## Evolution of open systems

Needless to say, there is more to evolutions of open systems than random isometries, and what follows is the most general scenario we will discuss.
Consider two interacting systems, \mathcal{A} and \mathcal{B}, but this time we do *not* assume that their interacting dynamics admits a control-target interpretation.
We will view \mathcal{A} as an auxiliary system, i.e. an ancilla, and focus on the evolution of system \mathcal{B}.

Let us pick an orthonormal basis |i\rangle of the Hilbert space \mathcal{H}_\mathcal{A} associated with the ancilla.
Any unitary transformation of the combined system \mathcal{AB} can then be written as
U
= \sum_{ij}|i\rangle\langle j|\otimes B_{ij}
=
\begin{bmatrix}
B_{11} & B_{12} & B_{13} & \ldots
\\B_{21} & B_{22} & B_{23} & \ldots
\\B_{31} & B_{32} & B_{33} & \ldots
\\\vdots & \vdots & \vdots & \ddots
\end{bmatrix}
where the B_{ij} are operators acting on the the Hilbert space \mathcal{H}_\mathcal{B} associated with system \mathcal{B}.
Note that the B_{ij} do *not* need to be unitary, but, for the overall transformation U to be unitary, they must satisfy
\begin{aligned}
\sum_i B_{ik}^\dagger B_{il}
&= \delta_{kl} \mathbf{1}_\mathcal{AB}
\\\sum_i B_{ki}B_{li}^\dagger
&= \delta_{kl} \mathbf{1}_\mathcal{B}
\end{aligned}
\tag{$\star$}
where \mathbf{1}_\mathcal{AB} and \mathbf{1}_\mathcal{B} are the identity operators on \mathcal{H}_\mathcal{A}\otimes\mathcal{H}_\mathcal{B} and \mathcal{H}_\mathcal{B}, respectively.
These two conditions correspond to the requirement that both column and row vectors must be orthonormal for a matrix to be unitary, except that here U is a block matrix, and the entries B_{ij} are complex matrices rather than complex numbers, so some care must be taken with the order of multiplication.
Again, the evolution of the system \mathcal{B} depends on both U *and* on the initial state of the auxiliary system \mathcal{A}.

Without any loss of generality, we may assume that system \mathcal{A} is in a pure state which can be chosen to be one of the basis states |i\rangle, say |k\rangle.
In this case, U acts by
U\colon |k\rangle\otimes|\psi\rangle \longmapsto
\sum_i |i\rangle\otimes B_{ik}|\psi\rangle
\tag{$\ddagger$}
for an arbitrary state |\psi\rangle of \mathcal{B}.

The resulting density operator for \mathcal{B} is found by taking the density operator of the output state of \mathcal{AB}, which is
\sum_{ij} |i\rangle\langle j|\otimes B_{ik}|\psi\rangle\langle\psi|B_{jk}^\dagger
and then tracing out \mathcal{A}, obtaining
\begin{aligned}
\operatorname{tr}_\mathcal{A} \left(
\sum_{ij} |i\rangle\langle j|\otimes B_{ik}|\psi\rangle\langle\psi|B_{jk}^\dagger
\right)
&= \sum_{ij} \langle i|j\rangle\cdot B_{ik}|\psi\rangle\langle\psi|B_{jk}^\dagger
\\&= \sum_i B_{ik}|\psi\rangle\langle\psi|B_{ik}^\dagger
\end{aligned}
where we have used the fact that \langle i|j\rangle=\delta_{ij}.
In general, for any input state \rho, we obtain the map
\rho\longmapsto
\rho'= \sum_i B_{ik}\rho B^\dagger_{ik} \equiv \sum_i B_{i}\rho B^\dagger_{i}
where, in the last expression on the right-hand size, we have dropped index k (remember, it was there only to remind us about the initial state of the ancilla).
Since the overall transformation U is unitary, recall that the B_i satisfy \sum_i B_i^\dagger B_i=\mathbf{1}.
This normalisation conditions guarantees that the trace is preserved.

In summary, we can think about a quantum evolution of subsystem \mathcal{B} as a sequence of the three distinct operations:
\begin{aligned}
\rho
\longmapsto &\underbrace{|k\rangle\langle k|\otimes\rho}_{\text{add ancilla}}
\\\longmapsto &\underbrace{U(|k\rangle\langle k|\otimes\rho) U^\dagger}_{\text{unitary evolution}}
\\\longmapsto &\underbrace{\operatorname{tr}_\mathcal{A} \left[U(|k\rangle\langle k|\otimes\rho) U^\dagger\right]}_{\text{discard ancilla}}
= \sum_i B_{i}\rho B_{i}^\dagger
=\rho'.
\end{aligned}

In summary:

- 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.
- 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.
- The output system in this scenario does not have to be the original input system, but usually it is.

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 *non-unitary* character of this evolution.