THE DENSITY OPERATOR :
We define the
density operator or Somatrix e as an ensemble of quantum states
{ I ✗ is} with probabilities { pi } for every density matrix y .
,
y → { pi 140>3 ,
; where { His} c- te
and
pi E R ; pi ≥ o ti and § pi = 1 .
✗ & pi Iti Xvi I
} Remember that this decomposition is not
=
e wires in the Hilbert space unique and also { 14 is} need not be an
LCH) , where all the wheat
operators in Herne .
{ if C- LCH) orthogonal basis .
So .
what are the
properties of p ?
+
? 1)
+ *
i) e is Hermitian :
y
=
pi ( Hi ✗ +i
=
& pi mi Xxii ( :
pi ER it i
,
) =
y
it) e
is
positive semidefinite :
Any Hermitian linear operator ME LCH) is
:
PʳᵈᵗʰʳhwᵈHw* if <∅ 1m10> ≥ 0, it 1 ∅> Ese .
-}
"
'
y_÷
"
Dues# 1 Prove that < ∅ IM / to> ER if M is
-
. :
-
-
i Hermitian .
.
" '
Now, 1414101> =
? pilot to ✗ till>
& pi KO Iti> 12 ≥
.
,
0--5011 mkt> ≥ 0
cos = 0
,
as
pi ≥ 0
Both 1015 and MIP) too on it i
the same side in some
heuristic sense . M does
If we
diagonalize if and write its edgar decomposition
✓
not
flop the state 14> •
for its spectral decomposition) , then by extension ofthe
above arguments ,
we need all its eigenvalues to be positive .
§ pi Hi ✗ til § di leixeil ; where { di , leis}
•
if = =
• .
is the eigenvalues and
eigenstates off
d. This
Then if is positive semidefinite if di ≥ 0
,
it holds
true for all positive semidefinite matrices or
operators .
) e has unit trace ly) (& pi Iti Xxii) § pi Tr( Hi ✗til)*
"
ni : Tr = Tr =
71
=
1 ( as § pi =
1 by definition)
Dues # 2 : Prove that Trott ✗41) =
1- in words .
Dues # 3 : Show that the one can define if =
§ pili ,
where ei is a set
of mixed state density matrices in @ → ✗ i ≠ 14 ixuil .
We define the
density operator or Somatrix e as an ensemble of quantum states
{ I ✗ is} with probabilities { pi } for every density matrix y .
,
y → { pi 140>3 ,
; where { His} c- te
and
pi E R ; pi ≥ o ti and § pi = 1 .
✗ & pi Iti Xvi I
} Remember that this decomposition is not
=
e wires in the Hilbert space unique and also { 14 is} need not be an
LCH) , where all the wheat
operators in Herne .
{ if C- LCH) orthogonal basis .
So .
what are the
properties of p ?
+
? 1)
+ *
i) e is Hermitian :
y
=
pi ( Hi ✗ +i
=
& pi mi Xxii ( :
pi ER it i
,
) =
y
it) e
is
positive semidefinite :
Any Hermitian linear operator ME LCH) is
:
PʳᵈᵗʰʳhwᵈHw* if <∅ 1m10> ≥ 0, it 1 ∅> Ese .
-}
"
'
y_÷
"
Dues# 1 Prove that < ∅ IM / to> ER if M is
-
. :
-
-
i Hermitian .
.
" '
Now, 1414101> =
? pilot to ✗ till>
& pi KO Iti> 12 ≥
.
,
0--5011 mkt> ≥ 0
cos = 0
,
as
pi ≥ 0
Both 1015 and MIP) too on it i
the same side in some
heuristic sense . M does
If we
diagonalize if and write its edgar decomposition
✓
not
flop the state 14> •
for its spectral decomposition) , then by extension ofthe
above arguments ,
we need all its eigenvalues to be positive .
§ pi Hi ✗ til § di leixeil ; where { di , leis}
•
if = =
• .
is the eigenvalues and
eigenstates off
d. This
Then if is positive semidefinite if di ≥ 0
,
it holds
true for all positive semidefinite matrices or
operators .
) e has unit trace ly) (& pi Iti Xxii) § pi Tr( Hi ✗til)*
"
ni : Tr = Tr =
71
=
1 ( as § pi =
1 by definition)
Dues # 2 : Prove that Trott ✗41) =
1- in words .
Dues # 3 : Show that the one can define if =
§ pili ,
where ei is a set
of mixed state density matrices in @ → ✗ i ≠ 14 ixuil .
