nLab positive operator

Definition

On Hilbert spaces

A positive operator is a linear operator $A$ on a Hilbert space $(H, \langle -,-\rangle)$ such that the quadratic form

$$v \mapsto \langle v, A v\rangle$$

for $v \in H$ is positive semi-definite.

Proposition

Every positive operator $A$ on a Hilbert space is self-adjoint.

Proof

Let $B = \frac{1}{2}(A + A^\dagger)$ and $C = \frac{1}{2}i(A^\dagger - A)$. Then $B$ and $C$ are self-adjoint, and $A = B + iC$. Now, $\langle v, A v \rangle = \langle v, B v \rangle + i \langle v, C v \rangle$ is real for any $v$, so $\langle v, C v \rangle = 0$ for all $v$. Hence $C = 0$ and $A = B$.

In $C^\ast$-algebras

More generally:

Definition

An element $A$ of an (abstract) C*-algebra is called positive if it is self-adjoint and its spectrum is contained in $[0, \infinity)$.

Here, ‘positive’ means positive semidefinite; see at inner product for the family of variations of this notion. (The relevant inner product here is that associated with the quadratic form above: $v, w \mapsto \langle v, A w\rangle$.)

An element $a \in A$ is positive if and only if it is of the form $a = b^* b$. This statement is not true for more general *-algebras over the complex numbers.

In dagger-categories

Yet more generally, in a dagger category a morphism $f$ is positive if it is of the form $f = g^\dagger \circ g$ for some morphism $g$.

(eg. Selinger 2005 Def. 4.1)

Related concepts

• positive operator valued measure

• normal operator

• self-adjoint operator

• state on an operator algebra

• density matrix

• quantum effect

• quantum operation

References

Discussion in dagger-compact categories with an eye twoards completely positive maps on spaces of density matrices:

• Peter Selinger, Dagger-compact closed categories and completely positive maps, Electronic Notes in Theoretical Computer Science, 170 (2007) 139-163 [doi:10.1016/j.entcs.2006.12.018, pdf]