Entanglement is the crucial ingredient of quantum many-body physics, and characterizing and quantifying entanglement in the closed-system dynamics of quantum simulators remains a challenge in today’s era of intermediate-scale quantum devices. Here we discuss an efficient tomographic protocol for reconstructing reduced density matrices and entanglement spectra for spin systems. The key step is a parametrization of the reduced density matrix in terms of an entanglement Hamiltonian involving only quasilocal few-body terms. This ansatz is fitted to, and can be independently verified from, a small number of randomized measurements. By analysing data from trapped-ion quantum simulators for quench dynamics of a one-dimensional long-range Ising model, we demonstrate the ability of the protocol to measure the time evolution of the entanglement spectrum, in agreement with theoretical expectations. Furthermore, we develop the protocol as a testbed for predictions of entanglement structure in quantum field theories, which we illustrate for conformal field theory in quench dynamics, as well as the Bisognano–Wichmann theorem for ground states. In theoretical simulations, we demonstrate favourable scaling of sampling efficiency with subsystem size. Although the post-processing might ultimately be exponential, our protocol addresses the bottleneck of exponential sampling complexity in the investigation of entanglement structure in quantum simulation, and brings subsystems of tens of spins into reach for present experiments
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All data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
Georgescu, I. M., Ashhab, S. & Nori, F. Quantum simulation. Rev. Mod. Phys. 86, 153–185 (2014).
Browaeys, A. & Lahaye, T. Many-body physics with individually controlled Rydberg atoms. Nat. Phys. 16, 132–142 (2020).
Monroe, C. et al. Programmable quantum simulations of spin systems with trapped ions. Rev. Mod. Phys. 93, 025001 (2021).
Brydges, T. et al. Probing Rényi entanglement entropy via randomized measurements. Science 364, 260–263 (2019).
Kokail, C. et al. Self-verifying variational quantum simulation of lattice models. Nature 569, 355–360 (2019).
Wilkinson, S. A. & Hartmann, M. J. Superconducting quantum many-body circuits for quantum simulation and computing. Appl. Phys. Lett. 116, 230501 (2020).
King, A. D. et al. Observation of topological phenomena in a programmable lattice of 1,800 qubits. Nature 560, 456–460 (2018).
Cirac, J. I. & Zoller, P. Goals and opportunities in quantum simulation. Nat. Phys. 8, 264–266 (2012).
Amico, L., Fazio, R., Osterloh, A. & Vedral, V. Entanglement in many-body systems. Rev. Mod. Phys. 80, 517–576 (2008).
Zeng, B., Chen, X., Zhou, D.-L. & Wen, X.-G. Quantum Information Meets Quantum Matter (Springer, 2019)
Regnault, N. in Topological Aspects of Condensed Matter Physics (eds Chamon, C. et al.) 165–210 (Oxford Univ. Press, 2017).
Dalmonte, M., Vermersch, B. & Zoller, P. Quantum simulation and spectroscopy of entanglement Hamiltonians. Nat. Phys. 14, 827–831 (2018).
Zhu, W., Huang, Z., He, Y.-C. & Wen, X. Entanglement Hamiltonian of many-body dynamics in strongly correlated systems. Phys. Rev. Lett. 124, 100605 (2020).
Chang, P.-Y., Chen, X., Gopalakrishnan, S. & Pixley, J. H. Evolution of entanglement spectra under generic quantum dynamics. Phys. Rev. Lett. 123, 190602 (2019).
Calabrese, P. & Cardy, J. Quantum quenches in 1 + 1 dimensional conformal field theories. J. Stat. Mech. Theor. Exp. 2016, 064003 (2016).
Wen, X., Ryu, S. & Ludwig, A. W. Entanglement Hamiltonian evolution during thermalization in conformal field theory. J. Stat. Mech. Theor. Exp. 2018, 113103 (2018).
Orús, R. Tensor networks for complex quantum systems. Nat. Rev. Phys. 1, 538–550 (2019).
Acharya, A., Kypraios, T. & Guţă, M. A comparative study of estimation methods in quantum tomography. J. Phys. A 52, 234001 (2019).
