In quantum mechanics, negativity is a measure of quantum entanglement which is easy to compute. It is a measure deriving from the PPT criterion for separability. It has shown to be an entanglement monotone and hence a proper measure of entanglement.
The negativity of a subsystem
A
can be defined in terms of a density matrix
ρ
as:
N
(
ρ
)
≡
|
|
ρ
Γ
A
|
|
1
−
1
2
where:
ρ
Γ
A
is the partial transpose of
ρ
with respect to subsystem
A
|
|
X
|
|
1
=
Tr
|
X
|
=
Tr
X
†
X
is the trace norm or the sum of the singular values of the operator
X
.
An alternative and equivalent definition is the absolute sum of the negative eigenvalues of
ρ
Γ
A
:
N
(
ρ
)
=
∑
i
|
λ
i
|
−
λ
i
2
where
λ
i
are all of the eigenvalues.
Is a convex function of
ρ
:
N
(
∑
i
p
i
ρ
i
)
≤
∑
i
p
i
N
(
ρ
i
)
Is an entanglement monotone:
N
(
P
(
ρ
i
)
)
≤
N
(
ρ
i
)
where
P
(
ρ
)
is an arbitrary LOCC operation over
ρ
The logarithmic negativity is an entanglement measure which is easily computable and an upper bound to the distillable entanglement. It is defined as
E
N
(
ρ
)
≡
log
2
|
|
ρ
Γ
A
|
|
1
where
Γ
A
is the partial transpose operation and
|
|
⋅
|
|
1
denotes the trace norm.
It relates to the negativity as follows:
E
N
(
ρ
)
:=
log
2
(
2
N
+
1
)
The logarithmic negativity
can be zero even if the state is entangled (if the state is PPT entangled).
does not reduce to the entropy of entanglement on pure states like most other entanglement measures.
is additive on tensor products:
E
N
(
ρ
⊗
σ
)
=
E
N
(
ρ
)
+
E
N
(
σ
)
is not asymptotically continuous. That means that for a sequence of bipartite Hilbert spaces
H
1
,
H
2
,
…
(typically with increasing dimension) we can have a sequence of quantum states
ρ
1
,
ρ
2
,
…
which converges to
ρ
⊗
n
1
,
ρ
⊗
n
2
,
…
(typically with increasing
n
i
) in the trace distance, but the sequence
E
N
(
ρ
1
)
/
n
1
,
E
N
(
ρ
2
)
/
n
2
,
…
does not converge to
E
N
(
ρ
)
.
is an upper bound to the distillable entanglement
