In mathematics (specifically linear algebra, operator theory, and functional analysis) as well as physics, a linear operator
acting on an inner product space is called positive-semidefinite (or non-negative) if, for every
,
and
, where
is the domain of
. Positive-semidefinite operators are denoted as
. The operator is said to be positive-definite, and written
, if
for all
.[1]
Many authors define a positive operator
to be a self-adjoint (or at least symmetric) non-negative operator. We show below that for a complex Hilbert space the self adjointness follows automatically from non-negativity. For a real Hilbert space non-negativity does not imply self adjointness.
In physics (specifically quantum mechanics), such operators represent quantum states, via the density matrix formalism.
Cauchy–Schwarz inequality
Take the inner product
to be anti-linear on the first argument and linear on the second and suppose that
is positive and symmetric, the latter meaning that
. Then the non negativity of
![{\displaystyle {\begin{aligned}\langle A(\lambda x+\mu y),\lambda x+\mu y\rangle =|\lambda |^{2}\langle Ax,x\rangle +\lambda ^{*}\mu \langle Ax,y\rangle +\lambda \mu ^{*}\langle Ay,x\rangle +|\mu |^{2}\langle Ay,y\rangle \\[1mm]=|\lambda |^{2}\langle Ax,x\rangle +\lambda ^{*}\mu \langle Ax,y\rangle +\lambda \mu ^{*}(\langle Ax,y\rangle )^{*}+|\mu |^{2}\langle Ay,y\rangle \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dfa6fe16545b068eb9ae69e1de9b19ffd2883b04)
for all complex
and
shows that
![{\displaystyle \left|\langle Ax,y\rangle \right|^{2}\leq \langle Ax,x\rangle \langle Ay,y\rangle .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a1a7cb26194214f7cef753559ad03c8d439c807)
It follows that
If
is defined everywhere, and
then
On a complex Hilbert space, if an operator is non-negative then it is symmetric
For
the polarization identity
![{\displaystyle {\begin{aligned}\langle Ax,y\rangle ={\frac {1}{4}}({}&\langle A(x+y),x+y\rangle -\langle A(x-y),x-y\rangle \\[1mm]&{}-i\langle A(x+iy),x+iy\rangle +i\langle A(x-iy),x-iy\rangle )\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8723ccc4576e55f1744625b2c81b2f99f465f3c6)
and the fact that
for positive operators, show that
so
is symmetric.
In contrast with the complex case, a positive-semidefinite operator on a real Hilbert space
may not be symmetric. As a counterexample, define
to be an operator of rotation by an acute angle
Then
but
so
is not symmetric.
If an operator is non-negative and defined on the whole Hilbert space, then it is self-adjoint and bounded
The symmetry of
implies that
and
For
to be self-adjoint, it is necessary that
In our case, the equality of domains holds because
so
is indeed self-adjoint. The fact that
is bounded now follows from the Hellinger–Toeplitz theorem.
This property does not hold on
Partial order of self-adjoint operators
A natural partial ordering of self-adjoint operators arises from the definition of positive operators. Define
if the following hold:
and
are self-adjoint ![{\displaystyle B-A\geq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b82610462af3b50f2b81f9cfab48b1f30fd25047)
It can be seen that a similar result as the Monotone convergence theorem holds for monotone increasing, bounded, self-adjoint operators on Hilbert spaces.[2]
Application to physics: quantum states
The definition of a quantum system includes a complex separable Hilbert space
and a set
of positive trace-class operators
on
for which
The set
is the set of states. Every
is called a state or a density operator. For
where
the operator
of projection onto the span of
is called a pure state. (Since each pure state is identifiable with a unit vector
some sources define pure states to be unit elements from
States that are not pure are called mixed.
References
- ^ Roman 2008, p. 250 §10
- ^ Eidelman, Yuli, Vitali D. Milman, and Antonis Tsolomitis. 2004. Functional analysis: an introduction. Providence (R.I.): American mathematical Society.
- Conway, John B. (1990), Functional Analysis: An Introduction, Springer Verlag, ISBN 0-387-97245-5