## 7.4 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.
Note that when we fix the initial state of system ^{107}
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**^{108}**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} . In quantum information science we often refer to this approach as*“the Church of the Larger Hilbert Space”*.**Kraus**^{109}**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 between the input and output Hilbert spaces\rho \longmapsto \rho'= \sum_i E_i\rho E_i^\dagger. 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. This operator–sum decomposition is not unique, since the operatorsE_i (known as the**Kraus operators**or**effects**) depend on the choice of basis in the ancilla. The Kraus operators must satisfy the normalisation condition\sum_i E^\dagger_iE_i=\mathbf{1} , also known as the**completeness relation**.

We can easily switch between these two equivalent representations:

- 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 .

In summary:

All linear transformations of density operators that can be written in the Stinespring (or, equivalently, Kraus) form represent *physically realisable operations*, and we call them **quantum channels**.^{110}

We note again that the Kraus decomposition is *not unique*: the operators

In summary:

Suppose ^{111} to ensure that * \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

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. Isometries preserve inner products, and therefore also preserve both the norm and the metric based upon the norm.↩︎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.↩︎Quantum channels are also known as

**superoperators**— this way physicists remind themselves that these transformations take operators to operators.↩︎If you wish, instead of appending zeros, you may view

R_{ji} as an isometry from the the smaller to the larger set of Kraus operators.↩︎