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 trK 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 tk in terms of Stinespring's unitary U as

atkb⟩ :  = ⟨a ⊗ kUb ⊗ 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 tk : H → H satisfy the completeness relation ktk † tk = 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

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

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

Category:Handbook of Quantum Information Category:Mathematical Structure

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Monday, October 26, 2015 - 17:56