What Is a Quantum Channel

A closed quantum system evolves by a unitary $U$ , and a measurement is described by a set of operators. But most physical processes are neither: a qubit couples to its environment, a photon is lost, a spin dephases. To describe what can happen to a quantum state in full generality we need a broader object than a unitary. That object is the quantum channel.

States are density operators

We work with density operators rather than state vectors, because the output of a noisy process is generally a mixed state. A density operator $\rho$ on a Hilbert space $\mathcal{H}$ is

\rho = \rho^\dagger \succeq 0, \qquad \operatorname{tr}\rho = 1,

i.e. Hermitian, positive semidefinite (no negative eigenvalues, so all "probabilities" are nonnegative), and unit trace (probabilities sum to one). Pure states $|\psi\rangle$ are the special case $\rho = |\psi\rangle\langle\psi|$ , for which $\rho^2 = \rho$ .

What a channel must do

A channel is a map $\mathcal{E}$ that sends density operators on an input space to density operators on an output space. To be physically sensible it must turn legitimate states into legitimate states even when applied to part of a larger system. Three requirements capture this.

Linearity. Probabilistic mixtures must be respected: if the input is $p\rho_1 + (1-p)\rho_2$ , the output must be $p\,\mathcal{E}(\rho_1) + (1-p)\,\mathcal{E}(\rho_2)$ . A nonlinear map would let you distinguish how a mixture was prepared, which is operationally meaningless.
Trace preservation. Probabilities must keep summing to one: $\operatorname{tr}\mathcal{E}(\rho) = \operatorname{tr}\rho$ for all $\rho$ . (A trace-non-increasing map describes a process conditioned on an outcome; a deterministic channel preserves the trace exactly.)
Complete positivity. $\mathcal{E}$ must map positive operators to positive operators — and so must $\mathcal{E} \otimes \mathbb{1}_R$ for every ancillary system $R$ . Mere positivity is not enough; the transpose map is positive but not completely positive, and is therefore not a channel.

A map satisfying all three is completely positive and trace preserving — CPTP — and quantum channel is just a name for a CPTP map. The next two lessons unpack complete positivity and trace preservation in detail; the rest of the module develops the concrete representations and examples.

\boxed{\;\mathcal{E} \text{ is a quantum channel} \iff \mathcal{E} \text{ is linear, completely positive, and trace preserving.}\;}

Where channels come from

Every channel arises from a simple physical picture: couple the system $S$ to an environment $E$ prepared in some fixed state, evolve the pair unitarily, then ignore (trace out) the environment:

\mathcal{E}(\rho_S) = \operatorname{tr}_E\!\big[\,U\,(\rho_S \otimes |0\rangle\langle 0|_E)\,U^\dagger\,\big].

This is the Stinespring/dilation viewpoint, and a theorem we meet later guarantees the converse: every CPTP map can be written this way. So "quantum channel" and "unitary on a larger system, followed by discarding the environment" are exactly the same class of processes. Closed-system unitary evolution is the special case where the environment is trivial.

Examples to keep in mind

The identity channel $\mathcal{E}(\rho)=\rho$ — a perfect, noiseless wire.
A unitary channel $\mathcal{E}(\rho)=U\rho U^\dagger$ — reversible evolution.
The completely depolarizing channel $\mathcal{E}(\rho)=\tfrac{I}{d}$ — total loss of information, output is always maximally mixed.
Amplitude damping — models spontaneous emission / energy loss to a cold bath.

These span the range from "nothing happens" to "everything is lost," and the machinery of this module lets us describe, compare, and quantify all of them on equal footing.

The takeaway

A quantum channel is the most general physically allowed transformation of a quantum state: a linear, completely positive, trace-preserving map on density operators. Equivalently, it is what you get by coupling to an environment, evolving unitarily, and discarding the environment. Everything that follows — Kraus operators, the Choi state, dilations, fidelities — is a tool for working with this single object.