Stinespring’s dilation and Kraus’s ambiguity
Once we start playing with adding physical systems and increasing the dimension of the underlying Hilbert space, it is convenient to switch from unitaries to isometries.
This is more for mathematical simplicity than physical insight, but it is always good to declutter our equations a bit if we can.
Recall that any unitary transformation of the combined system \mathcal{AB} can 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}, and where the B_{ij} are not necessarily unitary, but (in order for the overall transformation U to be unitary) 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}
Also recall that, when we fix the initial state of system \mathcal{A} to be |k\rangle, we know that U acts by
U\colon |k\rangle\otimes|\psi\rangle \longmapsto
\sum_i |i\rangle\otimes B_{ik}|\psi\rangle
for an arbitrary state |\psi\rangle of \mathcal{B}.
This allows us to define an isometry V\colon\mathcal{H}_\mathcal{B}\to\mathcal{H}_\mathcal{A}\otimes\mathcal{H}_\mathcal{B} by
V\colon |\psi\rangle
\longmapsto \sum_i|i\rangle\otimes E_i|\psi\rangle
where E_i\coloneqq B_{ik}, which satisfy
\sum_i E_i^\dagger E_i=\mathbf{1}_{\mathcal{B}}.
The matrix representation of an isometry is a rectangular matrix given by selecting only a few of the columns from a unitary matrix;
here, with |k\rangle fixed, it is only the k-th column of the block matrix U that determines the evolution of \mathcal{B}, as shown in Figure 9.1.
Let us now rephrase our derivation of the evolution of system \mathcal{B} using isometries.
Note that the isometry V in Figure 9.1 acts by
|\psi\rangle\langle\psi|
\longmapsto
V|\psi\rangle\langle\psi|V^\dagger
= \sum_{i,j} |i\rangle\langle j| \otimes E_i|\psi\rangle\langle\psi| E_j^\dagger.
We trace out \mathcal{A} (recalling that \operatorname{tr}|i\rangle\langle j| = \langle i|j\rangle=\delta_{ij}) and express the evolution of system \mathcal{B} (which is allowed to have a mixed input state \rho, since these can always be expressed as statistical mixtures of pure states |\psi\rangle) as
\rho
\longmapsto
\rho' = \operatorname{tr}_\mathcal{A} V\rho V^\dagger =\sum_i E_i\rho E_i^\dagger,
where \sum_iE_i^\dagger E_i=\mathbf{1}.
This expression shows two different ways of looking at quantum evolutions, and both have their own name.
Stinespring dilation.
Any quantum channel \mathcal{E} can be thought of as arising from a unitary evolution on a dilated system.
When we combine tensoring and the unitary evolution into an isometry V, we can express the action of the channel \mathcal{E} as
\rho \longmapsto \rho'= \operatorname{tr}_\mathcal{A} V\rho V^\dagger,
where we trace out a suitably chosen ancilla \mathcal{A}.
This is the approach that we discussed in Section 9.4.
In quantum information science, we often refer to this approach as the Church of the Larger Hilbert Space.
Kraus representation (a.k.a. operator-sum decomposition).
It is often more convenient to not deal with a larger Hilbert space, but to instead work with operators directly between the input and output Hilbert spaces, avoiding the middle one completely:
\rho \longmapsto \rho'= \sum_i E_i\rho E_i^\dagger
where the Kraus operators (or effects) E_i satisfy the normalisation condition \sum_i E^\dagger_iE_i=\mathbf{1} (also known as the completeness relation).
Here we avoid dragging in the ancilla, which can be a good thing, since ancillas typically represent environments that can be very large and complex.
Note that this operator–sum decomposition is not unique, since the Kraus operators E_i depend on the choice of basis in the ancilla.
These two representations are equivalent, and we can easily switch between them:
- We have already seen how to go from a unitary evolution U on a larger system to an isometry V, and then to a map on density operators represented by a set of Kraus operators E_i (as in Figure 9.1).
- Conversely, once we have an operator-sum representation of the channel with a set of Kraus operators E_i, we can introduce an ancilla of dimension equal to the number of Kraus operators, and use the orthonormal basis |i\rangle to form the isometry V=\sum_i|i\rangle\otimes E_i.
In terms of matrices, this corresponds to simply “stacking up” the matrices E_i to form the block column (as shown in Figure 9.1), which gives us the matrix representation of V.
If we want to go further, from an isometry V to a unitary U, then the next step is somewhat arbitrary: we can choose all the remaining block columns of U however we please, as long as we end up with a unitary matrix U.
All linear transformations of density operators that can be written in Stinespring (or, equivalently, Kraus) form represent physically realisable operations — we call them quantum channels, or superoperators (since they send operators to operators).
We note again that the Kraus decomposition is not unique: the operators E_i depend on the choice of the ancilla basis.
Indeed, let |e_i\rangle and |f_j\rangle be two orthonormal bases in the Hilbert space associated with the ancilla.
Then V can be expressed as
\begin{aligned}
V
&= \sum_i|e_i\rangle\otimes E_i
\\&= \sum_{i,j} |f_j\rangle\langle f_j|e_i\rangle\otimes E_i
\\&= \sum_{j} |f_j\rangle \otimes \sum_i \underbrace{\langle f_j|e_i\rangle}_{R_{ji}} E_i
\\&= \sum_{j} |f_j\rangle \otimes F_j
\end{aligned}
where we have used the fact that \sum_j |f_j\rangle\langle f_j|=\mathbf{1}, and where R_{ji}=\langle f_j|e_i\rangle is a unitary matrix connecting the two orthonormal bases (and also the two sets of the Kraus operators) via F_j=\sum_i R_{ji} E_i.
So we have a set of Kraus operators E_i associated with basis |e_i\rangle and another, unitarily related, set of Kraus operators F_j associated with basis |f_j\rangle, and the two sets describe the same isometry, and hence the same quantum channel.
This correspondence goes both ways: if two channels \mathcal{E} and \mathcal{F} have their Kraus operators related by some unitary R_{ji}, then the two channels are identical:
\begin{aligned}
\mathcal{F}(\rho)
&= \sum_j F_j\rho F^\dagger_j
\\&= \sum_{i,j,k} R_{ji}E_i \rho E^\dagger_k R^\star_{jk}
\\&=\sum_{i,k} \underbrace{\left(\sum_j R_{jk}^\star R_{ji}\right)}_{\delta_{ki}} E_i\rho E^\dagger_k
\\&= \sum_i E_i\rho E^\dagger_i
\\&= \mathcal{E}(\rho).
\end{aligned}
In summary:
Suppose E_1,\ldots,E_n and F_1,\ldots,F_m are Kraus operators associated with quantum channels \mathcal{E} and \mathcal{F}, respectively.
We can append zero operators to the shorter list to ensure that n=m (or we could view R_{ij} as an isometry instead of a unitary).
Then \mathcal{E} and \mathcal{F} describe the same channel if and only if F_j=\sum_i R_{ji} E_i for some unitary R.
In particular, this unitary equivalence of the Kraus operators implies that the identity channel \rho\mapsto\rho'=\mathbf{1}\rho\mathbf{1} can only have Kraus operators that are proportional to the identity.
