Physics 212: Statistical mechanics II, Fall 2006
Lecture XXX
Quantum information
Here we give a quick overview of a few ideas from quantum information that have applications
in statistical physics. An interesting property of quantum mechanics is that a pure state

Ψ
AB
i
of
a system partitioned into subsystems
A
and
B
does not always imply that
A
and
B
are in pure
quantum states. If
A
and
B
are individually in pure quantum states, we say that

Ψ
AB
i
is a product
state:

Ψ
AB
i
=

ψ
A
i ⊗ 
ψ
B
i
.
(1)
An example is the state of two spinhalf degrees of freedom in which both spins
A
and
B
are up:
 ↑
A
↑
B
i ≡  ↑
A
i ⊗  ↑
B
i
is a product state. However, many (in fact “most”) states are not product
states: a familiar example is the singlet state of two spins:
1
√
2
(
 ↑
A
↓
B
i   ↓
A
↑
B
i
)
.
(2)
Note that the
S
z
= 0 symmetric triplet state would also be entangled, even though it is part of the
same angular momentum multiplet as the product (unentangled states) with
S
z
=
±
¯
h
.
When the overall pure state of
AB
,

Ψ
AB
i
, is not a product state, the description of a subsystem
(say
A
) must be as a density matrix rather than a pure state. The “reduced density matrix” for
subsystem
A
is deﬁned as (here

ψ
i
i
are a basis for
A
, and

φ
i
i
are a basis for
B
):
h
ψ
i

ρ
A

ψ
j
i ≡
X
k
(
h
ψ
i
 ⊗ h
φ
k

)

Ψ
AB
ih
Ψ
AB

(

ψ
j
i ⊗ 
φ
k
i
)
.
(3)
We now show that the same value of
S
is obtained whether we keep subsystem
A
and trace
over
B
, or vice versa:
S
(
ρ
A
) =
S
(
ρ
B
)
.
(4)
Proof: We will show a slightly stronger statement, that under the conditions assumed above, the
two reduced density matrices have the same nonzero eigenvalues. Since the two in general have
diﬀerent dimensionalities, there may be more zero eigenvalues in one than in the other, but the
entropy depends only on the nonzero eigenvalues and hence will be the same.
This result follows from the Schmidt decomposition, that

Ψ
AB
i
can be written, for some
orthonormal bases for
A
and
B
, as

Ψ
AB
i
=
n
X
i
=1
λ
i

i
A
i
i
B
i
.
(5)
Here
λ
i
are real numbers with
n
X
i
=1
λ
i
2
= 1
,
(6)
and the number
n
of nonzero values is known as the Schmidt number. The Schmidt decomposition
implies that
λ
i
are the eigenvalues both of
ρ
A
and
ρ
B
, as
ρ
A
=
X
i
λ
i
2

i
A
ih
i
A

,
ρ
B
=
X
i
λ
i
2

i
B
ih
i
B

.
(7)
1
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View Full DocumentTo prove the Schmidt decomposition, we start from arbitrary bases for
A
and
B
: then

Ψ
AB
i
=
a
jk

j
A
i
k
B
i
,
(8)
where
j
and
k
are summed over, and
a
is not a square matrix unless
A
and
B
happen to have the
same dimensionality. The “singular value decomposition” of
a
jk
implies

Ψ
AB
i
=
u
ji
d
ii
v
ik

j
A
i
k
B
i
,
(9)
where
d
is a (again, not generally square) diagonal matrix with real elements, and
u
and
v
are
unitary matrices. We identify
λ
i
=
d
ii
,
u
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