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, \mathcal{A} and \mathcal{B}, but this time 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 on¹⁷⁶ 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_{i,j}|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 state¹⁷⁷, 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_{i,j} |i\rangle\langle j|\otimes B_{ik}|\psi\rangle\langle\psi|B_{jk}^\dagger and then tracing out \mathcal{A}, obtaining¹⁷⁸ \begin{aligned} \operatorname{tr}_\mathcal{A} \left( \sum_{i,j} |i\rangle\langle j|\otimes B_{ik}|\psi\rangle\langle\psi|B_{jk}^\dagger \right) &= \sum_{i,j} \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} In general, for any input state \rho, we obtain the map \begin{aligned} \rho\longmapsto\rho' &= \sum_i B_{ik}\rho B^\dagger_{ik} \\&\eqqcolon \sum_i B_{i}\rho B^\dagger_{i} \end{aligned} 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 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 f\colon S\to T between sets. Then we can decompose f into an injection (\hookrightarrow) and a surjection (\twoheadrightarrow) in two different ways:

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 \operatorname{Im}(f)) is unique (up to unique isomorphism); in the second, the middle set (namely S\sqcup(T\setminus\operatorname{Im}(f))) is not unique (we can use any set given by taking S and adding an extra arbitrary element for each element of T that is not in the image of f).

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.↩︎