Every channel can be written with a few operators on the system alone: $\hat{\rho}\to\sum_{l}\hat{M}_{l}\hat{\rho}\hat{M}_{l}^{\dagger}$, with $\sum_{l}\hat{M}_{l}^{\dagger}\hat{M}_{l}=\hat{1}$. For a system meeting an environment that starts in $\ket{e_{0}}$, they are $\hat{M}_{l}=\bra{e_{l}}\hat{U}\ket{e_{0}}$, one for each state the environment could be found in afterwards. Kraus proved the converse: every channel has this form, with at most $d^{2}$ operators, and sets related by a unitary describe the same channel. Two examples carry most of the physics of a qubit. Phase damping shrinks the coherences and leaves the populations alone. Amplitude damping lets the excited state $\ket{1}$ fall to the ground state $\ket{0}$ with probability $p$, as spontaneous decay does, and drives every state towards the ground state.

A pinball enters a machine, bounces off pins it cannot see, and leaves by one of several exits. The Kraus operators are the exits, each with its own effect on the ball; averaging over the exits gives the channel.

A cut through the Bloch ball under amplitude damping with $p=0.4$, the ground state $\ket{0}$ at the top. The sphere is carried onto an ellipsoid pulled up towards the ground state: $z$ lengths shrink by $1-p$, sideways ones by , and the centre moves up by .

Every channel can be written as

• Kraus’s theorem: every channel on a $d$-dimensional system has this form, with at most $d^{2}$ operators; sets related by a unitary mixing $\hat{M}_{k}'=\sum_{l}u_{kl}\hat{M}_{l}$ give the same channel. Stated here without proof.

Evolve the system with the environment, which starts in $\ket{e_{0}}$, and take the partial trace in a basis $\ket{e_{l}}$ of the environment. Each term of the trace sandwiches the system’s state between the operator and its adjoint, and the unitarity of makes these operators add up, as , to the identity.

The whole evolves; the environment starts in $\ket{e_{0}}$.

The partial trace in a basis $\ket{e_{l}}$ of the environment.

Each $\hat{M}_{l}$ is an operator on the system.

Completeness in the environment, then $\hat{U}^{\dagger}\hat{U}=\hat{1}$.

Amplitude damping lets the excited state fall to the ground state with probability $p$. The jump is the Kraus operator $\sqrt{p}\ket{0}\bra{1}$.

Probabilities add to one exactly when the Kraus operators satisfy $\sum\hat{M}^{\dagger}\hat{M}=1$.

Phase damping records whether the qubit is 0 or 1 and does nothing else: the populations stay and the coherences shrink by $1-p$.

Amplitude damping multiplies the coherence by $\sqrt{1-p}$, since only $\hat{M}_{0}$ keeps it, with on one side and on the other.

Great work! You now know how to:

• derive the Kraus form of a channel from a system and its environment
• apply phase damping and amplitude damping
• check that a set of Kraus operators keeps the trace