7.5 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.¹³⁰ 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 7.1.

Figure 7.1: For k=2, the second block column is selected. The matrix representation of the isometry V on the right-hand side look like a column vector, but remember that the entries E_i\coloneqq B_{ik} are matrices.

Let us now rephrase our derivation of the evolution of system \mathcal{B} using isometries. Note that the isometry V in Figure 7.1 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.

• Stinespring¹³¹ 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 7.4. In quantum information science, we often refer to this approach as the Church of the Larger Hilbert Space.

• Kraus¹³² 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 7.1).
• 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 7.1), 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 the Stinespring (or, equivalently, Kraus) form represent physically realisable operations — we call them quantum channels, or superoperators.¹³³

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 list¹³⁴ to ensure that n=m.

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.

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130. Recall that a map V is an isometry if V^\dagger V=\mathbf{1}. For example, adding a system in state |k\rangle gives an isometry V\colon|\psi\rangle\mapsto|k\rangle\otimes|\psi\rangle, and the combination of adding a system in a fixed state followed by a unitary evolution of the combined system is also an isometry.↩︎

131. William Forrest “Woody” Stinespring (1929–2012) was an American mathematician specialising in operator theory.↩︎

132. Karl Kraus (1938–1988) was a German physicist known for his contributions to the mathematical foundations of quantum theory. His book States, effects, and operations (Lecture Notes in Physics, Vol. 190, Springer-Verlag, Berlin 1983) is an early account of the notion of complete positivity in physics.↩︎

133. Physicists like the terminology “superoperators”, since it makes it clear that these transformations send operators to operators.↩︎

134. If you wish, instead of appending zeros, you may view R_{ji} as an isometry (instead of a unitary) from the the smaller to the larger set of Kraus operators.↩︎