Symmetry enforces entanglement at high temperatures

TL;DR

This work addresses the fate of quantum entanglement in thermal states of local Hamiltonians under symmetry constraints. It develops two complementary symmetry frameworks—the strong-symmetry canonical ensemble $\rho_{\beta,\Lambda}$ and the weak-symmetry Gibbs ensemble with superselection—showing that Abelian on-site symmetries generically prevent sudden death of entanglement at high temperature. Central results include the SEC/EC framework for persistent symmetric entanglement, the local indistinguishability between ensembles, and the equivalence of EC and NC, together with explicit analyses of the thermal cluster chain and fermionic systems. In fermionic settings, the paper proves that fermionic negativity persists for canonical and Gibbs ensembles, and resolves conjectures about Type II fermionic states, thereby highlighting a robust role of symmetry in protecting quantum resources at finite and asymptotically high temperatures.

Abstract

Many-body quantum systems with local interactions undergo ``sudden death of entanglement" at high temperatures, whereby thermal states become classical mixtures of product states. We investigate whether symmetry constraints can prevent this phenomenon. We prove that strongly symmetric thermal states (canonical ensemble) of generic Hamiltonians with on-site Abelian symmetries remain entangled with non-zero entanglement negativity at arbitrarily high temperatures, under mild conditions on the symmetry actions and the charge sector of the strong symmetry. Our results extend to weakly symmetric thermal states (Gibbs ensemble) under superselection rules, which restrict state decompositions to be symmetric. In particular, we show that fermionic Gibbs states evade sudden death of entanglement and have persistent fermionic negativity at high temperatures, proving along the way some existing conjectures about fermionic entanglement. These findings demonstrate that global symmetry correlations can preserve quantum entanglement despite thermal decoherence, providing new insights into the interplay between symmetry and quantum information in thermal equilibrium.

Figures (3)

• Figure 1: Schematic paths of generic many-body thermal states in relation to the set of (symmetrically) separable states SEP (SymSEP). The Gibbs ensemble $\rho_\beta \propto e^{-\beta H}$ is separable from $\beta = 0$ to $\beta_{\textsf{SDOE}}$ due to the sudden death of entanglement, but it becomes symmetrically entangled as soon as $\beta > 0$. Similarly, the canonical ensemble $\rho_{\beta, \Lambda} \propto e^{-\beta H} \Pi_\Lambda$ is also entangled for $\beta > 0$. For a more accurate representation of SEP and SymSEP in a slice of the space of quantum states, see Fig. \ref{['fig:geometry_twoqubits']} in Appendix \ref{['appsec:two_qubit']}.
• Figure 2: Entanglement negativity $E_N$ for the Gibbs and canonical ensembles of the cluster chain Hamiltonian $H = \sum_{i=1}^N Z_{i-1} X_i Z_{i+1}$ under $\mathbb{Z}_2$ symmetry generated by $\prod_i X_i$ in the limit $N \to \infty$. Both curves are independent of the entangling interval size $|A| \geq 2$, and the canonical one, of the charge sector.
• Figure 3: Subset of quantum states of two qubits formed by convex mixtures of the Bell states $\ket{\Psi_-} \propto \ket{01} -\ket{10}$ and $\ket{\Phi_-} = \ket{00} - \ket{11}$, and the maximally mixed $X_1X_2 = +1$ state $\frac{1}{2} \Pi_+ = \frac{1}{2} \mathinner{\ket{00}\!\bra{00}} + \frac{1}{2} \mathinner{\ket{11}\!\bra{11}}$. The dashed line traces the path of the Heisenberg antiferromagnetic thermal states $e^{-\beta \mathbf{S}_1 \cdot \mathbf{S}_2}$, or, equivalently, of the Werner states $\lambda \mathinner{\ket{\Psi_-}\!\bra{\Psi_-}} + (1-\lambda) \frac{1}{4} \mathbb{1}$. Meanwhile, the thick black line going from $\frac{1}{2} \Pi_-$ to $\ket{\Psi_-}$ traces the path of the canonical ensemble $e^{- \beta \mathbf{S}_1 \cdot \mathbf{S}_2} \Pi_{-1}$ in the $X_1 X_2 = -1$ sector. The SEP region in cyan consists of separable states, with the blue SymSEP line inside it indicating the symmetrically separable ones. Their loci were found via the Peres-Horodecki criterion, since it is necessary and sufficient for two-qubit systems Peres_1996Horodecki_1996. This illustration was inspired by Fig. 16.8(b) of bengtsson_geometry_2017.

Theorems & Definitions (36)

• Theorem 1: Persistence of entanglement: informal
• Theorem 2: Persistence of entanglement
• Definition 1
• Theorem 3
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof