ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

20 1. Characterizing entanglement
isotropic states in arbitrary dimension [231, 257], and for symmetric twomode
Gaussian states [95]. The additivity of the entanglement of formation
is currently an open problem [1].
• Entanglement cost.— The entanglement cost EC(ϱ) [24] quantifies how
much Bell pairs |Φ〉 = (|00〉 + |11〉)/ √ 2 one has to spend to create the
entangled state ϱ by means of LOCC. It is defined as the asymptotic ratio
between the minimum number M of used Bell pairs, and the number N
of output copies of ϱ,
EC(ϱ) = min
M
lim
{LOCC} N→∞
in
. (1.36)
N out
The entanglement cost, a difficult quantity to be computed in general
[252], is equal to the asymptotic regularization of the entanglement of
formation [110], EC(ϱ) = limN→∞[EF (ϱ ⊗N )/N], and would coincide with
EF (ϱ) if the additivity of the latter was proven.
• Distillable entanglement.— The converse of the entanglement cost is the
distillable entanglement [24], which is defined as the asymptotic fraction
M/N of Bell pairs which can be extracted from N copies of the state ϱ
by using the optimal LOCC distillation protocol,
ED(ϱ) = max
lim
{LOCC} N→∞
M out
. (1.37)
N in
The distillable entanglement vanishes for bound entangled states. The
quantity EC(ϱ)−ED(ϱ) can be regarded as the undistillable entanglement.
It is strictly nonzero for all entangled mixed states [276], meaning that
LOCC manipulation of quantum states is asymptotically irreversible apart
from the case of pure states (one loses entanglement units in the currency
exchange!).
• Relative entropy of entanglement.— An intuitive way to measure entanglement
is to consider the minimum “distance” between the state ϱ and the
convex set D ⊂ H of separable states. In particular, the relative entropy
of entanglement ER(ϱ) [244] is the entropic distance [i.e. the quantum
relative entropy Eq. (1.18)] between ϱ and the closest separable state σ ⋆ ,
ER(ϱ) = min
σ ⋆ ∈D Tr [ϱ (log ϱ − log σ⋆ )] . (1.38)
Note that the closest separable state σ ⋆ is typically not the corresponding
product state ϱ ⊗ = ϱ1⊗ϱ2; the relative entropy between ϱ and ϱ ⊗ is indeed
the mutual information Eq. (1.17), a measure of total correlations which
therefore overestimates entanglement (being nonzero also on separable,
classically correlated states).
Let us recall that the all the above mentioned entanglement measures coincide
on pure states, EF (|ψ〉) = EC(|ψ〉) = ER(|ψ〉) = ED(|ψ〉) ≡ EV (|ψ〉), while for
generic mixed states the following chain of analytic inequalities holds [120, 69, 117],
EF (ϱ) ≥ EC(ϱ) ≥ ER(ϱ) ≥ ED(ϱ) . (1.39)
• Negativities.— An important class of entanglement measures is constituted
by the negativities, which quantify the violation of the PPT criterion for