# Determining a local Hamiltonian from a single eigenstate

Xiao-Liang Qi1,2 and Daniel Ranard1

1Stanford Institute for Theoretical Physics, Stanford University, Stanford CA 94305 USA
2Institute for Advanced Study, Princeton NJ 08540 USA

### Abstract

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 $\textit{k}$-correlation spectrum" of a state, which reveals properties of local correlations in the state and may be of independent interest.

### ► References

[1] J. R. Garrison and T. Grover, Does a single eigenstate encode the full hamiltonian d?,'' Physical Review X, vol. 8, no. 2, p. 021026, 2018. https:/​/​doi.org/​10.1103/​PhysRevX.8.021026.
https:/​/​doi.org/​10.1103/​PhysRevX.8.021026

[2] J. M. Deutsch, Quantum statistical mechanics in a closed system,'' Physical Review A, vol. 43, no. 4, p. 2046, 1991. https:/​/​doi.org/​10.1103/​PhysRevA.43.2046.
https:/​/​doi.org/​10.1103/​PhysRevA.43.2046

[3] M. Srednicki, Chaos and quantum thermalization,'' Physical Review E, vol. 50, no. 2, p. 888, 1994. https:/​/​doi.org/​10.1103/​PhysRevE.50.888.
https:/​/​doi.org/​10.1103/​PhysRevE.50.888

[4] H. L. Haselgrove, M. A. Nielsen, and T. J. Osborne, Quantum states far from the energy eigenstates of any local hamiltonian,'' Physical review letters, vol. 91, no. 21, p. 210401, 2003. https:/​/​doi.org/​10.1103/​PhysRevLett.91.210401.
https:/​/​doi.org/​10.1103/​PhysRevLett.91.210401

[5] J. Chen, Z. Ji, Z. Wei, and B. Zeng, Correlations in excited states of local hamiltonians,'' Phys. Rev. A, vol. 85, p. 040303, Apr 2012. https:/​/​doi.org/​10.1103/​PhysRevA.85.040303.
https:/​/​doi.org/​10.1103/​PhysRevA.85.040303

[6] M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y.-K. Liu, Efficient quantum state tomography,'' Nature communications, vol. 1, p. 149, 2010. https:/​/​doi.org/​10.1038/​ncomms1147.
https:/​/​doi.org/​10.1038/​ncomms1147

[7] B. Swingle and I. H. Kim, Reconstructing quantum states from local data,'' Physical review letters, vol. 113, no. 26, p. 260501, 2014. https:/​/​doi.org/​10.1103/​PhysRevLett.113.260501.
https:/​/​doi.org/​10.1103/​PhysRevLett.113.260501

[8] V. Zauner, D. Draxler, L. Vanderstraeten, J. Haegeman, and F. Verstraete, Symmetry breaking and the geometry of reduced density matrices,'' New Journal of Physics, vol. 18, no. 11, p. 113033, 2016. https:/​/​doi.org/​10.1088/​1367-2630/​18/​11/​113033.
https:/​/​doi.org/​10.1088/​1367-2630/​18/​11/​113033

[9] H. Kim, M. C. Bañuls, J. I. Cirac, M. B. Hastings, and D. A. Huse, Slowest local operators in quantum spin chains,'' Physical Review E, vol. 92, no. 1, p. 012128, 2015. https:/​/​doi.org/​10.1103/​PhysRevE.92.012128.
https:/​/​doi.org/​10.1103/​PhysRevE.92.012128

[10] T. O'Brien, D. A. Abanin, G. Vidal, and Z. Papić, Explicit construction of local conserved operators in disordered many-body systems,'' Physical Review B, vol. 94, no. 14, p. 144208, 2016. https:/​/​doi.org/​10.1103/​PhysRevB.94.144208.
https:/​/​doi.org/​10.1103/​PhysRevB.94.144208

[11] P. Zanardi, P. Giorda, and M. Cozzini, Information-theoretic differential geometry of quantum phase transitions,'' Physical review letters, vol. 99, no. 10, p. 100603, 2007. https:/​/​doi.org/​10.1103/​PhysRevLett.99.100603.
https:/​/​doi.org/​10.1103/​PhysRevLett.99.100603

