16
2 Elements of Quantum Information Theory
partition {I j }
k
j=1 if it can be written as a convex combination of product states on
the k partitions:
ρ =
i
p i ρ
i
I 1
⊗ · · · ⊗ ρ
i
I k
.
(2.29)
If every I j comprises only one index, then the state (2.29) is called fully separable.
If a state is not fully separable, then it is entangled. If the partition is only composed
of two subsets I 1 and I 2 , the state (2.29) is called biseparable. If a state cannot be
expressed as a convex combination of biseparable states, then it is called genuine
multipartite entangled (GME).
An example of a pure GME state which plays a major role in multipartite quantum
cryptographic protocols is the Greenberger–Horne–Zeilinger (GHZ) state [11]:
|GHZ N =
1
√
2
|0
⊗N
+ |1
⊗N
.
(2.30)
A remarkable application of the density operator formalism is the ability to
describe subsystems of composite systems through the reduced density operator.
This is particularly useful when the global state is entangled and the states of its
subsystems are not immediately intelligible.
Definition 2.5 (Reduced density operator) Let ρ AB be the state of a bipartite quantum system. Then the reduced density operator representing the state on subsystem
A is given by:
ρ A = Tr B [ρ AB ],
(2.31)
where Tr B is the partial trace on subsystem B.
The partial trace is defined as the regular trace (2.8) but only acts on the subsystems
indicated in the subscript. For instance, given two orthonormal bases {|a i } i and
{|b j } j for the two subsystems A and B, we can express ρ AB in the outer product
notation and compute the action of the partial trace as follows:
Tr B [ρ AB ] =
i, j,k,l
r (i, j),(k,l) Tr B [|a i a k | ⊗ |b j b l |]
=
i, j,k,l
r (i, j),(k,l) b l |b j |a i a k |,
(2.32)
where r (i, j),(k,l) = =a i , b j |ρ AB |a k , b l are the matrix elements of ρ AB .
Definition 2.5 is justified by the fact that the reduced density operator ρ A provides
the correct measurement statistics for measurements made on subsystem A.
2 Elements of Quantum Information Theory
partition {I j }
k
j=1 if it can be written as a convex combination of product states on
the k partitions:
ρ =
i
p i ρ
i
I 1
⊗ · · · ⊗ ρ
i
I k
.
(2.29)
If every I j comprises only one index, then the state (2.29) is called fully separable.
If a state is not fully separable, then it is entangled. If the partition is only composed
of two subsets I 1 and I 2 , the state (2.29) is called biseparable. If a state cannot be
expressed as a convex combination of biseparable states, then it is called genuine
multipartite entangled (GME).
An example of a pure GME state which plays a major role in multipartite quantum
cryptographic protocols is the Greenberger–Horne–Zeilinger (GHZ) state [11]:
|GHZ N =
1
√
2
|0
⊗N
+ |1
⊗N
.
(2.30)
A remarkable application of the density operator formalism is the ability to
describe subsystems of composite systems through the reduced density operator.
This is particularly useful when the global state is entangled and the states of its
subsystems are not immediately intelligible.
Definition 2.5 (Reduced density operator) Let ρ AB be the state of a bipartite quantum system. Then the reduced density operator representing the state on subsystem
A is given by:
ρ A = Tr B [ρ AB ],
(2.31)
where Tr B is the partial trace on subsystem B.
The partial trace is defined as the regular trace (2.8) but only acts on the subsystems
indicated in the subscript. For instance, given two orthonormal bases {|a i } i and
{|b j } j for the two subsystems A and B, we can express ρ AB in the outer product
notation and compute the action of the partial trace as follows:
Tr B [ρ AB ] =
i, j,k,l
r (i, j),(k,l) Tr B [|a i a k | ⊗ |b j b l |]
=
i, j,k,l
r (i, j),(k,l) b l |b j |a i a k |,
(2.32)
where r (i, j),(k,l) = =a i , b j |ρ AB |a k , b l are the matrix elements of ρ AB .
Definition 2.5 is justified by the fact that the reduced density operator ρ A provides
the correct measurement statistics for measurements made on subsystem A.