JHEP10(2018)152
We call it multipartite entanglement of purification and write ∆
P
(ρ
A
1
:···:A
n
) = ∆
P
(A
1
:
··· : A
n
) unless we need to specify a given state. ∆
P
can be regarded as the value of sum of
quantum entanglement in an optimal purification |ψi
A
1
A
0
1
···A
n
A
0
n
, between one of n-parties
and the other n −1 parts. For example, for tripartite state ρ
ABC
, it can be represented as
∆
P
(A : B : C)
:
= min
|ψi
AA
0
BB
0
CC
0
[S
AA
0
+ S
BB
0
+ S
CC
0
], (2.13)
and the entanglement entropies S
AA
0
, S
BB
0
and S
CC
0
in the brackets characterize quantum
entanglement between AA
0
: BB
0
CC
0
, BB
0
: AA
0
CC
0
and CC
0
: AA
0
BB
0
, respectively. We
also note that a purification that gives the optimal value in (2.13) may not be unique in
general, as is so for the entanglement of purification E
P
.
2.2 Other measures
Besides the entanglement of purification, there have been a lot of measures of quantum
and/or classical correlations which quantify bipartite correlations for mixed states - in-
cluding mutual information and squashed entanglement [44, 45]. Their generalization for
multipartite cases have also been proposed in the literature [39, 40]. To see that multipar-
tite entanglement is not just a sum of bipartite one, let us consider the famous GHZ state
in a 3-qubit system:
|GHZi =
1
√
2
(|000i + |111i). (2.14)
For this state, the three qubits are strongly correlated by genuine tripartite entanglement.
The point is that, after one of the subsystems is traced out, the remaining bipartite state
is just a separable state and there is no quantum entanglement. This is an example which
shows that the structure of multipartite quantum entanglement is much richer than bipar-
tite ones.
To quantify an amount of correlations for multipartite states, there have been several
proposals by generalizing the mutual information. One well known quantity, the so-called
tripartite information is defined as (based on Ven diagram)
˜
I
3
(A : B : C) := S
A
+ S
B
+ S
C
− S
AB
− S
BC
− S
CA
+ S
ABC
= I(A : C) + I(A : B) − I(A : BC). (2.15)
This quantity plays an important role in holography and has been studied e.g. in [32, 35–37].
The tripartite information
˜
I
3
, however, can take all of negative, zero, or positive values in
generic quantum systems, and especially it always vanishes for all pure states. These facts
make it hard to be regarded as a standard measure of correlations.
An alternative approach is introduced in [39, 40] and the authors called the general-
ization as multipartite mutual information, defined by the relative entropy between a given
original state and its local product state:
I(A : B : C) := S(ρ
ABC
||ρ
A
⊗ ρ
B
⊗ ρ
C
) = S
A
+ S
B
+ S
C
− S
ABC
, (2.16)
where S(ρ||σ) = Trρ log ρ − Trρ log σ. This definition is motivated by an expression of the
bipartite mutual information,
I(A : B) = S(ρ
AB
||ρ
A
⊗ ρ
B
). (2.17)
– 5 –
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