Gross, D., Liu, Y.-K., Flammia, S. T., Becker, S. & Eisert, J. Quantum state tomography via compressed sensing. Phys. Rev. Lett. 105, 150401 (2010).
Cramer, M. et al. Efficient quantum state tomography. Nat. Commun. 1, 149 (2010).
Smolin, J. A., Gambetta, J. M. & Smith, G. Efficient method for computing the maximum-likelihood quantum state from measurements with additive Gaussian noise. Phys. Rev. Lett. 108, 070502 (2012).
Torlai, G. et al. Neural-network quantum state tomography. Nat. Phys. 14, 447–450 (2018).
Choo, K., von Keyserlingk, C. W., Regnault, N. & Neupert, T. Measurement of the entanglement spectrum of a symmetry-protected topological state using the IBM quantum computer. Phys. Rev. Lett. 121, 086808 (2018).
Haah, J., Harrow, A. W., Ji, Z., Wu, X. & Yu, N. Sample-optimal tomography of quantum states. IEEE Trans. Inf. Theory 63, 5628–5641 (2017).
O’Donnell, R. & Wright, J. Efficient quantum tomography. In Proc. Forty-eighth Annual ACM Symposium on Theory of Computing, STOC ’16, 899–912 (ACM, 2016); https://doi.org/10.1145/2897518.2897544
Brando, F. G. S. L., Kueng, R. & Frana, D. S. Fast and robust quantum state tomography from few basis measurements. Preprint at https://arxiv.org/pdf/2009.08216.pdf (2020).
Anshu, A., Arunachalam, S., Kuwahara, T. & Soleimanifar, M. Sample-efficient learning of quantum many-body systems. Preprint at https://arxiv.org/pdf/2004.07266.pdf (2020).
Elben, A. et al. Cross-platform verification of intermediate scale quantum devices. Phys. Rev. Lett. 124, 010504 (2020).
Liang, Y.-C. et al. Quantum fidelity measures for mixed states. Rep. Prog. Phys. 82, 076001 (2019).
Bisognano, J. J. & Wichmann, E. H. On the duality condition for a Hermitian scalar field. J. Math. Phys. 16, 985–1007 (1975).
Bisognano, J. J. & Wichmann, E. H. On the duality condition for quantum fields. J. Math. Phys. 17, 303–321 (1976).
Calabrese, P. & Cardy, J. Entanglement entropy and conformal field theory. J. Phys. A 42, 504005 (2009).
Swingle, B. & Senthil, T. Geometric proof of the equality between entanglement and edge spectra. Phys. Rev. B 86, 045117 (2012).
Alba, V., Haque, M. & Läuchli, A. M. Boundary-locality and perturbative structure of entanglement spectra in gapped systems. Phys. Rev. Lett. 108, 227201 (2012).
Eisler, V. & Peschel, I. Analytical results for the entanglement Hamiltonian of a free-fermion chain. J. Phys. A 50, 284003 (2017).
Itoyama, H. & Thacker, H. B. Lattice Virasoro algebra and corner transfer matrices in the Baxter eight-vertex model. Phys. Rev. Lett. 58, 1395–1398 (1987).
Giudici, G., Mendes-Santos, T., Calabrese, P. & Dalmonte, M. Entanglement Hamiltonians of lattice models via the Bisognano–Wichmann theorem. Phys. Rev. B 98, 134403 (2018).
Hislop, P. D. & Longo, R. Modular structure of the local algebras associated with the free massless scalar field theory. Commun. Math. Phys. 84, 71–85 (1982).
Cardy, J. & Tonni, E. Entanglement Hamiltonians in two-dimensional conformal field theory. J. Stat. Mech. Theor. Exp. 2016, 123103 (2016).
Elben, A. et al. Mixed-state entanglement from local randomized measurements. Phys. Rev. Lett. 125, 200501 (2020).
Deutsch, J. M. Quantum statistical mechanics in a closed system. Phys. Rev. A 43, 2046–2049 (1991).
Srednicki, M. Chaos and quantum thermalization. Phys. Rev. E 50, 888–901 (1994).
Rigol, M., Dunjko, V. & Olshanii, M. Thermalization and its mechanism for generic isolated quantum systems. Nature 452, 854–858 (2008).
