<s>
In	O
the	O
theory	O
of	O
quantum	O
communication	O
,	O
the	O
quantum	B-Device
capacity	I-Device
is	O
the	O
highest	O
rate	O
at	O
which	O
quantum	O
information	O
can	O
be	O
communicated	O
over	O
many	O
independent	O
uses	O
of	O
a	O
noisy	O
quantum	O
channel	O
from	O
a	O
sender	O
to	O
a	O
receiver	O
.	O
</s>
<s>
The	O
quantum	B-Device
capacity	I-Device
theorem	O
is	O
important	O
for	O
the	O
theory	O
of	O
quantum	B-Device
error	I-Device
correction	I-Device
,	O
and	O
more	O
broadly	O
for	O
the	O
theory	O
of	O
quantum	B-Architecture
computation	I-Architecture
.	O
</s>
<s>
The	O
theorem	O
giving	O
a	O
lower	O
bound	O
on	O
the	O
quantum	B-Device
capacity	I-Device
of	O
any	O
channel	O
is	O
colloquially	O
known	O
as	O
the	O
LSD	O
theorem	O
,	O
after	O
the	O
authors	O
Lloyd	O
,	O
Shor	O
,	O
and	O
Devetak	O
who	O
proved	O
it	O
with	O
increasing	O
standards	O
of	O
rigor	O
.	O
</s>
<s>
We	O
prove	O
the	O
theorem	O
for	O
this	O
special	O
case	O
by	O
exploiting	O
random	O
stabilizer	B-Device
codes	I-Device
and	O
correcting	O
only	O
the	O
likely	O
errors	O
that	O
the	O
channel	O
produces	O
.	O
</s>
<s>
Theorem	O
(	O
hashing	B-Device
bound	I-Device
)	O
.	O
</s>
<s>
There	O
exists	O
a	O
stabilizer	O
quantum	B-Device
error-correcting	I-Device
code	I-Device
that	O
achieves	O
the	O
hashing	O
limit	O
for	O
a	O
Pauli	O
channel	O
of	O
the	O
following	O
form:where	O
and	O
is	O
the	O
entropy	O
of	O
this	O
probability	O
vector	O
.	O
</s>
<s>
Also	O
,	O
we	O
consider	O
the	O
expectation	O
of	O
the	O
error	O
probability	O
under	O
a	O
random	O
choice	O
of	O
a	O
stabilizer	B-Device
code	I-Device
.	O
</s>
<s>
We	O
conclude	O
that	O
as	O
long	O
as	O
the	O
rate	O
,	O
the	O
expectation	O
of	O
the	O
error	O
probability	O
becomes	O
arbitrarily	O
small	O
,	O
so	O
that	O
there	O
exists	O
at	O
least	O
one	O
choice	O
of	O
a	O
stabilizer	B-Device
code	I-Device
with	O
the	O
same	O
bound	O
on	O
the	O
error	O
probability	O
.	O
</s>
