Source-linked AI summary
Determining a local Hamiltonian from a single eigenstate
Xiao-Liang Qi, Daniel Ranard
TL;DR
The paper asks whether one eigenstate suffices to determine a local Hamiltonian, develops a correlation-matrix reconstruction based on local two-point correlations, and finds generic exact recovery for finite-size systems. It also studies sensitivity, restricted-region data, and correlation-spectrum structure, while warning that infinite-size higher-dimensional extrapolation is more complicated.
Problem
The paper asks whether knowledge of a single ground or excited eigenstate, or its local correlation functions, is sufficient to uniquely determine a local Hamiltonian.
Method
The paper constructs the k-correlation matrix from connected correlations of range-k local observables and uses its kernel and spectrum to analyze Hamiltonian reconstruction.
Results
For finite-size systems, a generic local Hamiltonian may be recovered from a single eigenstate using only two-point correlations, with polynomial-time reconstruction.
Takeaways & Limitations
The k-correlation spectrum indicates whether reconstruction is unique, measures sensitivity to error, and reveals additional local-correlation structure such as momentum bands.
Takeaways & Limitations
Infinite-size extrapolation is more complicated, especially in dimensions d ≥2 where topological phases can make relevant Hamiltonian subsets non-dense or disconnected.
Abstract
from arXiv · showhide
We ask whether the knowledge of a single eigenstate of a local Hamiltonian is sufficient to uniquely determine the Hamiltonian. We present evidence that the answer is "yes" for generic local Hamiltonians, given either the ground state or an excited eigenstate. In fact, knowing only the two-point equal-time correlation functions of local observables with respect to the eigenstate should generically be sufficient to exactly recover the Hamiltonian for finite-size systems, with numerical algorithms that run in a time that is polynomial in the system size. We also investigate the large-system limit, the sensitivity of the reconstruction to error, and the case when correlation functions are only known for observables on a fixed sub-region. Numerical demonstrations support the results for finite one-dimensional spin chains (though caution must be taken when extrapolating to infinite-size systems in higher dimensions). For the purpose of our analysis, we define the "$k$-correlation spectrum" of a state, which reveals properties of local correlations in the state and may be of independent interest.
1 Introduction
The paper argues that a single eigenstate generically determines a finite-size local Hamiltonian, and develops the k-correlation spectrum as both a reconstruction tool and a descriptor of local correlations.
- Problem and scope: For finite-range local lattice Hamiltonians, almost all Hamiltonians may be uniquely recovered from any single eigenstate.The argument applies to broad classes of finite-size systems and can extend to certain k-local Hamiltonians.
- Heuristic argument: A parameter-counting argument suggests generic injectivity because local-Hamiltonian dimension grows polynomially while Hilbert-space dimension grows exponentially with system size.The paper uses this heuristic before developing a constructive method.
- Generic uniqueness: Counterexamples arise from nongeneric structures such as uncoupled sites or local conserved quantities, but under an stated assumption they form a measure-zero set.These examples show why the generic qualifier is necessary.
- Constructive method: The k-correlation matrix is built from connected correlations of range-k local observables, providing a constructive route to recover the Hamiltonian from one eigenstate.Its local-operator basis uses operators supported on spatially contiguous regions of size k.
- Correlation spectrum: The correlation spectrum can be computed without a Hamiltonian, and its zero eigenspace characterizes local Hamiltonians sharing the state as an eigenstate.At least one zero indicates existence of a nonzero traceless local Hamiltonian; exactly one zero gives uniqueness up to scaling.
- Evidence and practical scope: Numerical studies on one-dimensional spin chains support exact recovery from any single eigenstate, while the smallest nonzero correlation-spectrum eigenvalue controls sensitivity to eigenstate error.The paper also discusses approximate recovery when observations are restricted to a sub-region.
- Additional implications: The correlation spectrum also reveals structure beyond reconstruction, including momentum-organized bands and a band gap for ground states of gapped translation-invariant systems.The paper notes possible experimental use of measured local correlation functions to approximate the Hamiltonian.
