<s>
In	O
linear	B-Language
algebra	I-Language
and	O
functional	B-Application
analysis	I-Application
,	O
the	O
partial	B-Algorithm
trace	I-Algorithm
is	O
a	O
generalization	O
of	O
the	O
trace	O
.	O
</s>
<s>
Whereas	O
the	O
trace	O
is	O
a	O
scalar	O
valued	O
function	O
on	O
operators	O
,	O
the	O
partial	B-Algorithm
trace	I-Algorithm
is	O
an	O
operator-valued	O
function	O
.	O
</s>
<s>
The	O
partial	B-Algorithm
trace	I-Algorithm
has	O
applications	O
in	O
quantum	O
information	O
and	O
decoherence	O
which	O
is	O
relevant	O
for	O
quantum	O
measurement	O
and	O
thereby	O
to	O
the	O
decoherent	O
approaches	O
to	O
interpretations	O
of	O
quantum	O
mechanics	O
,	O
including	O
consistent	O
histories	O
and	O
the	O
relative	O
state	O
interpretation	O
.	O
</s>
<s>
For	O
any	O
space	O
,	O
let	O
denote	O
the	O
space	O
of	O
linear	B-Architecture
operators	I-Architecture
on	O
.	O
</s>
<s>
The	O
partial	B-Algorithm
trace	I-Algorithm
over	O
is	O
then	O
written	O
as	O
.	O
</s>
<s>
The	O
associated	O
linear	B-Architecture
operator	I-Architecture
on	O
V	O
is	O
independent	O
of	O
the	O
choice	O
of	O
bases	O
and	O
is	O
by	O
definition	O
the	O
partial	B-Algorithm
trace	I-Algorithm
.	O
</s>
<s>
To	O
see	O
that	O
the	O
conditions	O
above	O
determine	O
the	O
partial	B-Algorithm
trace	I-Algorithm
uniquely	O
,	O
let	O
form	O
a	O
basis	O
for	O
,	O
let	O
form	O
a	O
basis	O
for	O
,	O
let	O
be	O
the	O
map	O
that	O
sends	O
to	O
(	O
and	O
all	O
other	O
basis	O
elements	O
to	O
zero	O
)	O
,	O
and	O
let	O
be	O
the	O
map	O
that	O
sends	O
to	O
.	O
</s>
<s>
It	O
is	O
the	O
partial	B-Algorithm
trace	I-Algorithm
of	O
linear	B-Architecture
transformations	I-Architecture
that	O
is	O
the	O
subject	O
of	O
Joyal	O
,	O
Street	O
,	O
and	O
Verity	O
's	O
notion	O
of	O
Traced	O
monoidal	O
category	O
.	O
</s>
<s>
Another	O
case	O
of	O
this	O
abstract	O
notion	O
of	O
partial	B-Algorithm
trace	I-Algorithm
takes	O
place	O
in	O
the	O
category	O
of	O
finite	O
sets	O
and	O
bijections	O
between	O
them	O
,	O
in	O
which	O
the	O
monoidal	O
product	O
is	O
disjoint	O
union	O
.	O
</s>
<s>
The	O
partial	B-Algorithm
trace	I-Algorithm
generalizes	O
to	O
operators	O
on	O
infinite	O
dimensional	O
Hilbert	O
spaces	O
.	O
</s>
<s>
converges	O
in	O
the	O
strong	B-Algorithm
operator	I-Algorithm
topology	I-Algorithm
of	O
L(V )	O
,	O
it	O
is	O
independent	O
of	O
the	O
chosen	O
basis	O
of	O
W	O
.	O
The	O
partial	B-Algorithm
trace	I-Algorithm
TrW(T )	O
is	O
defined	O
to	O
be	O
this	O
operator	O
.	O
</s>
<s>
The	O
partial	B-Algorithm
trace	I-Algorithm
of	O
a	O
self-adjoint	O
operator	O
is	O
defined	O
if	O
and	O
only	O
if	O
the	O
partial	B-Algorithm
traces	I-Algorithm
of	O
the	O
positive	O
and	O
negative	O
parts	O
are	O
defined	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
finite	O
dimensional	O
Hilbert	O
spaces	O
,	O
there	O
is	O
a	O
useful	O
way	O
of	O
looking	O
at	O
partial	B-Algorithm
trace	I-Algorithm
involving	O
integration	O
with	O
respect	O
to	O
a	O
suitably	O
normalized	O
Haar	O
measure	O
μ	O
over	O
the	O
unitary	O
group	O
U(W )	O
of	O
W	O
.	O
Suitably	O
normalized	O
means	O
that	O
μ	O
is	O
taken	O
to	O
be	O
a	O
measure	O
with	O
total	O
mass	O
dim(W )	O
.	O
</s>
<s>
The	O
operator	O
R	O
is	O
the	O
partial	B-Algorithm
trace	I-Algorithm
of	O
T	O
.	O
</s>
<s>
The	O
partial	B-Algorithm
trace	I-Algorithm
can	O
be	O
viewed	O
as	O
a	O
quantum	O
operation	O
.	O
</s>
<s>
The	O
partial	B-Algorithm
trace	I-Algorithm
of	O
ρ	O
with	O
respect	O
to	O
the	O
system	O
B	O
,	O
denoted	O
by	O
,	O
is	O
called	O
the	O
reduced	O
state	O
of	O
ρ	O
on	O
system	O
A	O
.	O
</s>
<s>
We	O
see	O
that	O
this	O
is	O
satisfied	O
if	O
is	O
as	O
defined	O
above	O
via	O
the	O
partial	B-Algorithm
trace	I-Algorithm
.	O
</s>
<s>
The	O
density	O
matrix	O
ρ	O
is	O
Hermitian	B-Algorithm
,	O
positive	B-Algorithm
semi-definite	I-Algorithm
,	O
and	O
has	O
a	O
trace	O
of	O
1	O
.	O
</s>
<s>
Its	O
easy	O
to	O
see	O
that	O
the	O
partial	B-Algorithm
trace	I-Algorithm
also	O
satisfies	O
these	O
conditions	O
.	O
</s>
<s>
The	O
space	O
of	O
observables	O
for	O
each	O
system	O
are	O
then	O
abelian	O
C*	B-Algorithm
-algebras	I-Algorithm
.	O
</s>
<s>
Thus	O
the	O
partial	B-Algorithm
trace	I-Algorithm
is	O
the	O
quantum	O
mechanical	O
equivalent	O
of	O
this	O
operation	O
.	O
</s>
