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(N) in any weak classical noises. For comparison, we also calculate the von Neumann entropy S_N∕2(N) of a subsystem composed of N∕2 spins as a measure of bipartite entanglement. We find that S_N∕2(N) 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(N) of the macroscopically entangled states with p=2 is as small as O(log N)⪡N∕2. Therefore larger S_N∕2(N) 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(N), is discussed to be due to spatial propagation of magnons.