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Stabilizer Thermal Eigenstates at Infinite Temperature

Akihiro Hokkyo

TL;DR

This work tackles the analytic construction of thermal-like energy eigenstates in nonintegrable many-body systems by employing stabilizer and graph-state formalisms. The authors prove a sharp no-go theorem: stabilizer eigenstates with zero energy of $2$-body Hamiltonians cannot achieve $k$-body microscopic thermal equilibrium for any $k \ge 4$, and they show this bound is tight via explicit two-body Hamiltonians that realize $3$-body MITE (and $O(N)$-local MITE). They connect the result to a structural condition linking stabilizer states to zero-energy eigenstates, revealing a fundamental constraint from few-body interactions. Furthermore, they present explicit constructions, including a star-shaped cluster-state example, to illustrate the range and limitations of stabilizer-based thermal eigenstates. The findings illuminate the interplay between interaction locality, entanglement structure, and the emergence of thermal behavior in quantum many-body systems.

Abstract

Understanding how to analyze highly entangled thermal eigenstates is a central challenge in the study of quantum many-body systems. In this Letter, we introduce a stabilizer-based approach to construct analytically tractable energy eigenstates of nonintegrable many-body Hamiltonians. Focusing on zero-energy eigenstates at infinite temperature, we prove a sharp no-go theorem: stabilizer eigenstates of two-body Hamiltonians cannot satisfy $k$-body microscopic thermal equilibrium for any $k\ge4$. We further show that this bound is tight by explicitly constructing two-body nonintegrable Hamiltonians whose stabilizer eigenstates reproduce thermal expectation values for all two-body and all three-body observables. Finally, we identify the structural origin of this limitation by characterizing the conditions under which a stabilizer state can appear as a zero-energy eigenstate of a Hamiltonian, thereby revealing a fundamental constraint imposed by the few-body nature of interactions.

Stabilizer Thermal Eigenstates at Infinite Temperature

TL;DR

This work tackles the analytic construction of thermal-like energy eigenstates in nonintegrable many-body systems by employing stabilizer and graph-state formalisms. The authors prove a sharp no-go theorem: stabilizer eigenstates with zero energy of -body Hamiltonians cannot achieve -body microscopic thermal equilibrium for any , and they show this bound is tight via explicit two-body Hamiltonians that realize -body MITE (and -local MITE). They connect the result to a structural condition linking stabilizer states to zero-energy eigenstates, revealing a fundamental constraint from few-body interactions. Furthermore, they present explicit constructions, including a star-shaped cluster-state example, to illustrate the range and limitations of stabilizer-based thermal eigenstates. The findings illuminate the interplay between interaction locality, entanglement structure, and the emergence of thermal behavior in quantum many-body systems.

Abstract

Understanding how to analyze highly entangled thermal eigenstates is a central challenge in the study of quantum many-body systems. In this Letter, we introduce a stabilizer-based approach to construct analytically tractable energy eigenstates of nonintegrable many-body Hamiltonians. Focusing on zero-energy eigenstates at infinite temperature, we prove a sharp no-go theorem: stabilizer eigenstates of two-body Hamiltonians cannot satisfy -body microscopic thermal equilibrium for any . We further show that this bound is tight by explicitly constructing two-body nonintegrable Hamiltonians whose stabilizer eigenstates reproduce thermal expectation values for all two-body and all three-body observables. Finally, we identify the structural origin of this limitation by characterizing the conditions under which a stabilizer state can appear as a zero-energy eigenstate of a Hamiltonian, thereby revealing a fundamental constraint imposed by the few-body nature of interactions.
Paper Structure (10 sections, 3 theorems, 26 equations, 2 figures)

This paper contains 10 sections, 3 theorems, 26 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Local and few-body microscopic thermal equilibrium
  3. Stabilizer eigenstates of local Hamiltonians.
  4. Main results
  5. Explicit construction saturating the bound.
  6. Origin of the no-go constraint
  7. Another example: star-shaped cluster states
  8. Conclusion and outlook
  9. Acknowledgments
  10. Note added

Key Result

Theorem 1

Any stabilizer eigenstate with zero energy of a $2$-body nonzero Hamiltonian is not in $k$-body MITE for any $k\ge 4$.

Figures (2)

  • Figure 1: Schematic illustration of the graph $\mathcal{G}_1$ (Eq. \ref{['eq:edge_graph']}) for $N=12$. Each vertex $i$ is connected to three vertices $i+N/2-1$, $i+N/2$, and $i+N/2+1$. The corresponding graph state is in $3$-body MITE and also in $5$-local MITE.
  • Figure 2: Schematic illustration of the graph $\mathcal{G}_2$ (Eq. \ref{['eq:edge_graph_cluster']}) for $N=9$. Each vertex is connected to the vertex at distance $(N-1)/2=4$. The corresponding graph state is in $2$-body MITE and also in $4$-local MITE.

Theorems & Definitions (5)

  • Theorem 1: No-go
  • Theorem 2: Achievability and tightness
  • Proposition 3
  • proof
  • proof : Proof of Theorem \ref{['thm:no-go']}