ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

1.1. Information contained in a quantum state 13
mixedness of a state ϱ by the amount of information it lacks, and as measures of
the overall degree of coherence of the state.
The generalized entropies Sp’s range from 0 for pure states to 1/(p − 1) for
completely mixed states with fully degenerate eigenspectra. We also mention that,
in the asymptotic limit of arbitrary large p, the function Tr ϱ p becomes a function
only of the largest eigenvalue of ϱ: more and more information about the state is
discarded in such an estimate for the degree of purity; considering for any nonpure
state Sp in the limit p → ∞, yields a trivial constant null function, with no
information at all about the state under exam. We also note that, for any given
quantum state, Sp is a monotonically decreasing function of p.
Finally, another important class of entropic measures includes the Rényi en-
tropies [194]
It can be shown that [207]
S R p =
log Tr ϱp
1 − p
, p > 1. (1.14)
lim
p→1+ Sp = lim
p→1+ SR p = − Tr (ϱ log ϱ) ≡ SV , (1.15)
so that also the Von Neumann entropy, Eq. (1.4), can be defined in terms of p-norms
and within the framework of generalized entropies.
1.1.4. Mutual information
The subadditivity property (1.6) of Von Neumann entropy is at the heart of the
measure typically employed in quantum information theory to quantify total —
classical and quantum — correlations in a quantum state, namely the mutual information
[101]
I(ϱ) = SV (ϱ1) + SV (ϱ2) − SV (ϱ) , (1.16)
where ϱ is the state of the global system and ϱ1,2 correspond to the reduced density
matrices. Mutual information quantifies the information we obtain on ϱ by looking
at the system in its entirety, minus the information we can extract from the separate
observation of the subsystems. It can in fact be written as relative entropy between
ϱ and the corresponding product state ϱ ⊗ = ϱ1 ⊗ ϱ2,
I(ϱ) = SR(ϱϱ ⊗ ) , (1.17)
where the relative entropy, a distance-like measure between two quantum states in
terms of information, is defined as [243]
SR(ϱσ) = −SV (ϱ) − Tr [ϱ log σ] = Tr [ϱ (log ϱ − log σ)] . (1.18)
If ϱ is a pure quantum state [SV (ϱ) = 0], the Von Neumann entropy of its
reduced states SV (ϱ1) = SV (ϱ2) quantifies the entanglement between the two parties,
as we will soon show. Being I(ϱ) = 2SV (ϱ1) = 2SV (ϱ2) in this case, one says
that the pure state also contains some classical correlations, equal in content to the
quantum part, SV (ϱ1) = SV (ϱ2).
In mixed states a more complex scenario emerges. The mere distinction between
classical correlations, i.e. producible by means of local operations and classical communication
(LOCC) only, and entanglement, due to a purely quantum interaction
between subsystems, is a highly nontrivial, and not generally accomplished yet,
task [112, 101].