The Fate of Entanglement

TL;DR

This work shows that, under broad physical evolutions—temperature rise, time dynamics, or increased particle separation—all forms of genuine multipartite entanglement tend to disappear, effectively leaving only bipartite or non-GME correlations. The authors frame the fate of entanglement with a geometric picture: separable states form a convex continent in state space, and typical evolutions push systems into its interior, yielding a finite entanglement range and dynamical sudden death times. They quantify entanglement via logarithmic negativity, geometric entanglement, and a 3-party W criterion, using certification tools like the Gilbert algorithm and a general separability criterion to rigorously bound separability in challenging cases. The analysis of explicit models—the frustrated icosahedral Ising system and a 1D spin chain—demonstrates rapid entanglement decay with temperature, time, and separation, and reveals a fundamental difference for fermions due to parity superselection, where a continent of genuine multipartite fermionic entanglement is absent and GME vanishes under typical evolution. These insights illuminate the intrinsic structure of entanglement in quantum matter and guide future work on robust entangled states and scalable quantum architectures.

Abstract

Quantum entanglement manifests itself in non-local correlations between the constituents of a system. In its simplest realization, a measurement on one subsystem is affected by a prior measurement on its partner, irrespective of their separation. For multiple parties, purely collective types of entanglement exist but their detection, even theoretically, remains an outstanding open question. Here, we argue that all forms of multipartite entanglement entirely disappear during the typical evolution of a physical state as it heats up, evolves in time in a large family of dynamical protocols, or as its parts become separated. We focus on the generic case where the system interacts with an environment. These results mainly follow from the geometry of the entanglement-free continent in the space of physical states, and hold in great generality. We illustrate these phenomena with a frustrated molecular quantum magnet in and out of equilibrium, and a quantum spin chain. In contrast, if the particles are fermions, such as electrons, another notion of entanglement exists that protects bipartite quantum correlations. However, genuinely collective fermionic entanglement disappears during typical evolution, thus sharing the same fate as in bosonic systems. These findings provide fundamental knowledge about the structure of entanglement in quantum matter and architectures, paving the way for its manipulation.

Figures (8)

• Figure 1: Evolution of multipartite entanglement.a) We consider a general state of $m$ subsystems, here illustrated for $m=4$, which can be in contact with an environment. b) The state is described by a density matrix $\rho(s)$ that evolves according to a parameter $s$ such as temperature, time or separation. The blue region represents the "sea" of entangled states, where deep-blue regions are more entangled than light-blue ones. The orange disk is the separable continent.
• Figure 2: Icosahedral molecule and its entanglement evolutions.a) Illustration of the quantum molecular magnet with icosahedral geometry. b) Dependence of the geometric entanglement $\mathcal{D}$, the 2-spin logarithmic negativity $\mathcal{E}$ and the genuine 3-party entanglement criterion $\mathcal{D}_b$ in the icosahedral molecule as a function of the transverse field at zero temperature. c) Evolution of the same quantities and the criterion $W$ as a function of temperature with $h=3$. d) Same quantities as a function of time in the quench protocol Ico-1, where the system is prepared in a product state and evolves under the anti-ferromagnetic Ising Hamiltonian with $h=3$. In panels b)- c)- d), the open shapes indicate that the corresponding distance measure is rigorously certified to vanish: the state is fully (bi)separable.
• Figure 3: Quench for the spin chain. Geometric entanglement $\mathcal{D}$ and $\mathcal{D}_b$ as a function of time after a quench in the spin chain defined in Eq. \ref{['eq:xyz']}. The open shapes indicate that the state is rigorously certified to be fully (bi)separable.
• Figure 4: Structure of the space of states for fermions. For fermions, because of the fermion-parity superselection rule, there is no separable continent. Instead, separable states form a zero-width sand beach with entangled states arbitrarily close. This beach is surrounded by a light-blue region of biseparable fermionic states, beyond which states possess GME.
• Figure 5: Test of the full-rank hypothesis. Minimal eigenvalue $\lambda_{\rm min}$ of the reduced density matrices for three adjacent spins in the icosahedral molecule at $h=3$ (top), and for four adjacent spins in the spin chain (bottom) for the whole spectrum, labeled by the corresponding energy $E$. We note that the groundstate has the smallest $\lambda_{\rm min}$ in the top panel, while it is the second smallest in the bottom one.