ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

14 1. Characterizing entanglement
We are now going to summarize the most relevant results to date concerning
the qualitative and quantitative characterization of entanglement.
1.2. Entanglement and non-locality
From a phenomenological point of view, the phenomenon of entanglement is fairly
simple. When two physical systems come to an interaction, some correlation of a
quantum nature is generated between the two of them, which persists even when
the interaction is switched off and the two systems are spatially separated 2 . If we
measure a local observable on the first system, its state collapses of course in an
eigenstate of that observable. Surprisingly, also the state of the second system,
wherever it is (in the ideal case of zero environmental decoherence), is modified
instantly. Responsible for this “spooky action at a distance” [73] is the non-classical
and non-local quantum correlation known as entanglement.
Suppose we have a bipartite or multipartite quantum state: well, the answer
to an apparently innocent question like
Does this state contain quantum correlations?
is extremely hard to be achieved [111, 13, 188]. The first step concerns a basic
understanding of what such a question really means.
One may argue that a system contains quantum correlations if the observables
associated to the different subsystems are correlated, and their correlations cannot
be reproduced with purely classical means. This implies that some form of inseparability
or non-factorizability is necessary to properly take into account those
correlations. For what concerns globally pure states of the composite quantum system,
it is relatively easy to check if the correlations are of genuine quantum nature.
In particular, it is enough to check if a Bell-CHSH inequality [19, 58] is violated
[96], to conclude that a pure quantum state is entangled. There are in fact many
different criteria to characterize entanglement, but all of them are practically based
on equivalent forms of non-locality in pure quantum states.
These equivalences fade when we deal with mixed states. At variance with a
pure state, a mixture can be prepared in (generally infinitely) many different ways.
Not being able to reconstruct the original preparation of the state, one cannot
extract all the information it contains. Accordingly, there is not a completely
general and practical criterion to decide whether correlations in a mixed quantum
state are of classical or quantum nature. Moreover, different manifestations of
quantum inseparability are in general not equivalent. For instance, one pays more
(in units of Bell singlets) to create an entangled mixed state ϱ — entanglement cost
[24] — than what one can get back from reconverting ϱ into a product of singlets
— distillable entanglement [24] — via LOCC [276]. Another example is provided
by Werner in a seminal work [264], where he introduced a parametric family of
mixed states (known as Werner states) which, in some range of the parameters,
are entangled (inseparable) without violating any Bell inequality on local realism,
and thus admitting a description in terms of local hidden variables. It is indeed
an open question in quantum information theory to prove whether any entangled
state violates some Bell-type inequality [1, 229].
2 Entanglement can be also created without direct interaction between the subsystems, via
the so-called entanglement swapping [22].