  Does bound entanglement naturally appear in quantum many-body systems? We
address this question by showing the existence of bound-entangled thermal
states for harmonic oscillator systems consisting of an arbitrary number of
particles. By explicit calculations of the negativity for different partitions,
we find a range of temperatures for which no entanglement can be distilled by
means of local operations, despite the system being globally entangled. We
offer an interpretation of this result in terms of entanglement-area laws,
typical of these systems. Finally, we discuss generalizations of this result to
other systems, including spin chains.
