In mathematics (specifically linear algebra , operator theory , and functional analysis ) as well as physics , a linear operator
A
{\displaystyle A}
inner product space is called positive-semidefinite (or non-negative ) if, for every
x
∈
Dom
(
A
)
{\displaystyle x\in \mathop {\text{Dom}} (A)}
⟨
A
x
,
x
⟩
∈
R
{\displaystyle \langle Ax,x\rangle \in \mathbb {R} }
⟨
A
x
,
x
⟩
≥
0
{\displaystyle \langle Ax,x\rangle \geq 0}
Dom
(
A
)
{\displaystyle \mathop {\text{Dom}} (A)}
domain of
A
{\displaystyle A}
A
≥
0
{\displaystyle A\geq 0}
positive-definite , and written
A
>
0
{\displaystyle A>0}
⟨
A
x
,
x
⟩
>
0
,
{\displaystyle \langle Ax,x\rangle >0,}
x
∈
D
o
m
(
A
)
∖
{
0
}
{\displaystyle x\in \mathop {\mathrm {Dom} } (A)\setminus \{0\}}
[ 1]
In physics (specifically quantum mechanics ), such operators represent quantum states , via the density matrix formalism.
Many authors define a positive operator
A
{\displaystyle A}
⟨
A
(
λ
x
+
μ
y
)
,
λ
x
+
μ
y
⟩
=
|
λ
|
2
⟨
A
x
,
x
⟩
+
λ
∗
μ
⟨
A
x
,
y
⟩
+
λ
μ
∗
⟨
A
y
,
x
⟩
+
|
μ
|
2
⟨
A
y
,
y
⟩
{\displaystyle \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 }
=
|
λ
|
2
⟨
A
x
,
x
⟩
+
λ
∗
μ
⟨
A
x
,
y
⟩
+
λ
μ
∗
(
⟨
A
x
,
y
⟩
)
∗
+
|
μ
|
2
⟨
A
y
,
y
⟩
{\displaystyle =|\lambda |^{2}\langle Ax,x\rangle +\lambda ^{*}\mu \langle Ax,y\rangle +\lambda \mu ^{*}(\langle Ax,y\rangle )^{*}+|\mu |^{2}\langle Ay,y\rangle }
λ
{\displaystyle \lambda }
μ
{\displaystyle \mu }
|
⟨
A
x
,
y
⟩
|
2
≤
⟨
A
x
,
x
⟩
⟨
A
y
,
y
⟩
.
{\displaystyle \left|\langle Ax,y\rangle \right|^{2}\leq \langle Ax,x\rangle \langle Ay,y\rangle .}
It follows that
Im
A
⊥
Ker
A
.
{\displaystyle \mathop {\text{Im}} A\perp \mathop {\text{Ker}} A.}
A
{\displaystyle A}
⟨
A
x
,
x
⟩
=
0
,
{\displaystyle \langle Ax,x\rangle =0,}
A
x
=
0.
{\displaystyle Ax=0.}
A ≥
0
{\displaystyle 0}
A is symmetricWithout loss of generality , let the inner product
⟨
⋅
,
⋅
⟩
{\displaystyle \langle \cdot ,\cdot \rangle }
anti-linear on the first argument and linear on the second. (If the reverse is true, then we work with
⟨
x
,
y
⟩
op
=
def
⟨
y
,
x
⟩
{\displaystyle \langle x,y\rangle _{\text{op}}{\stackrel {\text{def}}{=}}\ \langle y,x\rangle }
x
,
y
∈
Dom
A
,
{\displaystyle x,y\in \mathop {\text{Dom}} A,}
polarization identity
⟨
A
x
,
y
⟩
=
1
4
(
⟨
A
(
x
+
y
)
,
x
+
y
⟩
−
⟨
A
(
x
−
y
)
,
x
−
y
⟩
−
i
⟨
A
(
x
+
i
y
)
,
x
+
i
y
⟩
+
i
⟨
A
(
x
−
i
y
)
,
x
−
i
y
⟩
)
{\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}}}
and the fact that
⟨
A
x
,
x
⟩
=
⟨
x
,
A
x
⟩
,
{\displaystyle \langle Ax,x\rangle =\langle x,Ax\rangle ,}
⟨
A
x
,
y
⟩
=
⟨
x
,
A
y
⟩
,
{\displaystyle \langle Ax,y\rangle =\langle x,Ay\rangle ,}
A
{\displaystyle A}
In contrast with the complex case, a positive-semidefinite operator on a real Hilbert space
H
R
{\displaystyle H_{\mathbb {R} }}
A
:
R
2
→
R
2
{\displaystyle A:\mathbb {R} ^{2}\to \mathbb {R} ^{2}}
acute angle
φ
∈
(
−
π
/
2
,
π
/
2
)
.
