At the Limits of Criticality-Based Quantum Metrology: Apparent Super-Heisenberg Scaling Revisited - Publication - Bridge of Knowledge

Search

At the Limits of Criticality-Based Quantum Metrology: Apparent Super-Heisenberg Scaling Revisited

Abstract

We address the question of whether the super-Heisenberg scaling for quantum estimation is indeed realizable. We unify the results of two approaches. In the first one, the original system is compared with its copy rotated by the parameter-dependent dynamics. If the parameter is coupled to the one-body part of the Hamiltonian, the precision of its estimation is known to scale at most as N−1 (Heisenberg scaling) in terms of the number of elementary subsystems used N. The second approach compares the overlap between the ground states of the parameter-dependent Hamiltonian in critical systems, often leading to an apparent super-Heisenberg scaling. However, we point out that if one takes into account the scaling of time needed to perform the necessary operations, i.e., ensuring adiabaticity of the evolution, the Heisenberg limit given by the rotation scenario is recovered. We illustrate the general theory on a ferromagnetic Heisenberg spin chain example and show that it exhibits such super-Heisenberg scaling of ground-state fidelity around the critical value of the parameter (magnetic field) governing the one-body part of the Hamiltonian. Even an elementary estimator represented by a single-site magnetization already outperforms the Heisenberg behavior providing the N−1.5 scaling. In this case, Fisher information sets the ultimate scaling as N−1.75, which can be saturated by measuring magnetization on all sites simultaneously. We discuss universal scaling predictions of the estimation precision offered by such observables, both at zero and finite temperatures, and support them with numerical simulations in the model. We provide an experimental proposal of realization of the considered model via mapping the system to ultracold bosons in a periodically shaken optical lattice. We explicitly derive that the Heisenberg limit is recovered when the time needed for preparation of quantum states involved is taken into account.

Citations

  • 6 0

    CrossRef

  • 0

    Web of Science

  • 7 6

    Scopus

Authors (5)

Cite as

Full text

download paper
downloaded 49 times
Publication version
Accepted or Published Version
License
Creative Commons: CC-BY open in new tab

Keywords

Details

Category:
Articles
Type:
artykuł w czasopiśmie wyróżnionym w JCR
Published in:
Physical Review X no. 8, edition 2, pages 1 - 16,
ISSN: 2160-3308
Language:
English
Publication year:
2018
Bibliographic description:
Rams M., Sierant P., Dutta O., Horodecki P., Zakrzewski J.: At the Limits of Criticality-Based Quantum Metrology: Apparent Super-Heisenberg Scaling Revisited// Physical Review X. -Vol. 8, iss. 2 (2018), s.1-16
DOI:
Digital Object Identifier (open in new tab) 10.1103/physrevx.8.021022
Bibliography: test
  1. C. M. Caves, Quantum-Mechanical Noise in an Interfer- ometer, Phys. Rev. D 23, 1693 (1981). open in new tab
  2. D. J. Wineland, J. J. Bollinger, W. M. Itano, F. L. Moore, and D. J. Heinzen, Spin Squeezing and Reduced Quantum Noise in Spectroscopy, Phys. Rev. A 46, R6797 (1992). open in new tab
  3. S. L. Braunstein and C. M. Caves, Statistical Distance and the Geometry of Quantum States, Phys. Rev. Lett. 72, 3439 (1994). open in new tab
  4. V. Giovannetti, S. Lloyd, and L. Maccone, Quantum- Enhanced Measurements: Beating the Standard Quantum Limit, Science 306, 1330 (2004). open in new tab
  5. A. Uhlmann, The "Transition Probability" in the State Space of a *-Algebra, Rep. Math. Phys. 9, 273 (1976). open in new tab
  6. M. M. Taddei, B. M. Escher, L. Davidovich, and R. L. de Matos Filho, Quantum Speed Limit for Physical Processes, Phys. Rev. Lett. 110, 050402 (2013). open in new tab
  7. V. Giovannetti, S. Lloyd, and L. Maccone, Advances in Quantum Metrology, Nat. Photonics 5, 222 (2011). open in new tab
  8. G. Tóth and I. Apellaniz, Quantum Metrology from a Quantum Information Science Perspective, J. Phys. A 47, 424006 (2014). open in new tab
  9. R. Demkowicz-Dobrzanski, M. Jarzyna, and J. Kolodynski, Quantum Limits in Optical Interferometry, Prog. Opt. 60, 345 (2015). open in new tab
  10. S. M. Roy and S. L. Braunstein, Exponentially Enhanced Quantum Metrology, Phys. Rev. Lett. 100, 220501 (2008). open in new tab
  11. S. Boixo, S. T. Flammia, C. M. Caves, and J. M. Geremia, Generalized Limits for Single-Parameter Quantum Estimation, Phys. Rev. Lett. 98, 090401 (2007). open in new tab
  12. V. Giovannetti, S. Lloyd, and L. Maccone, Quantum Metrology, Phys. Rev. Lett. 96, 010401 (2006). open in new tab
  13. A. De Pasquale, D. Rossini, P. Facchi, and V. Giovannetti, Quantum Parameter Estimation Affected by Unitary Disturbance, Phys. Rev. A 88, 052117 (2013). open in new tab
  14. M. Skotiniotis, P. Sekatski, and W. Dür, Quantum Metrol- ogy for the Ising Hamiltonian with Transverse Magnetic Field, New J. Phys. 17, 073032 (2015). open in new tab
  15. S. Pang and T. A. Brun, Quantum Metrology for a General Hamiltonian Parameter, Phys. Rev. A 90, 022117 (2014). open in new tab
  16. J. M. E. Fraisse and D. Braun, Hamiltonian Extensions in Quantum Metrology, Quantum Meas. Quantum Metrol. 4, 8 (2017). open in new tab
  17. R. Demkowicz-Dobrzanski, J. Kołodyński, and M. Guta, The Elusive Heisenberg Limit in Quantum Enhanced Metrology, Nat. Commun. 3, 1063 (2012). open in new tab
  18. S. Alipour, M. Mehboudi, and A. T. Rezakhani, Quantum Metrology in Open Systems: Dissipative Cramér-Rao Bound, Phys. Rev. Lett. 112, 120405 (2014). open in new tab
  19. S. Alipour and A. T. Rezakhani, Extended Convexity of Quantum Fisher Information in Quantum Metrology, Phys. Rev. A 91, 042104 (2015). open in new tab
