arXiv:2108.09872 · Journal of Mathematical Physics

A covariant Stinespring theorem

Dominic Verdon

Published Sep 1, 2022• 8 citations

Abstract

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.

DOI: 10.1063/5.0071215

Related papers

Entanglement-symmetries of covariant channels

Dominic Verdon

Let G and G′ be monoidally equivalent compact quantum groups, and let H be a Hopf-Galois object realising a monoidal equivalence between these groups' representation categories. This monoidal equivalence induces an equivalence Chan( G ) → Chan( G′ ), where Chan( G ) is the category whose objects are finite-dimensional C∗ -algebras with an action of G and whose morphisms are covariant channels. We show that, if the Hopf-Galois object H has a finite-dimensional *-representation, then channels related by this equivalence can simulate each other using a finite-dimensional entangled resource. We use this result to calculate the entanglement-assisted capacities of certain quantum channels.

arXiv:2012.05761QuantumFeb 29, 20243 citations

Stinespring's construction as an adjunction

Arthur J. Parzygnat

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.

arXiv:1807.02533CompositionalityDec 20, 20197 citations

On a dilation of a some class of completely positive maps

Эльсаев Якуб Витальевич

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.

Tambov University Reports Series Natural and Technical SciencesJan 1, 20190 citations

Entanglement-equivalences of covariant channels

Dominic Verdon

We define two channels N and N' to be entanglement-equivalent if there exists an entanglement-assisted channel coding scheme for N from N' and vice versa. For example, the noiseless channel on a d^2-level classical system and the noiseless channel on a d-dimensional quantum system are entanglement-equivalent by teleportation and dense coding. We provide a construction of entanglement-equivalences of channels based on the principle of covariance. Let G be a compact quantum group of Kac type, let F': Rep(G) -> Hilb be a fibre functor accessible from the canonical fibre functor F: Rep(G) -> Hilb by a unitary pseudonatural transformation, and let G' be the compact quantum group of Kac type obtained from the fibre functor F' by Tannaka duality. Then for any G-covariant channel N we construct a G'-covariant channel N' and entanglement-assisted channel coding schemes witnessing an entanglement-equivalence of N and N'. This construction induces an equivalence of categories E: Chan(G) -> Chan(G'), where Chan(G) is the category whose objects are G-C*-algebras and whose morphisms are covariant channels. We show that all tight teleportation and dense coding schemes are a special case of our construction, using covariance for the action of the quantum permutation group of the algebra M_d(\mathbb{C}). Where G is a finite abelian group, we explicitly compute the effect of the equivalence E on algebras and channels. We thereby produce concrete new examples of entanglement-equivalences of channels in the case where G=Z_2 \times Z_2 and the noncanonical fibre functor is specified by the Pauli matrices.

Dec 10, 20201 citation

On some classes of bipartite unitary operators

Julien Deschamps, Ion Nechita, Clément Pellegrini

Abstract We investigate unitary operators acting on a tensor product space, with the property that the quantum channels they generate, via the Stinespring dilation theorem, are of a particular type, independently of the state of the ancilla system in the Stinespring relation. The types of quantum channels we consider are those of interest in quantum information theory: unitary conjugations, constant channels, unital channels, mixed unitary channels, positive partial transpose channels, and entanglement breaking channels. For some of the classes of bipartite unitary operators corresponding to the above types of channels, we provide explicit characterizations, necessary and/or sufficient conditions for membership, and we compute the dimension of the corresponding algebraic variety. Inclusions between these classes are considered, and we show that for small dimensions, many of these sets are identical.

arXiv:1509.06543Journal of Physics A Mathematical and TheoreticalJul 6, 201616 citations

Involutive Markov categories and the quantum de Finetti theorem

Fritz, Tobias, Antonio Lorenzin

Markov categories have recently emerged as a powerful high-level framework for probability theory and theoretical statistics. Here we study a quantum version of this concept, called involutive Markov categories. These are equivalent to Parzygnat's quantum Markov categories, but we argue that they offer a simpler and more practical approach. Our main examples of involutive Markov categories have pre-C*-algebras, including infinite-dimensional ones, as objects, together with completely positive unital maps as morphisms in the picture of interest. In this context, we prove a quantum de Finetti theorem for both the minimal and the maximal C*-tensor norms, and we develop a categorical description of such quantum de Finetti theorems which amounts to a universal property of state spaces.

arXiv:2312.09666Dec 15, 20230 citations