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.
If the symmetry, (an operator
J satisfying
J=J∗=J−1) which defines the Krein space, is replaced by a (not necessarily self-adjoint) unitary, then we have the notion of an
S-space which was introduced by Szafraniec. In this paper, we consider
S-spaces and study the structure of completely
U-positive maps between the algebras of bounded linear operators. We first give a Stinespring-type representation for a completely
U-positive map. On the other hand, we introduce Choi
U-matrix of a linear map and establish the equivalence of the Kraus
U-decompositions and Choi
U-matrices. Then we study properties of nilpotent completely
U-positive maps. We develop the
U-PPT criterion for separability of quantum
U-states and discuss the entanglement breaking condition of quantum
U-channels and explore
U-PPT squared conjecture. Finally, we give concrete examples of completely
U-positive maps and examples of
3⊗3 quantum
U-states which are
U-entangled and
U-separable.
We prove that a unital completely positive map between finite-dimensional C ⁎ -algebras is a homomorphism if and only if it is completely entropy-nonincreasing, where the relevant notion of entropy is a variant of von Neumann entropy. This adjusted von Neumann entropy is the negative of the relative entropy with respect to the uniform state on the C ⁎ -algebra, up to an additive constant. As an intermediate step, we prove that a unital completely positive map between finite-dimensional C ⁎ -algebras is a homomorphism if and only if its adjusted Choi operator is a projection. Both equivalences generalize familiar facts about stochastic maps between finite sets.
Let [Formula: see text] be a unital C*-subalgebra of the C*-algebra ℬ(ℋ) of all bounded operators on a complex separable Hilbert space ℋ. Let [Formula: see text] denote the convex set of all unital, linear, completely positive and normal maps of [Formula: see text] into itself. Using Stinespring's theorem, we present a criterion for an element [Formula: see text] to be extremal. When [Formula: see text], this criterion leads to an explicit description of the set of all extreme points of [Formula: see text]. We also obtain a quantum probabilistic analogue of the classical Birkhoff's theorem2 that every bistochastic matrix can be expressed as a convex combination of permutation matrices.
The space POVM H(X) of positive operator-valued probability measures on the Borel sets of a compact (or even locally compact) Hausdorff space X with values in B(H), the algebra of linear operators acting on a d-dimensional Hilbert space H, is studied from the perspectives of classical and nonclassical convexity through a transform Γ that associates any positive operator-valued measure ν with a certain completely positive linear map Γ(ν) of the homogeneous C*-algebra C(X)⊗B(H) into B(H). This association is achieved by using an operator-valued integral in which nonclassical random variables (that is, operator-valued functions) are integrated with respect to positive operator-valued measures and which has the feature that the integral of a random quantum effect is itself a quantum effect. A left inverse Ω for Γ yields an integral representation, along the lines of the classical Riesz representation theorem for linear functionals on C(X), of certain (but not all) unital completely positive linear maps φ:C(X)⊗B(H)→B(H). The extremal and C*-extremal points of POVM H(X) are determined.
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.