A generalization of the Choi-Jamiolkowski isomorphism for completely positive maps between operator algebras is introduced. Particular emphasis is placed on the case of normal unital completely positive maps defined between von Neumann algebras. This generalization is applied especially to the study of maps which are covariant under actions of a symmetry group. We highlight with the example of, e.g., phase-shift-covariant quantum channels the ease of this method in particular in the case of a compact symmetry group. We also discuss the case of channels which are covariant under actions of the Euclidean group of rigid motions in 3 dimensions.
This paper investigates the interplay between the properties of quantum states on the Hilbert space
ℓ2(κ) and the set-theoretic nature of the cardinal
κ.
We focus on the existence of singular
σ-additive states~--- functionals whose induced measures are
σ-additive yet vanish on singletons.
While the existence of such states is known to be equivalent to the Ulam measurability of
κ, their structural and dynamical properties remain largely unexplored.
We prove that any
σ-additive state on the diagonal algebra is representable as a Pettis integral over a singular
σ-additive measure, extending the classical representation theory to the non-normal sector.
Furthermore, we construct a class of quantum channels using
σ-complete ultrafilters that map normal states to singular
σ-additive states, effectively <<archiving>> information into the singular part of the state space.
We introduce the CP*-construction on a dagger compact closed category as a generalisation of Selinger's CPM-construction. While the latter takes a dagger compact closed category and forms its category of "abstract matrix algebras" and completely positive maps, the CP*-construction forms its category of "abstract C*-algebras" and completely positive maps. This analogy is justified by the case of finite-dimensional Hilbert spaces, where the CP*-construction yields the category of finite-dimensional C*-algebras and completely positive maps. The CP*-construction fully embeds Selinger's CPM-construction in such a way that the objects in the image of the embedding can be thought of as "purely quantum" state spaces. It also embeds the category of classical stochastic maps, whose image consists of "purely classical" state spaces. By allowing classical and quantum data to coexist, this provides elegant abstract notions of preparation, measurement, and more general quantum channels.
We introduce a construction that turns a category of pure state spaces and operators into a category of observable algebras and superoperators. For example, it turns the category of finite-dimensional Hilbert spaces into the category of finite-dimensional C*-algebras and completely positive maps. In particular, the new category contains both quantum and classical channels, providing elegant abstract notions of preparation and measurement. We also consider nonstandard models, that can be used to investigate which notions from algebraic quantum information theory are operationally justifiable.
We describe a categorical model of MALL (Multiplicative Additive Linear Logic) inspired by the Heisenberg-Schrödinger duality of finite-dimensional quantum theory. Proofs of formulas with positive logical polarity correspond to CPTP (completely positive trace-preserving) maps in our model, i.e. the quantum operations in the Schrödinger picture, whereas proofs of formulas with negative logical polarity correspond to CPU (completely positive unital) maps, i.e. the quantum operations in the Heisenberg picture. The mathematical development is based on noncommutative geometry and finite-dimensional von Neumann (co)algebras, which can be defined as special kinds of (co)monoid objects internal to the category of finite-dimensional operator spaces.
We investigate the structural and dynamical properties of a class of quantum channels on von Neumann algebras induced by averaging over operator groups via the Pettis integral.
Utilizing the classical Yosida--Hewitt decomposition, we focus on the interplay between the set-theoretic properties of the underlying space and the topological nature of the resulting quantum states.
We establish a suffucient condition under which a channel preserves
σ-additivity and exhibits a singularising property, completely suppressing the normal component of any incoming state.
In conjunction with the theory of Ulam real-valued measurable cardinals, this framework reveals a novel phenomenon: the existence of quantum channels that transform normal states into singular yet strictly
σ-additive states.
Furthermore, we analyze the structural constraints on the preservation of state purity imposed by the cardinality of the continuum, and extend our constructions to invariant measures on groups and their unitary representations, establishing the convergence of their Cesàro averages in the strong operator topology.