Related papers: CP$^{\infty}$ and beyond: 2-categorical dilation t…
We explore complementarity between output and environment of a quantum channel (or, more generally, CP map), making an observation that the output purity characteristics for complementary CP maps coincide. Hence, validity of the…
We introduce the geometric mean and the parallel sum of completely positive (CP) maps on von Neumann algebras, based on the Pusz--Woronowicz theory of positive sesquilinear forms. We provide a concrete characterization via a block matrix…
We describe the exponential map from an infinite-dimensional Lie algebra to an infinite-dimensional group of operators on a Hilbert space. Notions of differential geometry are introduced for these groups. In particular, the Ricci curvature,…
We relate notions of complementarity in three layers of quantum mechanics: (i) von Neumann algebras, (ii) Hilbert spaces, and (iii) orthomodular lattices. Taking a more general categorical perspective of which the above are instances, we…
It is shown that each linear operator on a separable Hilbert space which generates a finite type I von Neumann algebra has, up to unitary equivalence, a unique representation as a direct integral of inflations of mutually unitary…
This paper is devoted to the problem of recovering a potential $q$ in a domain in $\mathbb{R}^d$ for $d \geq 3$ from the Dirichlet to Neumann map. This problem is related to the inverse Calder\'on conductivity problem via the Liouville…
We study completely contractive representations of product systems of $C^*$-correspondences over semigroups. For a product system of $C^*$-correspondences over the semigroup $\mathbb{N}^2$, we prove that every such representation can be…
Quantum error correcting codes with finite-dimensional Hilbert spaces have yielded new insights on bulk reconstruction in AdS/CFT. In this paper, we give an explicit construction of a quantum error correcting code where the code and…
By considering a generalisation of the CPM construction, we develop an infinite hierarchy of probabilistic theories, exhibiting compositional decoherence structures which generalise the traditional quantum-to-classical transition.…
This paper unites two research lines. The first involves finding categorical models of quantum programming languages and their type systems. The second line concerns the program of quantization of mathematical structures, which amounts to…
We reformulate entanglement wedge reconstruction in the language of operator-algebra quantum error correction with infinite-dimensional physical and code Hilbert spaces. Von Neumann algebras are used to characterize observables in a…
Coecke and Heunen described completely positive maps in dagger monoidal categories and the {\sf CP}-infinity construction on these categories in order to construct a category of arbitrary dimensional quantum processes. This article…
Using new techniques for controlling the categoricity spectrum of a structure, we construct a structure with degree of categoricity but infinite spectral dimension, answering a question of Bazhenov, Kalimulin and Yamaleev. Using the same…
We define and study the $(\infty,2)$-category $\mathbf{Cat}_{\infty}(\mathcal{C})$ of $(\infty,1)$-categories internal to a general $(\infty,1)$-category $\mathcal{C}$ via an associated externalization construction. In the first part, we…
This article introduces Hilbert $*$-categories: an abstraction of categories with similar algebraic and analytic properties to the categories of real, complex, and quaternionic Hilbert spaces and bounded linear maps. Other examples include…
In this paper, we introduce the notion of a von Neumann category, as a generalization and categorification of von Neumann algebra. A von Neumann category is a premonoidal category with compatible dagger structure which embeds as a double…
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…
This study investigates Hermitian linear maps, focusing on their decomposition into completely positive (CP) maps and their extensions to CP maps using auxiliary spaces. We derive a precise lower bound on the Hilbert-Schmidt norm of the…
We reconstruct finite-dimensional quantum theory from categorical principles. That is, we provide properties ensuring that a given physical theory described by a dagger compact category in which one may `discard' objects is equivalent to a…
In this dissertation we study the category of completely positive normal contractive maps between von Neumann algebras. It includes an extensive introduction to the basic theory of $C^*$-algebras and von Neumann algebras.