nLab Bloch region

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)