== 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(𝜚) = tr_𝒦U(𝜚 ⊗ |0⟩⟨0|)U^†

for all 𝜚 ∈ S(ℋ), where tr_𝒦 denotes the partial trace on the 𝒦−system.

The ancilla space 𝒦 can be chosen such that dim 𝒦 ≤ 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 𝜚 ∈ S(ℋ). The K ≤ dim²ℋ Kraus operators t_k : ℋ → ℋ satisfy the completeness relation ∑_kt_k^†t_k = 1.

Purification of quantum states

Quantum states are channels 𝜚 : ℂ → S(ℋ) with one-dimensional input space ℂ (cf. Channel (CP map)). We may thus apply Stinespring's dilation theorem to conclude that 𝜚 can be given the representation

𝜚 = tr_𝒦|ψ⟩⟨ψ|,

where |ψ⟩ = U|0⟩ is a pure state on the combined system ℋ ⊗ 𝒦. In other words, every mixed state 𝜚 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 𝜚 = ∑_kp_k |k⟩⟨k|  ∈ S(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

• Channel (CP map)
• Measurements and preparations

Category:Handbook of Quantum Information Category:Mathematical Structure