2 Reconstructing the Hamiltonian using the correlation matrix
The paper reconstructs local Hamiltonians from a single eigenstate using the correlation matrix of local observables. Its correlation spectrum characterizes uniqueness, supports generic recovery in finite spin chains, and exposes sensitivity and large-system limitations.
- Correlation matrix: For a range-k local Hamiltonian, the correlation matrix is built from expectations of products of local-operator basis elements in the state.Its eigenvalue spectrum is basis-independent and can be computed directly from the state.
- Uniqueness criterion: A single zero in the correlation spectrum implies that the local Hamiltonian is unique up to overall scaling, whereas multiple zeros imply non-uniqueness.The null space consists of local Hamiltonians for which the state is an eigenstate.
- Numerical demonstrations: For 12-qubit random spin chains, every tested eigenstate produced a single zero and reconstructed the Hamiltonian to error about θ ≈10−10.The tests covered both disordered and translation-invariant cases.
- Generic recovery: Under the assumption that one Hamiltonian admits unique reconstruction, analyticity implies that non-reconstructable Hamiltonians form a measure-zero set.Finite-size numerical examples provide the required recoverable instances, including chains up to n = 12 qubits.
- Sensitivity and limits: In large systems, eigenvalues arbitrarily close to zero make reconstruction highly sensitive to numerical or eigenstate error, although translation invariance can permit restricting to q = 0.For translation-invariant systems, small lowest-band eigenvalues need not imply an error-prone reconstruction when only the q = 0 correlation matrix is used.
- Ground and excited states: The correlation spectrum differs between representative ground and excited states: ground states show a typical band gap, while mid-spectrum excited states show no such gap.The excited-state example has a single zero and accurately reconstructs the Hamiltonian; its lowest-band discontinuity implies non-exponentially decaying energy correlations.
3 Approximate reconstruction
Reconstruction error is controlled by the smallest nonzero correlation-spectrum eigenvalue, while restricted-region data can support approximate recovery under specific conditions. Large-system behavior remains established mainly for translation-invariant settings, with important unresolved cases.
- Error sensitivity: The reconstructed Hamiltonian’s error is inversely proportional to the smallest nonzero correlation-spectrum eigenvalue λ2.This sensitivity follows from perturbing the correlation matrix and applies when the eigenstate permits unique reconstruction.
- Large-system behavior: For translation-invariant systems, the relevant sensitivity is governed by the smallest nonzero eigenvalue in the zero-momentum sector, λq=0_2.If this eigenvalue approaches a nonzero limiting value, reconstruction sensitivity remains bounded in the large-system limit.
- Restricted-region data: Restricted-region correlations can recover the Hamiltonian locally or approximately when correlations decay sufficiently with subsystem size.For disordered systems the target is HA, whereas translation invariance can permit recovery of the full H; the method uses the reduced-state correlation matrix and its kernel.
- Restricted-region data: At nonzero energy density, the eigenstate thermalization hypothesis makes the reduced state approximately thermal, allowing approximate recovery of HA when m ≪ n.This does not imply that the two-point correlation matrix alone has a kernel suitable for the full-system eigenstate reconstruction method.
- Large-system behavior: For non-translation-invariant systems, recovery in the large-system limit remains unclear, and higher-dimensional translation-invariant cases lack rigorous understanding of generic ground states.These gaps limit extrapolation from finite systems and leave the behavior of correlation-spectrum zeros across phases unresolved.
4 Conclusion and discussion
The paper concludes that generic finite-size local Hamiltonians can be reconstructed from a single eigenstate using local two-point correlations and a correlation-matrix kernel. It also presents the correlation spectrum as a broader, observable-independent descriptor of local correlations and entanglement.
- Conclusion: For finite-size lattice systems, a generic local Hamiltonian can be recovered from a single eigenstate in polynomial time using two-point correlations of range-k observables.The kernel of the correlation matrix yields the reconstructed Hamiltonian, while the correlation spectrum diagnoses uniqueness and error sensitivity.
- Correlation spectrum: The correlation spectrum packages canonical information independent of any particular observable and provides more numerical data than a single quantity such as entanglement entropy.Its k-dependent spectra and eigen-operators may offer an independent framework for analyzing state correlations and entanglement.