Distillable entanglement and area laws in spin and harmonic-oscillator systems

We address the presence of nondistillable (bound) entanglement in natural many-body systems. In particular, we consider standard harmonic and spin-$\frac{1}{2}$ chains, at thermal equilibrium and characterized by few interaction parameters. The existence of bound entanglement is addressed by calculating explicitly the negativity of entanglement for different partitions. This allows us to individuate a range of temperatures for which no entanglement can be distilled by means of local operations, despite the system being globally entangled. We discuss how the appearance of bound entanglement can be linked to entanglement-area laws, typical of these systems. Various types of interactions are explored, showing that the presence of bound entanglement is an intrinsic feature of these systems. In the harmonic case, we analytically prove that thermal bound entanglement persists for systems composed by an arbitrary number of particles. Our results strongly suggest the existence of bound entangled states in the macroscopic limit also for spin-$\frac{1}{2}$ systems.

Figure 1

Log negativity $E_l$ as a function of the number of oscillators $n$ for half-half (a) and the even-odd (b) partitions. Temperatures are $T=0.45$ (solid line), $T=0.4$ (dashed line), and $T=0.35$ (dotted line), and $c=0.4$ (the same behavior is found for other values of $c$). For $T=0.45$ the state is bound entangled.

Figure 2

Threshold temperatures above which the log negativity is zero in the even-odd ($T_{\mathrm{th}}^{e:o}$, solid line) and half-half ($T_{\mathrm{th}}^{h:h}$, dotted line) partitions. The thresholds are plotted as a function of the coupling constant $c$ for the Harmonic chain with nearest-neighbor interactions composed by 800 oscillators. The intermediate line (dashed) shows the temperature $T_{\mathrm{th}}^{1:n-1}$ above which the state is separable in all the partitions $1:n-1$. Inset: $T_{\mathrm{th}}^{e:o}$ (solid line), $T_{\mathrm{th}}^{h:h}$ (dotted line), and $T_{\mathrm{th}}^{1:n-1}$ (dashed line) as a function of the number $n$ of oscillators composing the system (log-lin scale). The oscillators interact via nearest-neighbor couplings with $c=0.3$ (the same behavior is found for other values of $c$). The gap $T_{\mathrm{th}}^{e:o}-T_{\mathrm{th}}^{h:h}$ is seen to remain constant with the size of the system, apart from an initial transient.

Figure 3

$T_{\mathrm{th}}^{e:o}$ (solid line), $T_{\mathrm{th}}^{1:n-1}$ (dashed line), and $T_{\mathrm{th}}^{h:h}$ (dotted line) as a function of the coupling constant $\mu$ for the Harmonic chain with next-nearest-neighbor interactions in Eq. (7) composed by 200 oscillators.

Figure 4

The solid line represents $T_{\mathrm{th}}^{e:o}$ in the macroscopic limit, accordingly to Eq. (11): below this line the state of the system is entangled. The dashed line represents the upper bound to $T_{\mathrm{th}}^{h:h}$ as given by Eq. (31) (e.g., we considered $m=10$ and $s=3$): above this line the log negativity in the half-half partition is zero. In the shaded region then we can guarantee the presence of bound entanglement in the macroscopic limit.

Figure 5

Negativity in the even-odd (full symbols) and half-half (empty symbols) partitions for the thermal states of the Hamiltonian (32) with $B=1.9$ and $J=1$. The temperatures for each partition are set to be $T=2$ and 2.6 (see legend). 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. For $T=2.6$ we see that the state is bound entangled (the negativity in the half-half partition being zero).

Figure 6

Threshold temperatures above which the negativity is zero in the even-odd ($T_{\mathrm{th}}^{e:o}$, solid line), $1:n-1$ ($T_{\mathrm{th}}^{1:n-1}$, dotted line), and half-half ($T_{\mathrm{th}}^{h:h}$, dashed line) partitions. We plot the threshold temperature as a function of the coupling parameter $J$ of the Hamiltonian (32) with $n=10$ and $B=1.9$. Inset: Temperature above which the negativity in the even-odd ($T_{\mathrm{th}}^{e:o}$, solid line), $1:n-1$ ($T_{\mathrm{th}}^{1:n-1}$, dotted line), and half-half ($T_{\mathrm{th}}^{h:h}$, dashed line) partitions is zero as a function of the number $n$ of spins with Hamiltonian (32) with $J=1$ and $B=1.9$.

Figure 7

Negativity in the even-odd (solid line) and half-half (dashed line) partitions for the thermal states of Hamiltonian (32) with $n=10$, $J=1$, and $T=0.1$ as a function of $B$. For $B>2$ the ground state is the separable state $\mid 00\cdots 0⟩$, but some amount of entanglement still appears due to the mixture with entangled excited states.

Figure 8

Negativity in the even-odd (solid line) and half-half (dashed line) partitions for $n=10$, $J=1$, and $B=2.3$ for spin $\frac{1}{2}$ particles with Hamiltonian (32). As expected, the negativity in the even-odd partition is greater than the corresponding to the half-half partition, and bound entanglement appears in the temperature range $2.5<T<3.2$.