ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

4.3. Entanglement versus Entropic measures 61
symmetric Gaussian states, there is a unique interpretation for entanglement and
a unique ordering of entangled states belonging to that subset, as previously remarked.
4.3. Entanglement versus Entropic measures
Here we aim at a characterization of entanglement of two-mode Gaussian states and
in particular at unveiling its relationship with the degrees of information associated
with the global state of the system, and with the reduced states of each of the two
subsystems.
As extensively discussed in Chapter 1, the concepts of entanglement and information
encoded in a quantum state are closely related. Specifically, for pure
states bipartite entanglement is equivalent to the lack of information (mixedness)
of the reduced state of each subsystem. For mixed states, each subsystem has
its own level of impurity, and moreover the global state is itself characterized by
a nonzero mixedness. Each of these properties can be interpreted as information
on the preparation of the respective (marginal and global) states, as clarified in
Sec. 1.1. Therefore, by properly accessing these degrees of information one is intuitively
expected to deduce, to some extent, the status of the correlations between
the subsystems.
The main question we are posing here is
What can we say about the quantum correlations existing between
the subsystems of a quantum multipartite system in a mixed state,
if we know the degrees of information carried by the global and the
reduced states?
In this Section we provide an answer, which can be summarized as “almost everything”,
in the context of two-mode Gaussian states of CV systems. Based on our
published work [GA2, GA3, GA6], we will demonstrate, step by step, how the entanglement
— specifically, measured by the logarithmic negativity — of two-mode
Gaussian states can be accurately (both qualitatively and quantitatively) characterized
by the knowledge of global and marginal degrees of information, quantified
by the purities, or by the generalized entropies of the global state and of the reduced
states of the two subsystems.
4.3.1. Entanglement vs Information (I) – Maximal negativities at fixed global
purity
The first step towards giving an answer to our original question is to investigate the
properties of extremally entangled states at a given degree of global information.
Let us mention that, for two-qubit systems, the existence of maximally entangled
states at fixed mixedness (MEMS) was first found numerically by Ishizaka and
Hiroshima [126]. The discovery of such states spurred several theoretical works [246,
158], aimed at exploring the relations between different measures of entanglement
and mixedness [261] (strictly related to the questions of the ordering induced by
these different measures [255, 247], and of the volume of the set of mixed entangled
states [283, 282]).
Unfortunately, it is easy to show that a similar analysis in the CV scenario is
meaningless. Indeed, for any fixed, finite global purity µ there exist infinitely many
Gaussian states which are infinitely entangled. As an example, we can consider the