# The face generated by a point, generalized affine constraints, and quantum theory

@article{Weis2020TheFG, title={The face generated by a point, generalized affine constraints, and quantum theory}, author={Stephan Weis and Maksim E. Shirokov}, journal={arXiv: Functional Analysis}, year={2020} }

We analyze faces generated by points in an arbitrary convex set and their relative algebraic interiors, which are nonempty as we shall prove. We show that by intersecting a convex set with a sublevel or level set of a generalized affine functional, the dimension of the face generated by a point may decrease by at most one. We apply the results to the set of quantum states on a separable Hilbert space. Among others, we show that every state having finite expected values of any two (not… Expand

#### Figures from this paper

#### References

SHOWING 1-10 OF 32 REFERENCES

Minimum output entropy of a non-Gaussian quantum channel

- Physics
- 2016

We introduce a model of a non-Gaussian quantum channel that stems from the composition of two physically relevant processes occurring in open quantum systems, namely, amplitude damping and dephasing.… Expand

A course in convexity

- Mathematics, Computer Science
- Graduate studies in mathematics
- 2002

Convex sets at large Faces and extreme points Convex sets in topological vector spaces Polarity, duality and linear programming Convex bodies and ellipsoids Faces of polytopes Lattices and convex… Expand

Continuous Ensembles and the Capacity of Infinite-Dimensional Quantum Channels

- Mathematics
- 2006

This paper is devoted to the study of $\chi$-capacity, closely related to the classical capacity of infinite-dimensional quantum channels. For such channels generalized ensembles are defined as… Expand

Convergence Rates for Quantum Evolution and Entropic Continuity Bounds in Infinite Dimensions

- Mathematics, Physics
- Communications in Mathematical Physics
- 2019

By extending the concept of energy-constrained diamond norms, we obtain continuity bounds on the dynamics of both closed and open quantum systems in infinite dimensions, which are stronger than… Expand

Quantum Systems, Channels, Information: A Mathematical Introduction

- Computer Science
- 2019

This book gives an accessible, albeit mathematically rigorous and self-contained introduction to quantum information theory, starting from primary structures and leading to fundamental results and to exiting open problems. Expand

Energy-Constrained Private and Quantum Capacities of Quantum Channels

- Computer Science, Physics
- IEEE Transactions on Information Theory
- 2018

It is shown how the regularized, energy-constrained coherent information is equal to the capacity for the first two tasks and is an achievable rate for the latter two tasks, whenever the energy observable satisfies the Gibbs condition of having a well-defined thermal state for all temperatures and the channel satisfies a finite output-entropy condition. Expand

Operator E-norms and their use

- Mathematics, Physics
- 2018

We consider a family of norms (called operator E -norms) on the algebraB(H) of all bounded operators on a separable Hilbert space H induced by a positive densely defined operator G on H. Each norm of… Expand

Energy-constrained diamond norm with applications to the uniform continuity of continuous variable channel capacities

- Mathematics, Physics
- 2017

The channels, and more generally superoperators acting on the trace class operators of a quantum system naturally form a Banach space under the completely bounded trace norm (aka diamond norm).… Expand

Quantum Information Theory, Second edition, Cambridge, UK

- ISBN: 978-1-316-80997-6,
- 2017

Quantum state majorization at the output of bosonic Gaussian channels.

- Physics, Mathematics
- Nature communications
- 2014

It is proved that every output state of a phase-insensitive Gaussian channel is majorized by the output state corresponding to a coherent input, and that coherent states are the unique optimizers for the minimization of strictly concave output functionals. Expand