# nLab Bloch region

Contents

### Context

#### Measure and probability theory

measure theory

probability theory

# Contents

## Idea

In quantum physics and specifically in quantum computation/quantum information, Bloch region is the name of the convex space of density matrices on a given Hilbert space.

Equivalently this is the space of mixed quantum states for a quantum mechanical system with the lines in given Hilbert space as its pure states.

The superoperators that preserve Bloch regions (also for products, “complete positivity”) are called quantum operations.

## Properties

### Relation to the simplex

For $(n+1) \in \mathbb{N}$ a natural number, the Bloch region in the $(n+1)$-dimensional Hilbert space $\mathbb{C}^{n+1}$ is a quantum physics-analog of the space of probability measures on the finite set of $(n+1)$ elements. This in turn is equivalently the standard $n$-simplex $\Delta^n$.

## References

A quick introduction is in

• Greg Kuperberg, section 1.4 of A concise introduction to quantum probability, quantum mechanics, and quantum computation, 2005 (pdf)

Created on January 23, 2014 at 06:07:08. See the history of this page for a list of all contributions to it.