{\displaystyle \varphi \in (-\pi /2,\pi /2).}
⟨
A
x
,
x
⟩
=
‖
A
x
‖
‖
x
‖
cos
φ
>
0
,
{\displaystyle \langle Ax,x\rangle =\|Ax\|\|x\|\cos \varphi >0,}
A
∗
=
A
−
1
≠
A
,
{\displaystyle A^{*}=A^{-1}\neq A,}
A
{\displaystyle A}
A ≥
0
{\displaystyle 0}
A =
H
C
{\displaystyle H_{\mathbb {C} }}
A is self-adjoint and bounded The symmetry of
A
{\displaystyle A}
implies that
Dom
A
⊆
Dom
A
∗
{\displaystyle \mathop {\text{Dom}} A\subseteq \mathop {\text{Dom}} A^{*}}
A
=
A
∗
|
Dom
(
A
)
.
{\displaystyle A=A^{*}|_{\mathop {\text{Dom}} (A)}.}
A
{\displaystyle A}
Dom
A
=
Dom
A
∗
.
{\displaystyle \mathop {\text{Dom}} A=\mathop {\text{Dom}} A^{*}.}
domains holds because
H
C
=
Dom
A
⊆
Dom
A
∗
,
{\displaystyle H_{\mathbb {C} }=\mathop {\text{Dom}} A\subseteq \mathop {\text{Dom}} A^{*},}
A
{\displaystyle A}
A
{\displaystyle A}
Hellinger–Toeplitz theorem .
This property does not hold on
H
R
.
{\displaystyle H_{\mathbb {R} }.}
H
C
{\displaystyle H_{\mathbb {C} }}
A natural ordering of self-adjoint operators arises from the definition of positive operators. Define
B
≥
A
{\displaystyle B\geq A}
A
{\displaystyle A}
B
{\displaystyle B}
B
−
A
≥
0
{\displaystyle B-A\geq 0}
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
H
C
{\displaystyle H_{\mathbb {C} }}
S
{\displaystyle {\cal {S}}}
trace-class operators
ρ
{\displaystyle \rho }
H
C
{\displaystyle H_{\mathbb {C} }}
Trace
ρ
=
1.
{\displaystyle \mathop {\text{Trace}} \rho =1.}
set
S
{\displaystyle {\cal {S}}}
the set of states . Every
ρ
∈
S
{\displaystyle \rho \in {\cal {S}}}
state or a density operator . For
ψ
∈
H
C
,
{\displaystyle \psi \in H_{\mathbb {C} },}
‖
ψ
‖
=
1
,
{\displaystyle \|\psi \|=1,}
P
ψ
{\displaystyle P_{\psi }}
span of
ψ
{\displaystyle \psi }
pure state unit vector
ψ
∈
H
C
,
{\displaystyle \psi \in H_{\mathbb {C} },}
H
C
)
.
{\displaystyle H_{\mathbb {C} }).}
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.
![{\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}}}](/media/api/rest_v1/media/math/render/svg/8723ccc4576e55f1744625b2c81b2f99f465f3c6)