[12] C. Davis and W. M. Kahan, The rotation of eigenvectors by a perturbation. iii,'' SIAM Journal on Numerical Analysis, vol. 7, no. 1, pp. 1–46, 1970. https:/​/​doi.org/​10.1137/​0707001.
https:/​/​doi.org/​10.1137/​0707001

[13] A. Molnar, N. Schuch, F. Verstraete, and J. I. Cirac, Approximating gibbs states of local hamiltonians efficiently with projected entangled pair states,'' Physical review b, vol. 91, no. 4, p. 045138, 2015. https:/​/​doi.org/​10.1103/​PhysRevB.91.045138.
https:/​/​doi.org/​10.1103/​PhysRevB.91.045138

[14] J. Cho, Correlations in quantum spin systems from the boundary effect,'' New Journal of Physics, vol. 17, no. 5, p. 053021, 2015. https:/​/​doi.org/​10.1088/​1367-2630/​17/​5/​053021.
https:/​/​doi.org/​10.1088/​1367-2630/​17/​5/​053021

### Cited by

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[2] Maxime Dupont, Nicolas Macé, and Nicolas Laflorencie, "From eigenstate to Hamiltonian: Prospects for ergodicity and localization", Physical Review B 100 13, 134201 (2019).

[3] Eli Chertkov and Bryan K. Clark, "Computational Inverse Method for Constructing Spaces of Quantum Models from Wave Functions", Physical Review X 8 3, 031029 (2018).

[4] W. Zhu, Zhoushen Huang, and Yin-Chen He, "Reconstructing entanglement Hamiltonian via entanglement eigenstates", Physical Review B 99 23, 235109 (2019).

[5] Enrico M. Brehm, Diptarka Das, and Shouvik Datta, "Probing thermality beyond the diagonal", Physical Review D 98 12, 126015 (2018).

[6] Eyal Bairey, Itai Arad, and Netanel H. Lindner, "Learning a local Hamiltonian from local measurements", arXiv:1807.04564, Physical Review Letters 122 2, 020504 (2018).

[7] X. Turkeshi, T. Mendes-Santos, G. Giudici, and M. Dalmonte, "Entanglement-Guided Search for Parent Hamiltonians", Physical Review Letters 122 15, 150606 (2019).

[8] Shi-Yao Hou, Ningping Cao, Sirui Lu, Yi Shen, Yiu-Tung Poon, and Bei Zeng, "Determining system Hamiltonian from eigenstate measurements without correlation functions", arXiv:1903.06569.

[9] Martin Greiter, Vera Schnells, and Ronny Thomale, "Method to identify parent Hamiltonians for trial states", Physical Review B 98 8, 081113 (2018).

[10] Maxime Dupont and Nicolas Laflorencie, "Many-body localization as a large family of localized ground states", arXiv:1807.01313, Physical Review B 99 2, 020202 (2018).

[11] Keith R. Fratus and Syrian V. Truong, "Does a Single Eigenstate of a Hamiltonian Encode the Critical Behaviour of its Finite-Temperature Phase Transition?", arXiv:1810.11092.

[12] Xu-Yang Hou, Ziwen Huang, Hao Guo, Yan He, and Chih-Chun Chien, "BCS thermal vacuum of fermionic superfluids and its perturbation theory", Scientific Reports 8, 11995 (2018).

[13] Eyal Bairey, Chu Guo, Dario Poletti, Netanel H. Lindner, and Itai Arad, "Learning the dynamics of open quantum systems from local measurements", arXiv:1907.11154.

[14] G. J. Sreejith, M. Fremling, Gun Sang Jeon, and J. K. Jain, "Search for exact local Hamiltonians for general fractional quantum Hall states", Physical Review B 98 23, 235139 (2018).

[15] Kiryl Pakrouski, "Automatic design of Hamiltonians", arXiv:1907.05898.

[16] Vinit Kumar Singh and Jung Hoon Han, "Statistical recovery of the classical spin Hamiltonian", arXiv:1807.04884.

[17] W. Zhu, Zhoushen Huang, Yin-Chen He, and Xueda Wen, "Entanglement Hamiltonian of Many-body Dynamics in Strongly-correlated Systems", arXiv:1909.08808.

The above citations are from Crossref's cited-by service (last updated successfully 2020-02-16 14:53:48) and SAO/NASA ADS (last updated successfully 2020-02-16 14:53:49). The list may be incomplete as not all publishers provide suitable and complete citation data.