The Church of the larger Hilbert space

== Introduction == [[John Smolin]] coined [https://doi.org/10.1063/1.1333282 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 $T\; :\; S(\backslash mathcal\{H\})\; \backslash rightarrow\; S(\backslash mathcal\{H\})$ be a completely positive and trace-preserving map between states on a finite-dimensional Hilbert space $H$. Then there exists a Hilbert space $\backslash mathcal\{K\}$ and a unitary operation $U$ on $\backslash mathcal\{H\}\; \backslash otimes\; \backslash mathcal\{K\}$ such that :: $T(\backslash varrho)\; =\; tr\_\{\backslash mathcal\{K\}\}\; U^\{\}(\; \backslash varrho\; \backslash otimes\; |0\backslash rangle\; \backslash langle\; 0|)U^\{\backslash dagger\}$ :for all $\backslash varrho\; \backslash in\; S(\backslash mathcal\{H\})$, where $tr\_\{\backslash mathcal\{K\}\}$ denotes the [[partial trace]] on the $\backslash mathcal\{K\}-$system. : The ancilla space $\backslash mathcal\{K\}$ can be chosen such that $\backslash dim\; \backslash mathcal\{K\}\; \backslash leq\; \backslash dim^\{2\}\; \backslash mathcal\{H\}$. 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 $|k\backslash rangle$ of the ancilla space $\backslash mathcal\{K\}$ and define the '''Kraus operators''' $t\_k$ in terms of Stinespring's unitary $U$ as : $\backslash langle\; a|t\_k|b\; \backslash rangle\; :=\; \backslash langle\; a\; \backslash otimes\; k\; |U|b\; \backslash otimes\; 0\; \backslash rangle$ 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 $T\; :\; S(\backslash mathcal\{H\})\; \backslash rightarrow\; S(\backslash mathcal\{H\})$ can be given the form :: $T(\backslash varrho)\; =\; \backslash sum\_\{k=1\}^\{K\}\; t\_\{k\}^\{\}\; \backslash ,\; \backslash varrho\; \backslash ,\; t\_\{k\}^\{\backslash dagger\}$ : for all $\backslash varrho\; \backslash in\; S(\backslash mathcal\{H\})$. The $K\; \backslash leq\; \backslash dim^\{2\}\; \backslash mathcal\{H\}$ Kraus operators $t\_k\; :\; \backslash mathcal\{H\}\; \backslash rightarrow\; \backslash mathcal\{H\}$ satisfy the completeness relation $\backslash sum\_\{k\}\; t\_\{k\}^\{\backslash dagger\}\; t\_\{k\}^\{\}\; =\; \backslash mathbf\{1\}$. == Purification of quantum states == Quantum [[states]] are channels $\backslash varrho:\; \backslash mathbb\{C\}\; \backslash rightarrow\; S(\backslash mathcal\{H\})$ with one-dimensional input space $\backslash mathbb\{C\}$ (cf. [[Channel (CP map)]]). We may thus apply Stinespring's dilation theorem to conclude that $\backslash varrho$ can be given the representation : $\backslash varrho\; =\; tr\_\{\backslash mathcal\{K\}\}\; |\backslash psi\backslash rangle\backslash langle\; \backslash psi\; |$, where $|\backslash psi\; \backslash rangle\; =\; U\; |0\; \backslash rangle$ is a [[pure state]] on the combined system $\backslash mathcal\{H\}\; \backslash otimes\; \backslash mathcal\{K\}$. In other words, every mixed state $\backslash varrho$ can be thought of as arising from a pure state $|\backslash psi\; \backslash rangle$ 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 $\backslash varrho\; =\; \backslash sum\_k\; p\_k\; \backslash ,\; |k\backslash rangle\; \backslash langle\; k|\; \backslash ,\; \backslash in\; S(\backslash mathcal\{H\})$, such a '''purification''' is given by the state : $|\backslash psi\; \backslash rangle\; =\; \backslash sum\_k\; \backslash ,\; \backslash sqrt\{p\_k\}\; \backslash ,\; |k\; \backslash rangle\; \backslash otimes\; |k\backslash rangle\; \backslash ,\; \backslash in\; \backslash mathcal\{H\}\; \backslash otimes\; \backslash mathcal\{H\}$. == 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; [http://arxiv.org/abs/quant-ph/0202122 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]]