The Church of the larger Hilbert space

Contents

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(\mathcal{H}) \rightarrow S(\mathcal{H}) be a completely positive and trace-preserving map between states on a finite-dimensional Hilbert space H. Then there exists a Hilbert space \mathcal{K} and a unitary operation U on \mathcal{H} \otimes \mathcal{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 tr_{\mathcal{K}} denotes the partial trace on the \mathcal{K}-system.
The ancilla space \mathcal{K} can be chosen such that \dim \mathcal{K} \leq \dim^{2} \mathcal{H}. 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\rangle of the ancilla space  \mathcal{K} and define the Kraus operators tk in terms of Stinespring's unitary U as

 \langle a|t_k|b \rangle := \langle a \otimes k |U|b \otimes 0 \rangle

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(\mathcal{H}) \rightarrow S(\mathcal{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 \leq \dim^{2} \mathcal{H} Kraus operators t_k : \mathcal{H} \rightarrow \mathcal{H} satisfy the completeness relation  \sum_{k} t_{k}^{\dagger} t_{k}^{} = \mathbf{1}.


Purification of quantum states

Quantum states are channels  \varrho: \mathbb{C} \rightarrow S(\mathcal{H}) with one-dimensional input space \mathbb{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  |\psi \rangle = U |0 \rangle is a pure state on the combined system  \mathcal{H} \otimes \mathcal{K}. In other words, every mixed state \varrho can be thought of as arising from a pure state  |\psi \rangle 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

 |\psi \rangle = \sum_k \, \sqrt{p_k} \, |k \rangle \otimes |k\rangle \, \in \mathcal{H} \otimes \mathcal{H} .

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 C * algebras; Proc. Amer. Math. Soc. 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