<s>
If	O
additionally	O
one	O
has	O
then	O
is	O
called	O
an	O
antiunitary	B-Algorithm
operator	I-Algorithm
.	O
</s>
<s>
Antiunitary	B-Algorithm
operators	I-Algorithm
are	O
important	O
in	O
quantum	O
theory	O
because	O
they	O
are	O
used	O
to	O
represent	O
certain	O
symmetries	O
,	O
such	O
as	O
time	O
reversal	O
.	O
</s>
<s>
Due	O
to	O
Wigner	O
's	O
theorem	O
these	O
transformations	O
can	O
either	O
be	O
unitary	B-Algorithm
or	O
antiunitary	B-Algorithm
.	O
</s>
<s>
On	O
the	O
complex	O
plane	O
these	O
two	O
classes	O
correspond	O
(	O
up	O
to	O
translation	O
)	O
to	O
unitaries	O
and	O
antiunitaries	B-Algorithm
,	O
respectively	O
.	O
</s>
<s>
holds	O
for	O
all	O
elements	O
of	O
the	O
Hilbert	O
space	O
and	O
an	O
antiunitary	B-Algorithm
.	O
</s>
<s>
When	O
is	O
antiunitary	B-Algorithm
then	O
is	O
unitary	B-Algorithm
.	O
</s>
<s>
For	O
unitary	B-Algorithm
operator	I-Algorithm
the	O
operator	O
,	O
where	O
is	O
complex	O
conjugate	O
operator	O
,	O
is	O
antiunitary	B-Algorithm
.	O
</s>
<s>
The	O
reverse	O
is	O
also	O
true	O
,	O
for	O
antiunitary	B-Algorithm
the	O
operator	O
is	O
unitary	B-Algorithm
.	O
</s>
<s>
The	O
adjoint	O
of	O
an	O
antiunitary	B-Algorithm
is	O
also	O
antiunitary	B-Algorithm
and	O
(	O
This	O
is	O
not	O
to	O
be	O
confused	O
with	O
the	O
definition	O
of	O
unitary	B-Algorithm
operators	I-Algorithm
,	O
as	O
the	O
antiunitary	B-Algorithm
operator	I-Algorithm
is	O
not	O
complex	O
linear	O
.	O
)	O
</s>
<s>
The	O
complex	O
conjugate	O
operator	O
is	O
an	O
antiunitary	B-Algorithm
operator	I-Algorithm
on	O
the	O
complex	O
plane	O
.	O
</s>
<s>
The	O
operator	O
where	O
is	O
the	O
second	O
Pauli	O
matrix	O
and	O
is	O
the	O
complex	O
conjugate	O
operator	O
,	O
is	O
antiunitary	B-Algorithm
.	O
</s>
<s>
An	O
antiunitary	B-Algorithm
operator	I-Algorithm
on	O
a	O
finite-dimensional	O
space	O
may	O
be	O
decomposed	O
as	O
a	O
direct	O
sum	O
of	O
elementary	O
Wigner	O
antiunitaries	B-Algorithm
,	O
.	O
</s>
<s>
Note	O
that	O
the	O
above	O
decomposition	O
of	O
antiunitary	B-Algorithm
operators	I-Algorithm
contrasts	O
with	O
the	O
spectral	O
decomposition	O
of	O
unitary	B-Algorithm
operators	I-Algorithm
.	O
</s>
<s>
In	O
particular	O
,	O
a	O
unitary	B-Algorithm
operator	I-Algorithm
on	O
a	O
complex	O
Hilbert	O
space	O
may	O
be	O
decomposed	O
into	O
a	O
direct	O
sum	O
of	O
unitaries	O
acting	O
on	O
1-dimensional	O
complex	O
spaces	O
(	O
eigenspaces	O
)	O
,	O
but	O
an	O
antiunitary	B-Algorithm
operator	I-Algorithm
may	O
only	O
be	O
decomposed	O
into	O
a	O
direct	O
sum	O
of	O
elementary	O
operators	O
on	O
1	O
-	O
and	O
2-dimensional	O
complex	O
spaces	O
.	O
</s>
