2.4 Composite Systems and Entanglement
15
ρ AB =
i
p i |ψ
i
A ψ
i
A | ⊗ |ψ
i
B ψ
i
B |
=
i
p i |ψ
i
A , ψ
i
B ψ
i
A , ψ
i
B |
(2.24)
and is called a separable state. Note that product states are a particular case of
separable states and that the state (2.24) is the most general state obtained by LOCC.
Indeed, even if Alice and Bob prepare their systems in mixed states ρ
i
A and ρ
i
B , the
state of the composite system,
i p i ρ A i ⊗ ρ B i , can always be reduced to one of the
form (2.24).
However, not every composite quantum system is prepared with LOCC, hence its
state may not be a separable state (2.24) [9]. In that case, we say that the state of the
composite system is entangled.
Definition 2.3 (Separability, Entanglement) A quantum state ρ AB on H A ⊗ H B is
called separable if there exists a convex combination of pure product states |ψ
i
A ⊗
|ψ
i
B , with |ψ
i
A ∈ H A and |ψ
i
B ∈ H B , such that:
ρ AB =
i
p i |ψ
i
A , ψ
i
B ψ
i
A , ψ
i
B |.
(2.25)
Otherwise, ρ AB is called entangled.
Classifying whether a state is entangled or not is challenging. Consider for example the following pure states, known as Bell states:
|
±
=
|00 ± |11
√
2
,
(2.26)
whose density operators are clearly entangled according to Definition 2.3:
|
±
±
| =
1
2
[|0000| ± |0011| ± |1100| + |1111|] .
(2.27)
Surprisingly, their convex combination is not entangled:
1
2
|
+
+
| + |
−
−
|
=
1
2
[|0000| + |1111|] .
(2.28)
The definition of entanglement can be extended to a multipartite scenario. An N -
partite pure state is called fully separable if one can assign a single state vector to the
subsystem of each party, i.e. if it is a product state: |ψ
1
A ⊗ · · · ⊗ |ψ
N
A . Otherwise the
state is entangled. The more general definition valid for mixed states is the following.
Definition 2.4 (Multipartite Entanglement, [10]) Consider a set of N parties labelled
by the indices I = {1, 2, . . . , N } and a partition {I j }
k
j=1 of I, where I j are disjoint
subsets of I such that ∪ j I j = I. Then a state ρ is k-separable with respect to the
15
ρ AB =
i
p i |ψ
i
A ψ
i
A | ⊗ |ψ
i
B ψ
i
B |
=
i
p i |ψ
i
A , ψ
i
B ψ
i
A , ψ
i
B |
(2.24)
and is called a separable state. Note that product states are a particular case of
separable states and that the state (2.24) is the most general state obtained by LOCC.
Indeed, even if Alice and Bob prepare their systems in mixed states ρ
i
A and ρ
i
B , the
state of the composite system,
i p i ρ A i ⊗ ρ B i , can always be reduced to one of the
form (2.24).
However, not every composite quantum system is prepared with LOCC, hence its
state may not be a separable state (2.24) [9]. In that case, we say that the state of the
composite system is entangled.
Definition 2.3 (Separability, Entanglement) A quantum state ρ AB on H A ⊗ H B is
called separable if there exists a convex combination of pure product states |ψ
i
A ⊗
|ψ
i
B , with |ψ
i
A ∈ H A and |ψ
i
B ∈ H B , such that:
ρ AB =
i
p i |ψ
i
A , ψ
i
B ψ
i
A , ψ
i
B |.
(2.25)
Otherwise, ρ AB is called entangled.
Classifying whether a state is entangled or not is challenging. Consider for example the following pure states, known as Bell states:
|
±
=
|00 ± |11
√
2
,
(2.26)
whose density operators are clearly entangled according to Definition 2.3:
|
±
±
| =
1
2
[|0000| ± |0011| ± |1100| + |1111|] .
(2.27)
Surprisingly, their convex combination is not entangled:
1
2
|
+
+
| + |
−
−
|
=
1
2
[|0000| + |1111|] .
(2.28)
The definition of entanglement can be extended to a multipartite scenario. An N -
partite pure state is called fully separable if one can assign a single state vector to the
subsystem of each party, i.e. if it is a product state: |ψ
1
A ⊗ · · · ⊗ |ψ
N
A . Otherwise the
state is entangled. The more general definition valid for mixed states is the following.
Definition 2.4 (Multipartite Entanglement, [10]) Consider a set of N parties labelled
by the indices I = {1, 2, . . . , N } and a partition {I j }
k
j=1 of I, where I j are disjoint
subsets of I such that ∪ j I j = I. Then a state ρ is k-separable with respect to the