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arXiv:1807.02533 · Compositionality

Stinespring's construction as an adjunction

Arthur J. Parzygnat

Published Dec 20, 2019• 7 citations
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Abstract

Given a representation of a unital C∗ -algebra A on a Hilbert space H , together with a bounded linear map V:K→H from some other Hilbert space, one obtains a completely positive map on A via restriction using the adjoint action associated to V . We show this restriction forms a natural transformation from a functor of C∗ -algebra representations to a functor of completely positive maps. We exhibit Stinespring's construction as a left adjoint of this restriction. Our Stinespring adjunction provides a universal property associated to minimal Stinespring dilations and morphisms of Stinespring dilations. We use these results to prove the purification postulate for all finite-dimensional C∗ -algebras.

DOI
10.32408/compositionality-1-2

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Given a representation of a unital C∗ -algebra A on a Hilbert space H , together with a bounded linear map V:K→H from some other Hilbert space, one obtains a completely positive map on A via restriction using the adjoint action associated to V . We show this restriction forms a natural transformation from a functor of C∗ -algebra representations to a functor of completely positive maps. We exhibit Stinespring's construction as a left adjoint of this restriction. Our Stinespring adjunction provides a universal property associated to minimal Stinespring dilations and morphisms of Stinespring dilations. We use these results to prove the purification postulate for all finite-dimensional C∗ -algebras.
In this article we investigate sesquilinear forms defined on the Cartesian product of Hilbert C^*-module M over C^*-algebra B and taking values in B. The set of all such defined sesquilinear forms is denoted by S_B (M). We consider completely positive maps from locally C^*-algebra A to S_B (M). Moreover we assume that these completely positive maps are covariant with respect to actions of a group symmetry. This allow us to view these maps as generalizations covariant quantum instruments which are very important for the modern quantum mechanic and the quantum field theory. We analyze the dilation problem for these class of maps. In order to solve this problem we construct the minimal Stinespring representation and prove that every two minimal representations are unitarily equivalent.
We prove a finite-dimensional covariant Stinespring theorem for compact quantum groups. Let G be a compact quantum group, and let T≔Rep(G) be the rigid C*-tensor category of finite-dimensional continuous unitary representations of G. Let Mod(T) be the rigid C*-2-category of cofinite semisimple finitely decomposable T-module categories. We show that finite-dimensional G-C*-algebras can be identified with equivalence classes of 1-morphisms out of the object T in Mod(T). For 1-morphisms X:T→M1, Y:T→M2, we show that covariant completely positive maps between the corresponding G-C*-algebras can be “dilated” to isometries τ: X → Y ⊗ E, where E:M2→M1 is some “environment” 1-morphism. Dilations are unique up to partial isometry on the environment; in particular, the dilation minimizing the quantum dimension of the environment is unique up to a unitary. When G is a compact group, this recovers previous covariant Stinespring-type theorems.
In this work, we investigate the existence of a factorization for a unital completely positive map, between non-commutative probability space which do not change the expectation values of the events. These maps are called in literature stochastic maps. Using the Stinespring representations of completely positive map and assuming the existence of anti-unitary operator on Hilbert space related to these representations which satisfying some modular relations, we prove that stochastic maps with adjoint, admits a factorization.
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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.
Almost-idempotent quantum channels and approximate C∗C^*C∗-algebras
Let ΦΦΦ be a unital completely positive (UCP) map on the space of operators on some Hilbert space. We assume that ΦΦΦ is ηηη-idempotent, namely, ∥Φ2−Φ∥cb≤η\|Φ^2-Φ\|_{\mathrm{cb}} \leη∥Φ2−Φ∥cb​≤η, and construct an associated ε\varepsilonε-C∗C^*C∗ algebra (of almost-invariant observables) for ε=O(η)\varepsilon=O(η)ε=O(η). This type of structure has the axioms of a unital C∗C^*C∗ algebra but the associativity and other axioms involving the multiplication and the unit hold up to ε\varepsilonε. We prove that any finite-dimensional ε\varepsilonε-C∗C^*C∗ algebra AAA is O(ε)O(\varepsilon)O(ε)-isomorphic to some genuine C∗C^*C∗ algebra BBB. These bounds are universal, i.e. do not depend on the dimensionality or other parameters. When AAA comes from a finite-dimensional ηηη-idempotent UCP map ΦΦΦ, the O(η)O(η)O(η)-isomorphism and its inverse can be realized by UCP maps. This gives an approximate factorization of the quantum channel Φ∗Φ^*Φ∗ into a decoding channel, producing a state on BBB, and an encoding channel.