  20. A. Chenu, M. Beau, J. Cao, and A. del Campo, Quantum Simulation of Generic Many-Body Open System Dynamics Using Classical Noise, Phys. Rev. Lett. 118, 140403 (2017). open in new tab
  21. M. Beau and A. del Campo, Nonlinear Quantum Metrol- ogy of Many-Body Open Systems, Phys. Rev. Lett. 119, 010403 (2017). open in new tab
  22. R. Demkowicz-Dobrzański, J. Czajkowski, and P. Sekatski, Adaptive Quantum Metrology under General Markovian Noise, Phys. Rev. X 7, 041009 (2017). open in new tab
  23. P. Hyllus, W. Laskowski, R. Krischek, C. Schwemmer, W. Wieczorek, H. Weinfurter, L. Pezzé, and A. Smerzi, Fisher Information and Multiparticle Entanglement, Phys. Rev. A 85, 022321 (2012). open in new tab
  24. G. Tóth, Multipartite Entanglement and High-Precision Metrology, Phys. Rev. A 85, 022322 (2012). open in new tab
  25. R. Augusiak, J. Kołodyński, A. Streltsov, M. N. Bera, A. Acín, and M. Lewenstein, Asymptotic Role of Entan- glement in Quantum Metrology, Phys. Rev. A 94, 012339 (2016). open in new tab
  26. Ł. Czekaj, A. Przysiężna, M. Horodecki, and P. Horodecki, Quantum Metrology: Heisenberg Limit with Bound Entanglement, Phys. Rev. A 92, 062303 (2015). open in new tab
  27. P. Hauke, M. Heyl, L. Tagliacozzo, and P. Zoller, Meas- uring Multipartite Entanglement through Dynamic Susceptibilities, Nat. Phys. 12, 778 (2016). open in new tab
  28. I. Apellaniz, M. Kleinmann, O. Gühne, and G. Tóth, Optimal Witnessing of the Quantum Fisher Information with Few Measurements, Phys. Rev. A 95, 032330 (2017). open in new tab
  29. P. T. Ernst, S. Gotze, J. S. Krauser, K. Pyka, D.-S. Luhmann, D. Pfannkuche, and K. Sengstock, Probing Superfluids in Optical Lattices by Momentum-Resolved Bragg Spectroscopy, Nat. Phys. 6, 56 (2010). open in new tab
  30. D. Clément, N. Fabbri, L. Fallani, C. Fort, and M. Inguscio, J. Low Temp. Phys. 158, 5 (2010). open in new tab
  31. M. Tsang, Quantum Transition-Edge Detectors, Phys. Rev. A 88, 021801 (2013). open in new tab
  32. T. Macrì, A. Smerzi, and L. Pezzè, Loschmidt Echo for Quantum Metrology, Phys. Rev. A 94, 010102 (2016). open in new tab
  33. U. Marzolino and T. Prosen, Quantum Metrology with Nonequilibrium Steady States of Quantum Spin Chains, Phys. Rev. A 90, 062130 (2014). open in new tab
  34. U. Marzolino and T. Prosen, Computational Complexity of Nonequilibrium Steady States of Quantum Spin Chains, Phys. Rev. A 93, 032306 (2016). open in new tab
  35. M. A. Rajabpour, Multipartite Entanglement and Quantum Fisher Information in Conformal Field Theories, Phys. Rev. D 96, 126007 (2017). open in new tab
  36. D. Braun, G. Adesso, F. Benatti, R. Floreanini, U. Marzolino, M. W. Mitchell, and S. Pirandola, Quantum Enhanced Measurements without Entanglement, arXiv:1701.05152 [Rev. Mod. Phys. (to be published)]. open in new tab
  37. P. Zanardi and N. Paunković, Ground State Overlap and Quantum Phase Transitions, Phys. Rev. E 74, 031123 (2006). open in new tab
  38. W.-L. You, Y.-W. Li, and S.-J. Gu, Fidelity, Dynamic Structure Factor, and Susceptibility in Critical Phenom- ena, Phys. Rev. E 76, 022101 (2007). open in new tab
  39. P. Zanardi, M. G. A. Paris, and L. Campos Venuti, Quan- tum Criticality as a Resource for Quantum Estimation, Phys. Rev. A 78, 042105 (2008). open in new tab
  40. C. Invernizzi, M. Korbman, L. Campos Venuti, and M. G. A. Paris, Optimal Quantum Estimation in Spin Systems at Criticality, Phys. Rev. A 78, 042106 (2008). open in new tab
  41. G. Salvatori, A. Mandarino, and M. G. A. Paris, Quantum Metrology in Lipkin-Meshkov-Glick Critical Systems, Phys. Rev. A 90, 022111 (2014). open in new tab
