Related papers: Smooth manifold structure for extreme channels
A quantum control landscape is defined as the physical objective as a function of the control variables to be optimized. Analyzing the topology of these landscapes is important for understanding the origins of the increasing number of…
Quantum state and process tomography are typically analyzed under the assumption that devices emit independent and identically distributed (i.i.d.) states or channels. In realistic experiments, however, noise, drift, feedback, or…
The manifold of pure quantum states is a complex projective space endowed with the unitary-invariant geometry of Fubini and Study. According to the principles of geometric quantum mechanics, the detailed physical characteristics of a given…
The control of quantum system dynamics is generally performed by seeking a suitable applied field. The physical objective as a functional of the field forms the quantum control landscape, whose topology, under certain conditions, has been…
The motion of a quantum particle constrained to a two-dimensional non-compact Riemannian manifold with non-trivial metric can be described by a flat-space Schroedinger-type equation at the cost of introducing local mass and metric and…
This article addresses regularity of optimal transport maps for cost="squared distance" on Riemannian manifolds that are products of arbitrarily many round spheres with arbitrary sizes and dimensions. Such manifolds are known to be…
We study the generalization of quasipositive links from the three-sphere to arbitrary closed, orientable three-manifolds. Our main result shows that the boundary of any smooth, properly embedded complex curve in a Stein domain is a…
The simplest building blocks for quantum computations are the qubit-qubit quantum channels. In this paper, we analyze the structure of these channels via their Choi representation. The restriction of a quantum channel to the space of…
We establish conditions for a continuous map of nonzero degree between a smooth closed manifold and a negatively curved manifold of dimension greater than four to be homotopic to a smooth cover, and in particular a diffeomorphism when the…
Smooth structures on high dimensional manifolds are classified by maps to the infinite loop space $TOP/O$. The homotopy groups of this space are known to be finite. Given a compact Lie group $G$, this space can be regarded as an equivariant…
We classify closed, simply connected $n$-manifolds of non-negative sectional curvature admitting an isometric torus action of maximal symmetry rank in dimensions $2\leq n\leq 6$. In dimensions $3k$, $k=1,2$ there is only one such manifold…
We show that any topological, closed, oriented, non-spin $4$-manifold with fundamental group $\mathbb{Z}_{4k}$ and $\min(b_2^+, b_2^-)\geq 15$, has either none or infinitely many distinct smooth structures. Furthermore, we construct…
A `discrete differential manifold' we call a countable set together with an algebraic differential calculus on it. This structure has already been explored in previous work and provides us with a convenient framework for the formulation of…
We propose a quantum soft-covering problem for a given general quantum channel and one of its output states, which consists in finding the minimum rank of an input state needed to approximate the given channel output. We then prove a…
Manipulation of infinite dimensional quantum systems is important to controlling complex quantum dynamics with many practical physical and chemical backgrounds. In this paper, a general investigation is casted to the controllability problem…
We introduce an information-theoretic framework for smooth structures on topological manifolds, replacing coordinate charts with small-scale entropy data of local probability probes. A concise set of axioms identifies admissible coordinate…
We introduce and study two new classes of unital quantum channels. The first class describes a 2-parameter family of channels given by completely positive (CP) maps $M_3({\bf C}) \mapsto M_3({\bf C})$ which are both unital and…
The ground state of $^4$He confined in a system with the topology of a cylinder can display properties of a solid, superfluid and liquid crystal. This phase, which we call compactified supersolid (CSS), originates from wrapping the basal…
In 1994, Sakai introduced the property of $C^0$ transversality for two smooth curves in a two-dimensional manifold. This property was related to various shadowing properties of dynamical systems. In this short note, we generalize this…
Stabilizer circuits arise in almost every area of quantum computation and communication, so there is interest in studying them from an information-theoretic perspective, i.e. as quantum channels. We consider several natural approaches to…