In quantum information theory, the idea of a typical subspace plays an important role in the proofs of many coding theorems (the most prominent example being Schumacher compression). Its role is analogous to that of the typical set in classical information theory.
Consider a density operator
ρ
with the following spectral decomposition:
ρ
=
∑
x
p
X
(
x
)
|
x
⟩
⟨
x
|
.
The weakly typical subspace is defined as the span of all vectors such that the sample entropy
H
¯
(
x
n
)
of their classical label is close to the true entropy
H
(
X
)
of the distribution
p
X
(
x
)
:
T
δ
X
n
≡
span
{
|
x
n
⟩
:
|
H
¯
(
x
n
)
−
H
(
X
)
|
≤
δ
}
,
where
H
¯
(
x
n
)
≡
−
1
n
log
(
p
X
n
(
x
n
)
)
,
H
(
X
)
≡
−
∑
x
p
X
(
x
)
log
p
X
(
x
)
.
The projector
Π
ρ
,
δ
n
onto the typical subspace of
ρ
is defined as
Π
ρ
,
δ
n
≡
∑
x
n
∈
T
δ
X
n
|
x
n
⟩
⟨
x
n
|
,
where we have "overloaded" the symbol
T
δ
X
n
to refer also to the set of
δ
-typical sequences:
T
δ
X
n
≡
{
x
n
:
|
H
¯
(
x
n
)
−
H
(
X
)
|
≤
δ
}
.
The three important properties of the typical projector are as follows:
Tr
{
Π
ρ
,
δ
n
ρ
⊗
n
}
≥
1
−
ϵ
,
Tr
{
Π
ρ
,
δ
n
}
≤
2
n
[
H
(
X
)
+
δ
]
,
2
−
n
[
H
(
X
)
+
δ
]
Π
ρ
,
δ
n
≤
Π
ρ
,
δ
n
ρ
⊗
n
Π
ρ
,
δ
n
≤
2
−
n
[
H
(
X
)
−
δ
]
Π
ρ
,
δ
n
,
where the first property holds for arbitrary
ϵ
,
δ
>
0
and sufficiently large
n
.
Consider an ensemble
{
p
X
(
x
)
,
ρ
x
}
x
∈
X
of states. Suppose that each state
ρ
x
has the following spectral decomposition:
ρ
x
=
∑
y
p
Y
|
X
(
y
|
x
)
|
y
x
⟩
⟨
y
x
|
.
Consider a density operator
ρ
x
n
which is conditional on a classical sequence
x
n
≡
x
1
⋯
x
n
:
ρ
x
n
≡
ρ
x
1
⊗
⋯
⊗
ρ
x
n
.
We define the weak conditionally typical subspace as the span of vectors (conditional on the sequence
x
n
) such that the sample conditional entropy
H
¯
(
y
n
|
x
n
)
of their classical labels is close to the true conditional entropy
H
(
Y
|
X
)
of the distribution
p
Y
|
X
(
y
|
x
)
p
X
(
x
)
:
T
δ
Y
n
|
x
n
≡
span
{
|
y
x
n
n
⟩
:
|
H
¯
(
y
n
|
x
n
)
−
H
(
Y
|
X
)
|
≤
δ
}
,
where
H
¯
(
y
n
|
x
n
)
≡
−
1
n
log
(
p
Y
n
|
X
n
(
y
n
|
x
n
)
)
,
H
(
Y
|
X
)
≡
−
∑
x
p
X
(
x
)
∑
y
p
Y
|
X
(
y
|
x
)
log
p
Y
|
X
(
y
|
x
)
.
The projector
Π
ρ
x
n
,
δ
onto the weak conditionally typical subspace of
ρ
x
n
is as follows:
Π
ρ
x
n
,
δ
≡
∑
y
n
∈
T
δ
Y
n
|
x
n
|
y
x
n
n
⟩
⟨
y
x
n
n
|
,
where we have again overloaded the symbol
T
δ
Y
n
|
x
n
to refer to the set of weak conditionally typical sequences:
T
δ
Y
n
|
x
n
≡
{
y
n
:
|
H
¯
(
y
n
|
x
n
)
−
H
(
Y
|
X
)
|
≤
δ
}
.
The three important properties of the weak conditionally typical projector are as follows:
E
X
n
{
Tr
{
Π
ρ
X
n
,
δ
ρ
X
n
}
}
≥
1
−
ϵ
,
Tr
{
Π
ρ
x
n
,
δ
}
≤
2
n
[
H
(
Y
|
X
)
+
δ
]
,
2
−
n
[
H
(
Y
|
X
)
+
δ
]
Π
ρ
x
n
,
δ
≤
Π
ρ
x
n
,
δ
ρ
x
n
Π
ρ
x
n
,
δ
≤
2
−
n
[
H
(
Y
|
X
)
−
δ
]
Π
ρ
x
n
,
δ
,
where the first property holds for arbitrary
ϵ
,
δ
>
0
and sufficiently large
n
, and the expectation is with respect to the distribution
p
X
n
(
x
n
)
.
