Abstract We characterise absolutely dilatable completely positive maps on the space of all bounded operators on a Hilbert space that are also bimodular over a given von Neumann algebra as rotations by a suitable unitary on a larger Hilbert space followed by slicing along the trace of an additional ancilla. We define the local, quantum, and approximately quantum types of absolutely dilatable maps, according to the type of the admissible ancilla. We show that the local absolutely dilatable maps admit an exact factorisation through an abelian ancilla and show that they are limits in the point weak* topology of conjugations by unitaries in the commutant of the given von Neumann algebra. We show that the Connes Embedding Problem is equivalent to deciding if all absolutely dilatable maps are approximately quantum.
Abstract We characterise absolutely dilatable completely positive maps on the space of all bounded operators on a Hilbert space that are also bimodular over a given von Neumann algebra as rotations by a suitable unitary on a larger Hilbert space followed by slicing along the trace of an additional ancilla. We define the local, quantum, and approximately quantum types of absolutely dilatable maps, according to the type of the admissible ancilla. We show that the local absolutely dilatable maps admit an exact factorisation through an abelian ancilla and show that they are limits in the point weak* topology of conjugations by unitaries in the commutant of the given von Neumann algebra. We show that the Connes Embedding Problem is equivalent to deciding if all absolutely dilatable maps are approximately quantum.
CP∞ and beyond: 2-categorical dilation theory
The problem of extending the insights and techniques of categorical quantum mechanics to infinite-dimensional systems was considered in (Coecke and Heunen, 2016). In that work the CP∞-construction, which recovers the category of Hilbert spaces and quantum operations from the category of Hilbert spaces and bounded linear maps, was defined. Here we show that by a `horizontal categorification' of the CP∞-construction, one can recover the category of all von Neumann algebras and channels (normal unital completely positive maps) from the 2-category [W∗] of von Neumann algebras, bimodules and intertwiners. As an application, we extend Choi's characterisation of extremal channels between finite-dimensional matrix algebras to a characterisation of extremal channels between arbitrary von Neumann algebras.
The problem of extending the insights and techniques of categorical quantum mechanics to infinite-dimensional systems was considered in (Coecke and Heunen, 2016).In that work the CP ∞ -construction, which recovers the category of Hilbert spaces and quantum operations from the category of Hilbert spaces and bounded linear maps, was defined.Here we show that by a 'horizontal categorification' of the CP ∞construction, one can recover the category of all von Neumann algebras and channels (normal unital completely positive maps) from the 2-category of von Neumann algebras, bimodules and intertwiners.As an application, we extend Choi's characterisation of extremal channels between finite-dimensional matrix algebras to a characterisation of extremal channels between arbitrary von Neumann algebras.
Quantum supermaps are higher-order maps transforming quantum operations into quantum operations. Here we extend the theory of quantum supermaps, originally formulated in the finite dimensional setting, to the case of higher-order maps transforming quantum operations with input in a separable von Neumann algebra and output in the algebra of the bounded operators on a given separable Hilbert space. In this setting we prove two dilation theorems for quantum supermaps that are the analogues of the Stinespring and Radon-Nikodym theorems for quantum operations. Finally, we consider the case of quantum superinstruments, namely measures with values in the set of quantum supermaps, and derive a dilation theorem for them that is analogue to Ozawa's theorem for quantum instruments. The three dilation theorems presented here show that all the supermaps defined in this paper can be implemented by connecting devices in quantum circuits.
Quantum supermaps are higher-order maps transforming quantum operations into quantum operations. Here we extend the theory of quantum supermaps, originally formulated in the finite-dimensional setting, to the case of higher-order maps transforming quantum operations with input in a separable von Neumann algebra and output in the algebra of the bounded operators on a given separable Hilbert space. In this setting we prove two dilation theorems for quantum supermaps that are the analogues of the Stinespring and Radon-Nikodym theorems for quantum operations. Finally, we consider the case of quantum superinstruments, namely measures with values in the set of quantum supermaps, and derive a dilation theorem for them that is analogue to Ozawa's theorem for quantum instruments. The three dilation theorems presented here show that all the supermaps defined in this paper can be implemented by connecting devices in quantum circuits.
Channel--state duality identifies completely positive maps with bipartite states. Its generalisation to von Neumann algebras is known to produce states not on M⊗ˉN but on the tensor product of M with the \emph{opposite} algebra of N, or equivalently with a commutant; returning to N itself requires a transposition, and it has been observed that this step typically fails. We determine exactly when it does not fail. Our main result is that a Choi--Jamiołkowski type isomorphism which is natural in the first algebra exists if and only if the second algebra is ∗-anti-isomorphic to itself. Naturality cannot be dropped: whenever M is anti-isomorphic to itself such an isomorphism exists for trivial reasons, since the order structure of a predual is a Jordan invariant and cannot distinguish an algebra from its opposite. Consequently no such correspondence exists for the type III factors constructed by Connes. We show in addition that the domain admits no description in terms of properties of individual maps: the space of all normal maps with the operator norm is too large, and neither complete boundedness nor complete positivity cuts it down to the right size.
Motivated by applicability to quantum operations, quantum information, and quantum probability, we investigate the notion of absolute continuity for operator valued completely positive maps on C∗-algebras, previously introduced by Parthasarathy [in Athens Conference on Applied Probability and Time Series Analysis I (Springer-Verlag, Berlin, 1996), pp. 34–54]. We obtain an intrinsic definition of absolute continuity, we show that the Lebesgue decomposition defined by Parthasarathy is the maximal one among all other Lebesgue-type decompositions and that this maximal Lebesgue decomposition does not depend on the jointly dominating completely positive map, we obtain more flexible formulas for calculating the maximal Lebesgue decomposition, and we point out the nonuniqueness of the Lebesgue decomposition as well as a sufficient condition for uniqueness. In addition, we consider Radon–Nikodym derivatives for absolutely continuous completely positive maps that, in general, are unbounded positive self-adjoint operators affiliated to a certain von Neumann algebra, and we obtain a spectral approximation by bounded Radon–Nikodym derivatives. An application to the existence of the infimum of two completely positive maps is indicated, and formulas in terms of Choi’s matrices for the Lebesgue decomposition of completely positive maps in matrix algebras are obtained.