Exact Thermal Stabilizer Eigenstates at Infinite Temperature

Understanding how microscopic few-body interactions give rise to thermal behavior in isolated quantum many-body systems remains a central challenge in nonequilibrium statistical mechanics. While individual energy eigenstates are expected to reproduce thermal equilibrium values, analytic access to highly entangled thermal eigenstates of nonintegrable Hamiltonians remains scarce. In this Letter, we construct exact infinite-temperature eigenstates of generically nonintegrable two-body Hamiltonians using stabilizer states. These states can fully reproduce thermal expectation values for all spatially local observables, extending previously known Bell-pair-based constructions to a broader class. At the same time, we prove a sharp no-go theorem: stabilizer eigenstates of two-body Hamiltonians cannot satisfy microscopic thermal equilibrium for all four-body observables. This bound is tight, as we explicitly construct a translationally invariant Hamiltonian whose stabilizer eigenstate is thermal for all two-body and three-body observables as well as all spatially local observables. Our results suggest that reproducing higher-order thermal correlations requires nonstabilizer degrees of freedom, providing analytic insight into the interplay between interaction locality, microscopic thermal equilibrium, and quantum computational complexity.

💡 Research Summary

The paper tackles a long‑standing problem in nonequilibrium quantum statistical mechanics: how microscopic few‑body interactions can give rise to thermal behavior in isolated many‑body systems. While the eigenstate thermalization hypothesis (ETH) predicts that individual eigenstates of generic nonintegrable Hamiltonians reproduce thermal expectation values, analytic constructions of such highly entangled eigenstates have been scarce. The authors address this gap by constructing exact infinite‑temperature eigenstates of generic two‑body, nonintegrable Hamiltonians using stabilizer states—a class of highly entangled states well known in quantum information theory.

Stabilizer states are defined as simultaneous +1 eigenstates of an abelian subgroup G of the N‑qubit Pauli group that does not contain –I. Because G is maximal, the associated stabilizer state |ψ_G⟩ is unique (up to a global phase) and admits an efficient classical description. Graph states, a subclass of stabilizer states, are especially convenient: they are generated by applying controlled‑Z gates along the edges of a graph to a product of |+⟩ states. This representation allows exact evaluation of reduced density matrices and naturally yields volume‑law entanglement, making stabilizer states promising candidates for analytic thermal eigenstates.

The central results are two theorems. Theorem 1 (No‑go) states that any zero‑energy stabilizer eigenstate of a nontrivial two‑body Hamiltonian cannot satisfy microscopic thermal equilibrium (MITE) for any k‑body observables with k ≥ 4. In other words, stabilizer eigenstates can be thermal for one‑, two‑, and three‑body observables, but inevitably fail for four‑body observables. Theorem 2 (Achievability and Tightness) demonstrates that this bound is tight: the authors explicitly construct a translationally invariant, finite‑range, nonintegrable two‑body Hamiltonian whose zero‑energy stabilizer eigenstate is thermal for all two‑ and three‑body observables and for all contiguous subsystems of size O(N). Thus, k = 3 is the maximal body order for which stabilizer eigenstates of two‑body Hamiltonians can achieve MITE at infinite temperature.

The proof of the no‑go theorem hinges on a structural characterization (Proposition 3) of any traceless Hamiltonian that admits a given stabilizer state as a zero‑energy eigenstate. The proposition shows that such a Hamiltonian must be a sum of terms each factorizing a stabilizer element g ∈ G into a product of two Pauli strings P and Q (up to a phase), i.e. g = a_{P,Q} P Q. Consequently, each term in the Hamiltonian necessarily acts on at most |supp(P)| + |supp(Q)| qubits. Since the Hamiltonian is assumed to be two‑body, both |supp(P)| and |supp(Q)| are ≤ 2, which imposes a strict bound on the minimal support size δ(G) = min_{g≠I}|supp(g)| of the stabilizer group. MITE for k‑body observables requires δ(G) > k, but the two‑body factorization forces δ(G) ≤ ⌈(k+1)/2⌉, leading to a contradiction for k ≥ 4.

To illustrate the tightness, the authors present the Hamiltonian

  Ĥ = J ∑_{i∈