Macroscopic entanglement of many-magnon states

Abstract

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 wavenumbers. 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.

Visualization of superposition of macroscopically distinct states

“…They are separable states, because the mean-field approximation neglects the correlations between sites. On the other hand, the exact ground state is unique, symmetric, and has p = 2 [24,30,31,32,33]. We visualize the exact ground state.…”

“…It is known that re i and/or im i fluctuate(s) macroscopically (see Ref. [24] and Appendix A). We let such macroscopically fluctuating part(s) be an element(s) of S. In this way, we obtain a set of macroscopically fluctuating hermitian additive operators, e.g.,…”

“…Existence of a superposition of macroscopically distinct states in a many-body pure state can be identified by an index p (1 ≤ p ≤ 2) [21,22,23,24,27]: If a given pure state has p = 2, it contains a superposition of macroscopically distinct states [21,24].…”

“…Therefore a pure state having p = 2 contains a superposition of macroscopically distinct states (see Refs. [24,25] for detailed discussion). On the other hand, if p < 2, all additive operators 'have macroscopically definite values' in the sense that relative fluctuations of all additive operators vanish as N → ∞.…”

“…In quantum many-body systems, which include quantum computers [8,9,10,11], there are many states which contain superpositions of macroscopically distinct states [12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]. Existence of a superposition of macroscopically distinct states in a many-body pure state can be identified by an index p (1 ≤ p ≤ 2) [21,22,23,24,27]: If a given pure state has p = 2, it contains a superposition of macroscopically distinct states [21,24].…”