Detecting entanglement with Jarzynski’s equality

We present a method for detecting the entanglement of a state using nonequilibrium processes. A comparison of relative entropies allows us to construct an entanglement witness. The relative entropy can further be related to the quantum Jarzynski equality, allowing nonequilibrium work to be used in entanglement detection. To exemplify our results, we consider two different spin chains.

Figure 1

This is a two-dimensional representation of the multidimensional Hilbert space. The small oval represents the set of separable states, and the large oval the set of all states. ${\sigma }_{\mathrm{cm}}$ is a completely mixed state, and ${\sigma }_{\mathrm{css}}$ is the closest separable state to $\rho$. Any state along the pink curve, such as ${\sigma }^{\prime }$, has the same “distance” in terms of the relative entropy as ${\sigma }_{\mathrm{css}}$ to $\rho$.

Figure 2

This plots $J_z$ versus $B$ and $T$ when $N=3$, and shows the values for which we can detect entanglement in ${\rho }^{*}$. The state is entangled between the surface of the plot and the axes.

Figure 3

This plots $J_z$ versus $B$ and $T$ when $N=7$, and shows the values for which we can detect entanglement in ${\rho }^{*}$. The state is entangled between the surface of the plot and the axes.

Figure 4

This plots $D$, the Dzyaloshinskii-Moriya interaction, versus $B$ and $T$ when $n=3$. The larger transparent (purple) surface shows the values for which we can detect entanglement in ${\rho }^{*}$ at $J_z=0$, and the smaller (turquoise) surface shows this at $J_z=0.2$. The state is entangled between the surface of each plot and the axes.