Poisson states in quantum information

Mankei Tsang

By taking a Poisson limit for a sequence of rare quantum objects, I derive simple formulas for the Uhlmann fidelity, the quantum Chernoff quantity, the relative entropy, and the Helstrom information. I also present analogous formulas in classical information theory for a Poisson model. An operator called the intensity operator emerges as the central quantity in the formalism to describe Poisson states. It behaves like a density operator but is unnormalized. The formulas in terms of the intensity operators not only resemble the general formulas in terms of the density operators, but also coincide with some existing definitions of divergences between unnormalized positive-semidefinite matrices. Furthermore, I show that the effects of certain channels on Poisson states can be described by simple maps for the intensity operators.

Alberto Frigerio

Yidong Chen, Marius Junge

We introduce a noncommutative Poisson random measure on a von Neumann algebra. This is a noncommutative generalization of the classical Poisson random measure. We call this construction Poissonization. Poissonization is a functor from the category of von Neumann algebras with normal semifinite faithful weights to the category of von Neumann algebras with normal faithful states. Poissonization is a natural adaptation of the second quantization to the context of von Neumann algebras. The construction is compatible with normal (weight-preserving) homomorphisms and unital normal completely positive (weight-preserving) maps. We present two main applications of Poissonization. First Poissonization provides a new framework to construct algebraic quantum field theories that are not generalized free field theories. Second Poissonization permits straight-forward calculations of quantum relative entropies (and other quantum information quantities) in the case of type III von Neumann algebras.

Richard Jozsa

Masahito Hayashi, Satoshi Ishizaka, Akinori Kawachi +2 more

Marcin Makowski, Edward W. Piotrowski

The aim of this paper is to study the quantum-like approach to the description of the market in the context of the principle of minimum Fisher information. We wish to investigate the validity of using squeezed coherent states as market strategies. For this purpose, we focus on the representation of any squeezed coherent state with respect to the basis of the eigenvectors of the observable of market risk. We derive a formula for the probability of being the squeezed coherent state in one of these states. The distribution that we call generalized Poisson establishes the relation between the squeezed coherent states and their description in the language of risk in quantum terms. We provide a formula specifying the total risk of squeezed coherent strategy. Then, we propose a risk of risk concept that is in fact the second central moment of the generalized Poisson distribution. This is an important numerical characterization of squeezed coherent strategies. We provide its interpretations on the basis of the uncertainty relation for time and energy.