## 7.6 Composition of quantum channels

We mentioned that quantum channels are combinations of

- adding a physical system in a fixed state (via tensoring),
- discarding a physical system (taking a partial trace),
- and unitary transformations.

For consistency let us note that each of these operations admits an operator-sum decomposition. This is obvious for unitary evolution (

**Adding a system.**Any quantum system can be expanded by bringing in an auxiliary system in a fixed state|a\rangle . This transformation takes vectors in the Hilbert space associated with the original system and tensors them with a fixed vector|a\rangle in the Hilbert space associated with the auxiliary system:|\psi\rangle \longmapsto |a\rangle\otimes|\psi\rangle = (|a\rangle\otimes\mathbf{1}) |\psi\rangle. In terms of density operators, we write this “expansion” transformation as\begin{aligned} \rho \longmapsto \rho' &= |a\rangle\langle a|\otimes\rho \\&= (|a\rangle\otimes\mathbf{1})\rho (\langle a|\otimes\mathbf{1}) \\&= V\rho V^\dagger \end{aligned} whereV=\mathbf{1}\otimes|a\rangle . We note thatV^\dagger V = \mathbf{1}\otimes\langle a|a|=\rangle\mathbf{1} is the identity in the Hilbert space associated with the system, and soV is an isometry. Indeed, this transformation is an*isometric embedding*.**Discarding a system.**Conversely, given a composite system in state\rho , we can discard one of its subsystems. The partial trace over an auxiliary system can be written in the Kraus representation as\begin{aligned} \rho \longmapsto \rho' &= \operatorname{tr}_\mathcal{A}\rho \\&= (\operatorname{tr}\otimes\mathbf{1})\rho \\&= \sum_i (\langle i|\otimes\mathbf{1})\rho(|i\rangle\otimes\mathbf{1}) \\&= \sum_i E_i\rho E^\dagger_i \end{aligned} where the vectors|i\rangle form an orthonormal basis in the Hilbert space associated with the auxiliary system. Again, we can check that the Kraus operatorsE_i=\langle i|\otimes\mathbf{1} satisfy the completeness relation\sum_i E^\dagger_i E_i =\mathbf{1}\otimes\mathbf{1} (using the fact that\sum_i|i\rangle\langle i|=\mathbf{1} ).

Any *sequential* composition of two quantum channels ^{112} described by the Kraus operators

You might wonder why we explicitly called the above composition “sequential” — isn’t this how we always compose functions?
In actual fact, since we have access to tensor products, there is another sort of composition, namely **parallel** composition: if we have systems

Now that we know how to compose quantum channels in terms of Kraus operators, we can see that the Stinespring representation is perfectly consistent with the Kraus representation: the three basic operations that we are allowed to use to build channels in the Stinespring representation (i.e. adding a system, unitary evolution, and discarding a system) are all themselves quantum channels, in that they admit a Kraus decomposition.

Before moving on, we make a small (but important) remark:

When we compose quantum channels, each channel needs its own independent ancilla;
*do not share ancillas between different channels*.

For example, say we have three channels,

For more on this, see the appendix on Markov approximation.

Here we have tacitly assumed that the dimensions agree, i.e. that the output of

\mathcal{E} and the input of\mathcal{F} are of the same dimension, so that the composition makes sense.↩︎