## 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 *not* necessarily unitary, but (in order for the overall transformation

This allows us to define an isometry

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

Let us now rephrase our derivation of the evolution of system

**Stinespring**^{131}**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 isometryV , 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**.**Kraus**^{132}**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 operatorsE_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 isometryV , and then to a map on density operators represented by a set of Kraus operatorsE_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 isometryV=\sum_i|i\rangle\otimes E_i . In terms of matrices, this corresponds to simply “stacking up” the matricesE_i to form the block column (as shown in Figure 7.1), which gives us the matrix representation ofV . If we want to go further, from an isometryV to a unitaryU , then the next step is somewhat arbitrary: we can choose all the remaining block columns ofU however we please,*as long as*we end up with a unitary matrixU .

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

In summary:

Suppose ^{134} to ensure that

Then *if and only if*

In particular, this unitary equivalence of the Kraus operators implies that the identity channel

Recall that a map

V is an isometry ifV^\dagger V=\mathbf{1} . For example, adding a system in state|k\rangle gives an isometryV\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.↩︎William Forrest “Woody” Stinespring (1929–2012) was an American mathematician specialising in operator theory.↩︎

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

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.↩︎