Entanglement is considered a crucial resource in the performance of
quantum information processing, and it is very important to characterize
and understand it both qualitatively and quantitatively. However, even
today the notion of entanglement —with its counterpart: separability— is
not free of controversy and open problems exist. The first problem
arises because there are several complete sets of observables that allow
for a similar description of the system, but describe different
subsystems or partitions of the whole system (Zanardi 2001; Zanardi,
Lidar and Lloyd 2004; Caban et al. 2005; Harshman and Wickramasekara
2007). In algebraic terms, these sets generate the same total observable
algebra but induce different subalgebras. Entanglement depends on the
substructure of the full system and, therefore, is relative to the
selected set of observables. This freedom implies some ambiguity in the
notion of entanglement when it is associated to a system and, in order
to avoid it, many authors tried to establish a criterion (empirical,
operational, etc.) that favors one of all possible descriptions. Yet,
the problem can be circumvented by referring to a generalized version of
entanglement, where the correlations are defined with respect to a
certain subspace of observables, and irrespective of any preferred
decomposition of the system into subsystems (Barnum et al. 2004; Viola
and Barnum 2007; Derkacz, Gwozdz and Jakobczyk 2012). The same treatment
is valid for other types of genuine quantum correlations as discord.
This discussion, which we will advance in our work, has profound
implications on theoretical and experimental aspects of physics in
general, and in quantum information in particular. First, because the
relative nature of quantum correlations forces us to rethink the
orthodox classical limit of Quantum Mechanics via decoherence (Caponigro
and Giannetto 2010; Lychkovskiy 2013; Kastner 2014). The ability to
explain the emergence of classical behavior from quantum principles must
be investigated, because the picture concerning the resources for
information processing becomes more complex, and the potential for
important improvements is at hand (Verstraete and Cirac 2003; Bartlett,
Rudolph and Spekkens 2007; Thirring et al. 2011; Earman 2014).
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