What happens to a system over some time, seen alone, is a map $\hat{\rho}\to\mathcal{E}(\hat{\rho})$ from density matrices to density matrices: a quantum channel. It must be linear, keep the trace, and keep density matrices positive even when the system is one part of a larger whole — completely positive. Unitary evolution is a channel; so is a measurement whose result nobody reads; so is the depolarising channel, which with probability $p$ replaces the state by the completely mixed one and so shrinks the Bloch ball evenly towards its centre. Measurements seen through a larger system become positive operators adding to the identity — a POVM — which can do what projections cannot, such as recognise a state without ever being wrong, at the price of sometimes not answering.

A photocopier that blurs a little: every copy is still a page, but fine detail fades, and copies of copies fade further. A channel is such a machine for quantum states: it always returns a state, usually a less pure one.

A cut through the Bloch ball under the depolarising channel with $p=0.4$: every Bloch vector keeps $1-p=0.6$ of its length, so the sphere of pure states is carried onto the dashed sphere of radius 0.6.

A channel is a linear map of density matrices that keeps the trace and is completely positive. The depolarising channel is an example:

• Completely positive: $\mathcal{E}\otimes\mathcal{I}$ turns density matrices of the system with any partner into density matrices.
• Examples: $\hat{U}\hat{\rho}\hat{U}^{\dagger}$; a measurement along nobody reads, ; the depolarising channel, .

Measure the system and a partner together with projectors, the partner in a fixed state. Everything except the system’s state can be gathered into one operator on the system for each result. Those operators inherit Hermiticity, give probabilities that are never negative, and add to the identity because the projectors do.

Projectors $\hat{\Pi}_{m}$ on the whole, the partner in $\hat{\rho}_{E}$.

Gather everything but $\hat{\rho}$ into one operator on the system.

Take the adjoint, then move $\hat{\rho}_{E}$ round the trace over $E$.

The projectors add to the identity, and $\operatorname{tr}\hat{\rho}_{E}=1$.

For $\hat{\rho}=\ket{\psi}\bra{\psi}$ it is a probability.

A channel must be linear, keep the trace and be completely positive. It need not be reversible: most channels lose information.

The depolarising channel replaces the state by the completely mixed one with probability $p$, so every Bloch vector keeps $1-p$ of its length.

Unitary evolution is linear, keeps the trace and keeps every density matrix, with or without a partner, a density matrix.

Measuring along $z$ and forgetting the result leaves the populations and removes the coherences.

Great work! You now know how to:

• state what makes a map of density matrices a channel
• apply the depolarising channel and a measurement nobody reads
• derive a POVM from a measurement of a larger system