, degrading coherence
- Depolarising: replaces the state with the maximally mixed state with probability p
- Amplitude damping: models energy loss, |1> decaying to |0> (linked to T1)
- Phase damping: models pure dephasing, loss of coherence without energy loss (linked to T2)
How they act on the Bloch ball
Each channel deforms the Bloch ball. Depolarising shrinks it uniformly toward the centre. Amplitude damping pulls the ball toward the north pole (|0>) as excitations decay. Phase damping flattens it onto the z axis by erasing x and y components — the coherences. These pictures make the abstract Kraus maps tangible.
Why the formalism matters
Channels are the language for modelling, simulating, and correcting real noise. Error-correction thresholds are computed against channel models; benchmarking extracts effective channel parameters; and simulators propagate density matrices through channels to predict how a circuit degrades. They are how decoherence is made quantitative.