Garrison, J. R. & Grover, T. Does a single eigenstate encode the full Hamiltonian. Phys. Rev. X 8, 021026 (2018).
Guta, M., Kahn, J., Kueng, R. & Tropp, J. A. Fast state tomography with optimal error bounds. J. Phys. A 53, 204001 (2020).
Huang, H.-Y., Kueng, R. & Preskill, J. Predicting many properties of a quantum system from very few measurements. Nat. Phys. 16, 1050–1057 (2020).
Riofrío, C. A. et al. Experimental quantum compressed sensing for a seven-qubit system. Nat. Commun. 8, 15305 (2017).
Flammia, S. T. & Liu, Y.-K. Direct fidelity estimation from few Pauli measurements. Phys. Rev. Lett. 106, 230501 (2011).
da Silva, M. P., Landon-Cardinal, O. & Poulin, D. Practical characterization of quantum devices without tomography. Phys. Rev. Lett. 107, 210404 (2011).
Arute, F. et al. Quantum supremacy using a programmable superconducting processor. Nature 574, 505–510 (2019).
Dankert, C., Cleve, R., Emerson, J. & Livine, E. Exact and approximate unitary 2-designs and their application to fidelity estimation. Phys. Rev. A 80, 012304 (2009).
Qi, X.-L. & Ranard, D. Determining a local Hamiltonian from a single eigenstate. Quantum 3, 159 (2019).
Bairey, E., Arad, I. & Lindner, N. H. Learning a local Hamiltonian from local measurements. Phys. Rev. Lett. 122, 020504 (2019).
Wang, J. et al. Experimental quantum Hamiltonian learning. Nat. Phys. 13, 551–555 (2017).
Elben, A., Vermersch, B., Dalmonte, M., Cirac, J. I. & Zoller, P. Rényi entropies from random quenches in atomic Hubbard and spin models. Phys. Rev. Lett. 120, 050406 (2018).
LaRose, R., Tikku, A., O’Neel-Judy, É., Cincio, L. & Coles, P. J. Variational quantum state diagonalization. npj Quantum Inf. 5, 57 (2019).
Bravo-Prieto, C., García-Martín, D. & Latorre, J. I. Quantum singular value decomposer. Phys. Rev. A 101, 062310 (2020).
Sugiyama, T., Turner, P. S. & Murao, M. Precision-guaranteed quantum tomography. Phys. Rev. Lett. 111, 160406 (2013).
Elben, A., Vermersch, B., Roos, C. F. & Zoller, P. Statistical correlations between locally randomized measurements: a toolbox for probing entanglement in many-body quantum states. Phys. Rev. A 99, 052323 (2019).
Fishman, M., White, S. R. & Stoudenmire, E. M. The ITensor software library for tensor network calculations. Preprint at https://arxiv.org/pdf/2007.14822.pdf (2020).
We thank P. Calabrese, M. Dalmonte, G. Giudici, L.K. Joshi, B. Kraus, R. Kueng, C. Roos, L. Sieberer, J. Yu and W. Zhu for discussions, and members of the Innsbruck trapped-ion group for generously sharing the experimental data of ref. 4. Work at Innsbruck is supported by the European Union programme Horizon 2020 under grants 817482 (PASQuanS) and 731473 (FWF QuantERA via QTFLAG I03769), the US Air Force Office of Scientific Research (AFOSR) via IOE grant FA9550-19-1-7044 LASCEM and by the Simons Collaboration on Ultra-Quantum Matter, which is a grant from the Simons Foundation (651440, P.Z.). B.V. acknowledges funding from the Austrian Science Foundation (FWF, P 32597 N), and the French National Research Agency (ANR-20-CE47-0005 JCJC QRand). The computational results presented here have been achieved (in part) using the LEO HPC infrastructure of the University of Innsbruck. Numerical calculations were performed (in part) using the ITensor library60.
The authors declare no competing interests.
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Kokail, C., van Bijnen, R., Elben, A. et al. Entanglement Hamiltonian tomography in quantum simulation. Nat. Phys. 17, 936–942 (2021). https://doi.org/10.1038/s41567-021-01260-w