ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

1.3.3.1. Properties of entanglement monotones.
1.3. Theory of bipartite entanglement 19
• Nullification. E(ϱ) ≥ 0. If ϱ is separable, then E(ϱ) = 0.
• Normalization. For
a maximally entangled state in d × d dimension,
d−1
|Φ〉 = |i, i〉 / √ d, it should be
i=0
E(|Φ〉〈Φ|) = log d . (1.30)
• Local invariance. The measure E(ϱ) should be invariant under local
unitary transformations,
E
= E(ϱ) . (1.31)
( Û1 ⊗ Û2)
† †
ϱ ( Û 1 ⊗ Û 2 )
• LOCC monotonicity. The measure E(ϱ) should not increase on average
upon application of LOCC transformations,
E( Ô LOCC(ϱ)) ≤ E(ϱ) , (1.32)
• Continuity. The entanglement difference between two density matrices
infinitely close in trace norm should tend to zero,
ϱ − σ → 0 ⇒ E(ϱ) − E(σ) → 0 . (1.33)
Additional requirements (not strictly needed, and actually not satisfied even by
some ‘good’ entanglement measures), include: additivity on tensor product states,
E(ϱ ⊗N ) = N E(ϱ); convexity, E(λ ϱ + (1 − λ) σ) ≤ λ E(ϱ) + (1 − λ) E(σ) with
0 ≤ λ ≤ 1; reduction to the entropy of entanglement Eq. (1.25) on pure states.
The latter constraint is clearly not necessary, as it is enough for a quantifier E ′ (ϱ)
to be a strictly monotonic and convex function of another measure E ′′ (ϱ) which
satisfies the above listed properties, in order for E ′ (ϱ) to be regarded as a good
entanglement measure. Another interesting property a good entanglement measure
should satisfy, which becomes crucial in the multipartite setting, is monogamy in
the sense of Coffman-Kundu-Wootters [59]. In a tripartite state ϱABC,
E(ϱ A|(BC)) ≥ E(ϱ A|B) + E(ϱ A|C) . (1.34)
A detailed discussion about entanglement sharing and monogamy constraints [GA12]
will be provided in Sec. 1.4, as it embodies the central idea behind the results of
Part III of this Dissertation.
1.3.3.2. Entanglement measures. We will now recall the definition of some ‘popular’
entanglement measures, which have special relevance for our results obtained in the
continuous variable scenario. The author is referred to Refs. [188, 54] for better
and more comprehensive reviews.
• Entanglement of formation.— The entanglement of formation EF (ϱ) [24] is
the convex-roof extension [167] of the entropy of entanglement Eq. (1.25),
i.e. the weighted average of the pure-state entanglement,
EF (ϱ) = min
{pk, |ψk〉}
pk EV (|ψk〉) , (1.35)
k
minimized over all decompositions of the mixed state ϱ =
k pk|ψk〉〈ψk|.
An explicit solution of such nontrivial optimization problem is available
for two qubits [273], for highly symmetric states like Werner states and