Positive linear maps on operator algebras

Pinchings and positive linear maps

Let F be a closed sublattice of a Banach lattice G. We show that any positive linear mapping from F into L 1 ( μ ) {L^1}(\mu ) or C ( X ) C(X) for a Stonian space X has a positive norm preserving extension to G. A dual result for positive norm preserving liftings is also established. These results are applied to obtain extension and lifting theorems for order summable and majorizing linear mappings. We also obtain some partial results concerning positive extensions and liftings of compact linear mappings.

We propose a definition of purity for positive linear maps between Euclidean Jordan Algebras (EJA) that generalizes the notion of purity for quantum systems. We show that this definition of purity is closed under composition and taking adjoints and thus that the pure maps form a dagger category (which sets it apart from other possible definitions.) In fact, from the results presented in this paper, it follows that the category of EJAs with positive contractive linear maps is a dagger-effectus, a type of structure originally defined to study von Neumann algebras in an abstract categorical setting. In combination with previous work this characterizes EJAs as the most general systems allowed in a generalized probabilistic theory that is simultaneously a dagger-effectus. Using the dagger structure we get a notion of dagger-positive maps of the form f = g*g. We give a complete characterization of the pure dagger-positive maps and show that these correspond precisely to the Jordan algebraic version of the sequential product that maps (a,b) to sqrt(a) b sqrt(a). The notion of dagger-positivity therefore characterizes the sequential product.

A note on inequalities for positive linear maps

An effective method of investigation of positive maps on the set of positive definite operators

Furber and Jacobs have shown in their study of quantum computation that the category of commutative C*-algebras and PU-maps (positive linear maps which preserve the unit) is isomorphic to the Kleisli category of a comonad on the category of commutative C*-algebras with MIU-maps (linear maps which preserve multiplication, involution and unit). [Furber and Jacobs, 2013] In this paper, we prove a non-commutative variant of this result: the category of C*-algebras and PU-maps is isomorphic to the Kleisli category of a comonad on the subcategory of MIU-maps. A variation on this result has been used to construct a model of Selinger and Valiron's quantum lambda calculus using von Neumann algebras. [Cho and Westerbaan, 2016]

Shannon like inequalities for f-connections of positive linear maps and positive operators

We prove a theorem, which, while it fits naturally into the Segal–Weinless approach to quantization seems to have been overlooked in the literature: Let (D,σ) be a symplectic space, and 𝒯 (t) a one parameter group of symplectics on (D,σ). Let (ℋ, 2Im〈⋅ ‖ ⋅〉) be a complex Hilbert space considered as a real symplectic space, and U(t) a one-parameter unitary group on ℋ with strictly positive energy. Suppose there is a linear symplectic map K from D to ℋ with dense range, intertwining 𝒯 (t) and U(t). Then K is unique up to unitary equivalence.

Global and local continuity conditions for the output von Neumann entropy for positive maps between Banach spaces of trace-class operators in separable Hilbert spaces are obtained. Special attention is paid to completely positive maps: infinite dimensional quantum channels and operations.It is shown that as a result of some specific properties of the von Neumann entropy (as a function on the set of density operators) several results on the output entropy of positive maps can be obtained, which cannot be derived from the general properties of entropy type functions. In particular, it is proved that global continuity of the output entropy of a positive map follows from its finiteness. A characterization of positive linear maps preserving continuity of the entropy (in the following sense: continuity of the entropy on an arbitrary subset of input operators implies continuity of the output entropy on this subset) is obtained. A connection between the local continuity properties of two completely positive complementary maps is considered.Bibliography: 21 titles.

A normed ∗ \ast -algebra A \mathcal {A} is called a local C ∗ {C^\ast } -algebra, if all its maximal commutative ∗ \ast -subalgebras are C ∗ {C^\ast } -algebras. It is shown that any local C ∗ {C^\ast } -algebra dense in K ( H ) \mathcal {K}(\mathcal {H}) , the algebra of compact operators on the Hilbert space H \mathcal {H} equals K ( H ) \mathcal {K}(\mathcal {H}) . The same result holds also for local C ∗ {C^\ast } -algebras dense in A W ∗ A{W^\ast } -algebras without a II 1 {\text {II}_1} summand.

Chapter 6 gives a concise introduction into the theory of one-parameter groups or semigroups of operators with an emphasis on the interplay between groups and semigroups and their generators. In the first section, one-parameter unitary groups are investigated, and two fundamental theorems, Stone’s theorem and Trotter’s formula, are proved. Semigroups of operators are applied to Cauchy problems for abstract differential equations on Hilbert space. Then generators of semigroups of contractions on Banach spaces are studied, and the Hille–Yosida theorem is proved. Finally, generators of contraction semigroups on Hilbert space are characterized as m-dissipative operators.

Let A and B be unital JB-algebras. We study the extremal structure of the convex set S (A,B) of all identity preserving positive linear maps from A to B. We show that every unital Jordan homomorphism from A to B is an extreme point of S (A,B). An extreme point of S (A,B) need not be a homomorphism and we show that, given A, every extreme point of S (A,B) is a homomorphism for any B if, and only if, dim A ≤ 2. We also determine when S (A,B) is a simplex.

We generalize two classes of D-type positive linear maps by permutations. For the first class, we discuss their indecomposability and give conditions when they are indecomposable and decomposable. For the other class, we show that they are atomic.

Positive linear maps and quantum operations

This paper presents a novel semantics for a quantum programming language by\noperator algebras, which are known to give a formulation for quantum theory\nthat is alternative to the one by Hilbert spaces. We show that the opposite\ncategory of the category of W*-algebras and normal completely positive\nsubunital maps is an elementary quantum flow chart category in the sense of\nSelinger. As a consequence, it gives a denotational semantics for Selinger's\nfirst-order functional quantum programming language QPL. The use of operator\nalgebras allows us to accommodate infinite structures and to handle classical\nand quantum computations in a unified way.\n