# The Church of the larger Hilbert space

### Introduction

John Smolin coined the phrase "Going to the Church of the Larger Hilbert Space" for the dilation constructions of channels and states, which not only provide a neat characterization of the set of permissible quantum operations but are also a most useful tool in quantum information science.

According to **Stinespring's dilation theorem**, every completely positive and trace-preserving map, or channel, can be built from the basic operations of (1) tensoring with a second system in a specified state, (2) unitary transformation, and (3) reduction to a subsystem. Thus, any quantum operation can be thought of as arising from a unitary evolution on a larger (*dilated*) system. The auxiliary system to which one has to couple the given one is usually called the ancilla of the channel. Stinespring's representation comes with a bound on the dimension of the ancilla system, and is unique up to unitary equivalence.

### Stinespring's dilation theorem

We present Stinespring's theorem in a version adapted to completely positive and trace-preserving maps between finite-dimensional quantum systems. For simplicity, we assume that the input and output systems coincide. The theorem applies more generally to completely positive (not necessarily trace-preserving) maps between *C* * − algebras.

**Stinespring's dilation:**Let

*T*:

*S*(H) →

*S*(H) be a completely positive and trace-preserving map between states on a finite-dimensional Hilbert space

*H*. Then there exists a Hilbert space K and a unitary operation

*U*on H ⊗ K such that $T(\varrho) = tr_{\mathcal{K}} U^{}( \varrho \otimes |0\rangle \langle 0|)U^{\dagger}$ for all $\varrho \in S(\mathcal{H})$, where

*t*

*r*K denotes the partial trace on the K − system. The ancilla space K can be chosen such that dimK ≤ dim2H. This representation is unique up to unitary equivalence.

### Kraus decomposition

It is sometimes useful not to go to a larger Hilbert space, but to work with operators between the input and output Hilbert spaces of the channel itself. Such a representation can be immediately obtained from Stinespring's theorem: We introduce a basis ∣*k*⟩ of the ancilla space K and define the **Kraus operators** *t**k* in terms of Stinespring's unitary *U* as

*a*∣

*t*

*k*∣

*b*⟩ : = ⟨

*a*⊗

*k*∣

*U*∣

*b*⊗ 0⟩

The Stinespring representation then becomes the **operator-sum decomposition** or **Kraus decomposition** of the quantum channel *T*:

**Kraus decomposition:**Every completely positive and trace-preserving map

*T*:

*S*(H) →

*S*(H) can be given the form $T(\varrho) = \sum_{k=1}^{K} t_{k}^{} \, \varrho \, t_{k}^{\dagger}$ for all $\varrho \in S(\mathcal{H})$. The

*K*≤ dim2H Kraus operators

*t*

*k*: H → H satisfy the completeness relation ∑

*k*

*t*

*k*†

*t*

*k*=

**1**.

### Purification of quantum states

Quantum states are channels $\varrho: \mathbb{C} \rightarrow S(\mathcal{H})$ with one-dimensional input space C (cf. Channel (CP map)). We may thus apply Stinespring's dilation theorem to conclude that $\varrho$ can be given the representation

$\varrho = tr_{\mathcal{K}} |\psi\rangle\langle \psi |$,where ∣*ψ*⟩ = *U*∣0⟩ is a pure state on the combined system H ⊗ K. In other words, every mixed state $\varrho$ can be thought of as arising from a pure state ∣*ψ*⟩ on a larger Hilbert space. This special version of Stinespring's theorem is usually called the **GNS construction** of quantum states, after Gelfand and Naimark, and Segal.

For a given mixed state with spectral decomposition $\varrho = \sum_k p_k \, |k\rangle \langle k| \, \in S(\mathcal{H})$, such a **purification** is given by the state

### References and further reading

- M. A. Nielsen, I. L. Chuang:
*Quantum Computation and Quantum Information*; Cambridge University Press, Cambridge 2000 - K. Kraus:
*States, Effects, and Operations*; Springer, Berlin 1983 - E. B. Davies:
*Quantum Theory of Open Systems*; Academic Press, London 1976 - V. Paulsen:
*Completely Bounded Maps and Operator Algebras*; Cambridge University Press, Cambridge 2002 - M. Keyl:
*Fundamentals of Quantum Information Theory*; Phys. Rep.**369**(2002) 431-548; quant-ph/0202122 - W. F. Stinespring:
*Positive Functions on*; Proc. Amer. Math. Soc.*C** − algebras**6**(1955) 211 - I. M. Gelfand, M. A. Naimark:
*On the Imbedding of Normed Rings into the Ring of Operators in Hilbert space*; Mat. Sb.**12**(1943) 197 - I. E. Segal:
*Irreducible Representations of Operator Algebras*; Bull. Math. Soc.**61**(1947) 69

### See also

Category:Handbook of Quantum Information Category:Mathematical Structure