## 9.4 Evolution of open systems

Needless to say, there is more to evolutions of open systems than mere random isometries, and what follows is the most general scenario that we will come across in our study of quantum information.

Consider two interacting systems, *not* assume that their interacting dynamics admits a control-target interpretation.
We will view ^{176} the evolution of system

Let us pick an orthonormal basis *not* need to be unitary, but, for the overall transformation *and* on the initial state of the auxiliary system

Without any loss of generality, we may assume that system ^{177}, which can be chosen to be one of the basis states

The resulting density operator for ^{178}

In summary, we can think about a quantum evolution of subsystem

In words:

- First we pick up a system of interest which, in general, can be in a mixed state
\rho . It may be the case that this system is entangled with some other degrees of freedom or with some other physical systems, but these other entities will remain passive and will not enter any subsequent dynamics. - Then we dilate the system: we add an ancilla which is large enough to include everything our system will interact with, and also large enough to be in a pure state. The expansion ends when the composed system is (for all practical purposes) isolated and follows a unitary evolution
U . - We allow the expanded system to evolve under the unitary evolution.
- After the unitary evolution takes place, we discard the ancilla and focus on the system alone. In fact we do not have to discard exactly what we added: we can discard only part of the ancilla, or any other part of the dilated system.
^{179}

It is adding (i.e. tensoring) the auxiliary system in a fixed state, and then discarding it (via the partial trace), that is responsible for the seemingly *non-unitary* character of this evolution.

The next step is to use what we have learnt about isometries (namely that they are like unitaries but where the dimension is allowed to increase) to combine the first two of these operations (adding an ancilla and following some unitary evolution) into a single operation.
This will lead to the so-called **Stinespring dilation theorem**, as well as its ancilla-free counterpart, the **Kraus decomposition**.

This three-stage process (adding an ancilla, unitary evolution, and then tracing out the ancilla) might reasonably be called a “factorisation”, since it factors a (non-unitary) evolution into constituent parts: first something that looks a bit like an injection (since it maps a smaller space into a bigger one); then something that looks a bit like an isomorphism (since unitaries are invertible); and finally something that looks a bit like a surjection (since it maps a bigger space down to a smaller one). For now, let’s forget about this middle part of the factorisation (where we let our system evolve unitarily), and just keep the first and last part in mind as we look at the following construction.

Pick any function

S\twoheadrightarrow\operatorname{Im}(f)\hookrightarrow T S\hookrightarrow S\sqcup(T\setminus\operatorname{Im}(f))\twoheadrightarrow T

where the first is a surjection followed by an injection, and the second is an injection followed by a surjection.
In the first decomposition, the middle set (namely *not* unique (we can use any set given by taking

The first of these decompositions is probably much more familiar and friendly looking than the second, but it is indeed the second which is of interest to us here, since it is of the same form as our three-stage process: something injective-looking followed by something surjective-looking.
Indeed, as shown in Cunningham and Heunen’s “Purity through Factorisation”, arXiv:1705.07652, Stinespring dilation (which is roughly this three-stage process that we’ve been talking about) gives rise to a **weak factorisation system**, but *not* an **orthogonal** one.

These notions (weak and orthogonal factorisation systems) are absolutely fundamental to a large area of modern mathematics that deals with homotopy theory and “higher structures” using the language of **model categories**.

For now, when we write tensor products, we will place the ancilla

*first*and the system of interest*second*:\mathcal{H}_\mathcal{A}\otimes\mathcal{H}_\mathcal{B} . We do this to begin with simply because block matrices on tensor products are easier to interpret when written in this particular order. Later on we will revert to the more common convention in which the system of interest is placed*first*.↩︎If

\mathcal{A} were initially in a mixed state, we could always regard\mathcal{A} as a subsystem of some larger\widetilde{\mathcal{A}} that is in an entangled pure state.↩︎Recall that

\langle i|j\rangle=\delta_{ij} .↩︎Because of this, the output system in this scenario does not have to be the same as the original input system (e.g. it could be strictly larger), but usually it is.↩︎