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Metric graph properties lie in the heart of the analysis of complex networks, while in this paper we study their convexity through mathematical definition of a convex subgraph. A subgraph is convex if every geodesic path between the nodes…
Necessary and sufficient conditions for convexity and strong convexity, respectively, of sublevel sets that are defined by finitely many real-valued $C^{1,1}$-maps are presented. A novel characterization of strongly convex sets in terms of…
We investigate the classical capacity of two quantum channels with memory: a periodic channel with depolarizing channel branches, and a convex combination of depolarizing channels. We prove that the capacity is additive in both cases. As a…
It is well known that correlations predicted by quantum mechanics cannot be explained by any classical (local-realistic) theory. The relative strength of quantum and classical correlations is usually studied in the context of Bell…
Stochastic matrices and positive maps in matrix algebras proved to be very important tools for analysing classical and quantum systems. In particular they represent a natural set of transformations for classical and quantum states,…
It is known that any covering space of a topological group has the natural structure of a topological group. This article discusses a noncommutative generalization of this fact. A noncommutative generalization of the topological group is a…
Vector Addition Systems (VAS), aka Petri nets, are a popular model of concurrency. The reachability set of a VAS is the set of configurations reachable from the initial configuration. Leroux has studied the geometric properties of VAS…
We introduce a new framework for quantifying the complexity of quantum channels, grounded in a suitably chosen resource set. This class of convex functions is designed to analyze the complexity of both open and closed quantum systems. By…
In this paper, we consider the dynamical modeling of a class of quantum network systems consisting of qubits. Qubit probes are employed to measure a set of selected nodes of the quantum network systems. For a variety of applications, a…
We consider two apparently separated problems: in the first part of the paper we study the concept of a scalable (approximate) programmable quantum gate (SPQG). These are special (approximate) programmable quantum gates, with nice…
The interplay between the algebraic structure (operator algebras) for the quantum observables and the convex structure of the state space has been explored for a long time and most advanced results are due to Alfsen and Shultz. Here we…
A group $G$ is called to be acceptable (due to M. Larsen) if for any finite group $H$, two element-conjugate homomorphisms are globally conjugate. We answer the acceptability question for general linear, special linear, unitary, symplectic…
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…
If $G$ is a compact group, continuous normalized positive definite functions are in one-to-one correspondence with unital quantum channels acting as Fourier multipliers on the group von Neumann algebra $\mathrm{VN}(G)$. We study the convex…
Let $\P$ be any collection of paths of a graph $G=(V,E)$. For $S\subseteq V$, define $I(S)=S\cup\{v\mid v \ \mbox{lies in a path of} \ \P \ \mbox{with endpoints in} \ S\}$. Let $\C$ be the collection of fixed points of the function $I$,…
Quantum channels, which are completely positive and trace preserving mappings, can alter the dimension of a system; e.g., a quantum channel from a qubit to a qutrit. We study the convex set properties of dimension-altering quantum channels,…
Variational analysis presents a unified theory encompassing in particular both smoothness and convexity. In a Euclidean space, convex sets and smooth manifolds both have straightforward local geometry. However, in the most basic hybrid case…
We investigate the reachability problem in symmetric vector addition systems with states (VASS), where transitions are invariant under a group of permutations of coordinates. One extremal case, the trivial groups, yields general VASS. In…
Commutativity gadgets provide a technique for lifting classical reductions between constraint satisfaction problems to quantum-sound reductions between the corresponding nonlocal games. We develop a general framework for commutativity…
We analyze the quantum binary adder channel, i.e. the quantum generalization of the classical, and well-studied, binary adder channel: in this model qubits rather than classical bits are transmitted. This of course is as special case of the…