ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

1.3. Theory of bipartite entanglement 21
separability (see Sec. 1.3.2.1), i.e. how much the partial transposition of
ϱ fails to be positive. The negativity N (ϱ) [283, 74] is defined as
ϱ
N (ϱ) =
Ti
− 1 1 ,
2
(1.40)
where
Ô1
= Tr Ô † Ô (1.41)
is the trace norm of the operator Ô. The negativity is a computable
measure of entanglement, being
N (ϱ) = max 0, −
, (1.42)
where the λ −
k ’s are the negative eigenvalues of the partial transpose.
In continuous variable systems, the negativity is still a proper entanglement
measure [253], even though a related measure is more often used,
the logarithmic negativity EN (ϱ) [253, 74, 186],
k
λ −
k
EN (ϱ) = log ϱ Ti 1 = log [1 + 2N (ϱ)] . (1.43)
The logarithmic negativity is additive and, despite being non-convex, is
a full entanglement monotone under LOCC [186]; it is an upper bound
for the distillable entanglement [74], EN (ϱ) ≥ ED(ϱ), and it is the exact
entanglement cost under operations preserving the positivity of the
partial transpose [12]. The logarithmic negativity will be our measure of
choice for the quantification of bipartite entanglement of Gaussian states
(see Part II) and on it we will base the definition of a new entanglement
measure for continuous variable systems, the contangle [GA10], which will
be exploited in the analysis of distributed multipartite entanglement of
Gaussian states (see Part III).
• Squashed entanglement.— Another interesting entanglement measure is
the squashed entanglement Esq(ϱ) [55] which is defined as
1
Esq(ϱAB) = inf
E 2 I(ϱABE)
: trE{ϱABE} = ϱAB , (1.44)
where I(ϱABE) = S(ϱAE) + S(ϱBE) − S(ϱABE) − S(ϱE) is the quantum
conditional mutual information, which is often also denoted as I(A; B|E).
The motivation behind Esq comes from related quantities in classical cryptography
that determine correlations between two communicating parties
and an eavesdropper. The squashed entanglement is a convex entanglement
monotone that is a lower bound to EF (ϱ) and an upper bound to
ED(ϱ), and is hence automatically equal to EV (ϱ) on pure states. It
is also additive on tensor products, and continuous [5]. Cherry on the
cake, it is also monogamous i.e. it satisfies Eq. (1.34) for arbitrary systems
[133]. The severe drawback which affects this otherwise ideal measure
of entanglement is its computability: in principle the minimization
in Eq. (1.44) must be carried out over all possible extensions, including
infinite-dimensional ones, which is highly nontrivial. Maybe the task can
be simplified, in the case of Gaussian states, by restricting to Gaussian