Macroscopic entanglement of many-magnon states

We study macroscopic entanglement of various pure states of a one-dimensional $N$-spin system with $N⪢1$. Here, a quantum state is said to be macroscopically entangled if it is a superposition of macroscopically distinct states. To judge whether such superposition is hidden in a general state, we use an essentially unique index $p$: A pure state is macroscopically entangled if $p=2$, whereas it may be entangled but not macroscopically if $p<2$. This index is directly related to fundamental stabilities of many-body states. We calculate the index $p$ for various states in which magnons are excited with various densities and wave numbers. We find macroscopically entangled states $(p=2)$ as well as states with $p=1$. The former states are unstable in the sense that they are unstable against some local measurements. On the other hand, the latter states are stable in the senses that they are stable against any local measurements and that their decoherence rates never exceed $O\left(N\right)$ in any weak classical noises. For comparison, we also calculate the von Neumann entropy $S_{N∕2}\left(N\right)$ of a subsystem composed of $N∕2$ spins as a measure of bipartite entanglement. We find that $S_{N∕2}\left(N\right)$ of some states with $p=1$ is of the same order of magnitude as the maximum value $N∕2$. On the other hand, $S_{N∕2}\left(N\right)$ of the macroscopically entangled states with $p=2$ is as small as $O\left(\log N\right)⪡N∕2$. Therefore larger $S_{N∕2}\left(N\right)$ does not mean more instability. We also point out that these results are partly analogous to those for interacting many bosons. Furthermore, the origin of the huge entanglement, as measured either by $p$ or $S_{N∕2}\left(N\right)$, is discussed to be due to spatial propagation of magnons.

Figure 1

The maximum eigenvalue $e_{\max }$ of the VCM of two-magnon states with wave numbers $k_1$ and $k_2$ as functions of the number $N$ of spins. Because of the translational invariance of the system in the $k$ space, we take $k_1=0$ without loss of generality. The solid line represents the analytic expression $e_{\max }=1+(5N-12)∕N$, Eq. (27), which assumes that all wave numbers are equal.

Figure 2

The maximum eigenvalue $e_{\max }$ of the VCM of $m$-magnon states with $m=N∕2,N∕4$, and $N∕6$ as functions of the number $N$ of spins. The wave numbers of magnons are all different taking the values 0, $\pm 2\pi ∕N,\pm 4\pi ∕N$,…, respectively, i.e., the first Brillouin zone is continuously occupied from the bottom.

Figure 3

The maximum eigenvalue $e_{\max }$ of the VCM of $N∕2$-magnon states as functions of the number $N$ of spins. Most magnons have equal wave numbers, i.e., most magnons are “condensed.” The solid line represents the analytic expression $e_{\max }=1+N∕2$, Eq. (27), which assumes that all wave numbers are equal. The circle and crosse represent numerical results for the cases where the wave numbers of one or two magnons are different. Because of the translational invariance in the $k$ space, we take the wave number of the condensed magnons as 0 without loss of generality.

Figure 4

The von Neumann entropy of a subsystem, $S_{N∕2}\left(N\right)$, of three-magnon states with wave numbers $k_1,k_2$, and $k_3$ as functions of the number $N$ of spins. Because of the translational invariance of the system in the $k$ space, we take $k_2=0$ without loss of generality.

Figure 5

The von Neumann entropy of a subsystem, $S_{N∕2}\left(N\right)$, of $m$-magnon states with $m=N∕2,N∕4$, and $N∕6$ as functions of the number $N$ of spins. The wave numbers of magnons are all different taking the values 0,$\pm 2\pi ∕N,\pm 4\pi ∕N$,…, respectively, i.e., the first Brillouin zone is continuously occupied from the bottom. The lines represent the regression lines calculated with the least squares.