ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

12 1. Characterizing entanglement
where ϱ1,2 are the reduced density matrices ϱ1,2 = Tr2,1 ϱ associated to
subsystems S1,2. For states of the form ϱ ⊗ = ϱ1⊗ϱ2, Eq. (1.6) is saturated,
yielding that Von Neumann entropy is additive on tensor product states:
SV (ϱ1 ⊗ ϱ2) = SV (ϱ1) + SV (ϱ2) . (1.7)
The purity, Eq. (1.1), is instead multiplicative on product states, as the
trace of a product equates the product of the traces:
µ(ϱ1 ⊗ ϱ2) = µ(ϱ1) · µ(ϱ2) . (1.8)
• Araki–Lieb inequality [9]. In a bipartite system,
SV (ϱ) ≥ |SV (ϱ1) − SV (ϱ2)| . (1.9)
Properties (1.6) and (1.9) are typically grouped in the so-called triangle
inequality
|SV (ϱ1) − SV (ϱ2)| ≤ SV (ϱ) ≤ SV (ϱ1) + SV (ϱ2) . (1.10)
It is interesting to remark that Ineq. (1.6) is in sharp contrast with the analogous
property of classical Shannon entropy,
S(X, Y ) ≥ S(X), S(Y ) . (1.11)
Shannon entropy of a joint probability distribution is always greater than the Shannon
entropy of each marginal probability distribution, meaning that there is more
information in a global classical system than in any of its parts. On the other hand,
consider a bipartite quantum system in a pure state ϱ = |ψ〉〈ψ| . We have then
for Von Neumann entropies: SV (ϱ) = 0, while SV (ϱ1) (1.9)
= SV (ϱ2) ≥ 0. The global
state ϱ has been prepared in a well defined way, but if we measure local observables
on the subsystems, the measurement outcomes are unavoidably random and
to some extent unpredictable. We cannot reconstruct the whole information about
how the global system was prepared in the state ϱ (apart from the trivial instance
of ϱ being a product state ϱ = ϱ1 ⊗ ϱ2), by only looking separately at the two subsystems.
Information is rather encoded in non-local and non-factorizable quantum
correlations — entanglement — between the two subsystems. The comparison between
the relations Eq. (1.6) and Eq. (1.11) clearly evidences the difference between
classical and quantum information.
1.1.3. Generalized entropies
In general, the degree of mixedness of a quantum state ϱ can be characterized
completely by the knowledge of all the associated Schatten p–norms [18]
ϱp = ( Tr |ϱ| p ) 1
p = ( Tr ϱ p ) 1
p , with p ≥ 1. (1.12)
In particular, the case p = 2 is directly related to the purity µ, Eq. (1.1), as it is
essentially equivalent (up to normalization) to the linear entropy Eq. (1.2). The
p-norms are multiplicative on tensor product states and thus determine a family of
non-extensive “generalized entropies” Sp [17, 232], defined as
1 − Tr ϱp
Sp = , p > 1. (1.13)
p − 1
These quantities have been introduced independently Bastiaans in the context of
quantum optics [17], and by Tsallis in the context of statistical mechanics [232].
In the quantum arena, they can be interpreted both as quantifiers of the degree of