Thermal Bound Entanglement in Macroscopic Systems and Area Law

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.

Figure 1

$T_{\mathrm{dist}}^{\mathrm{e}\mathrm{\text{:}}\mathrm{o}}$ (solid line) and $T_{\mathrm{dist}}^{\mathrm{h}\mathrm{\text{:}}\mathrm{h}}$ (dashed line) as a function of the coupling constant $c$ for the Harmonic chain with nearest-neighbor interactions composed by 800 oscillators. Notice that the range of temperatures $T_{\mathrm{dist}}^{\mathrm{h}\mathrm{\text{:}}\mathrm{h}}<T<T_{\mathrm{dist}}^{\mathrm{e}\mathrm{\text{:}}\mathrm{o}}$ for which bound entanglement is guaranteed increases when the system approaches the critical point $c=0.5$. The intermediate dotted line represents the upper bound to $T_{\mathrm{dist}}^{\mathrm{h}\mathrm{\text{:}}\mathrm{h}}$ as given by In Eq. (13) (e.g., we considered $m=10$ and $s=3$) and valid in the macroscopic limit: above this line the log-negativity in the half-half partition is zero. The threshold $T_{\mathrm{dist}}^{\mathrm{e}\mathrm{\text{:}}\mathrm{o}}$ in the macroscopic limit [given by Eq. (4)] coincides with the one calculated numerically for 800 oscillators (see text for details). As a consequence, in the shaded region we can guarantee the presence of bound entanglement in the macroscopic limit. Inset: $T_{\mathrm{dist}}^{\mathrm{e}\mathrm{\text{:}}\mathrm{o}}$ (solid line) and $T_{\mathrm{dist}}^{\mathrm{h}\mathrm{\text{:}}\mathrm{h}}$ (dashed line) as a function of the number $n$ of oscillators composing the system (log-lin scale), for $c=0.3$ (the same behavior is found for other values of $c$). The gap $T_{\mathrm{dist}}^{\mathrm{e}\mathrm{\text{:}}\mathrm{o}}-T_{\mathrm{dist}}^{\mathrm{h}\mathrm{\text{:}}\mathrm{h}}$ is seen to remain constant with the size of the system, apart from an initial transient.

Figure 2

Negativity in the even-odd (full symbols) and half-half (empty symbols) partitions for thermal states of $n$ spin-one-half particles with nearest-neighbor Hamiltonian given by $H_{XX}=-\sum _{}^{}({\sigma }_i^x{\sigma }_{i+1}^x+{\sigma }_i^y{\sigma }_{i+1}^y)+B\sum _{}^{}{\sigma }_i^z$, and $B=1.9$. The temperatures for each partition are set to be, from top to bottom, $T=2$, 2.6. In the even-odd partition we can clearly see an increase of the negativity with respect to the system size, whereas it saturates in the half-half partition. Both behaviors are expected from the entanglement-area law. For $T=2.6$ we see that the state is bound entangled (the negativity in the half-half partition being zero).