  42. M. Bina, I. Amelio, and M. G. A. Paris, Dicke Coupling by Feasible Local Measurements at the Superradiant Quan- tum Phase Transition, Phys. Rev. E 93, 052118 (2016). open in new tab
  43. W. L. Boyajian, M. Skotiniotis, W. Dür, and B. Kraus, Compressed Quantum Metrology for the Ising Hamilto- nian, Phys. Rev. A 94, 062326 (2016). open in new tab
  44. M. Mehboudi, L. A. Correa, and A. Sanpera, Achieving Sub-Shot-Noise Sensing at Finite Temperatures, Phys. Rev. A 94, 042121 (2016). open in new tab
  45. I. Frérot and T. Roscilde, Quantum Critical Metrology, arXiv:1707.08804. open in new tab
  46. M. Hübner, Computation of Uhlmann's Parallel Transport for Density Matrices and the Bures Metric on Three- Dimensional Hilbert Space, Phys. Lett. A 179, 226 (1993). open in new tab
  47. L. Gong and P. Tong, Fidelity, Fidelity Susceptibility, and von Neumann Entropy to Characterize the Phase Diagram of an Extended Harper Model, Phys. Rev. B 78, 115114 (2008). open in new tab
  48. S.-J. Gu, H.-M. Kwok, W.-Q. Ning, and H.-Q. Lin, Fidelity Susceptibility, Scaling, and Universality in Quantum Critical Phenomena, Phys. Rev. B 77, 245109 (2008). open in new tab
  49. S. Greschner, A. K. Kolezhuk, and T. Vekua, Fidelity Susceptibility and Conductivity of the Current in One-Dimensional Lattice Models with Open or Periodic Boundary Conditions, Phys. Rev. B 88, 195101 (2013). open in new tab
  50. H.-Q. Zhou and J. P. Barjaktarevič, Fidelity and Quantum Phase Transitions, J. Phys. A 41, 412001 (2008). open in new tab
  51. M. M. Rams and B. Damski, Quantum Fidelity in the Thermodynamic Limit, Phys. Rev. Lett. 106, 055701 (2011); Scaling of Ground-State Fidelity in the Thermo- dynamic Limit: XY Model and Beyond, Phys. Rev. A 84, 032324 (2011). open in new tab
  52. L. Campos Venuti and P. Zanardi, Quantum Critical Scaling of the Geometric Tensors, Phys. Rev. Lett. 99, 095701 (2007). open in new tab
  53. D. Schwandt, F. Alet, and S. Capponi, Quantum Monte Carlo Simulations of Fidelity at Magnetic Quantum Phase Transitions, Phys. Rev. Lett. 103, 170501 (2009). open in new tab
  54. A. F. Albuquerque, F. Alet, C. Sire, and S. Capponi, Quantum Critical Scaling of Fidelity Susceptibility, Phys. Rev. B 81, 064418 (2010). open in new tab
  55. A. Polkovnikov and V. Gritsev, Universal Dynamics Near Quantum Critical Points, Understanding Quantum Phase Transitions, arXiv:0910.3692. open in new tab
  56. S Sachdev, Quantum Phase Transitions (Cambridge University Press, Cambridge, England, 1999).
  57. S. L. Sondhi, S. M. Girvin, J. P. Carini, and D. Shahar, Continuous Quantum Phase Transitions, Rev. Mod. Phys. 69, 315 (1997). open in new tab
  58. M. A. Continentino, Quantum Scaling in Many-Body Systems (World Scientific Publishing, Singapore, 2001). open in new tab
  59. C. De Grandi, V. Gritsev, and A. Polkovnikov, Quench Dynamics Near a Quantum Critical Point, Phys. Rev. B 81, 012303 (2010); Quench Dynamics Near a Quantum Critical Point: Application to the Sine-Gordon Model, Phys. Rev. B 81, 224301 (2010). open in new tab
  60. Y. Chen, Z. D. Wang, Y. Q. Li, and F. C. Zhang, Spin- Orbital Entanglement and Quantum Phase Transitions in a Spin-Orbital Chain with SUð2Þ × SUð2Þ Symmetry, Phys. Rev. B 75, 195113 (2007). open in new tab
  61. B. Damski, Fidelity Susceptibility of the Quantum Ising Model in a Transverse Field: The Exact Solution, Phys. Rev. E 87, 052131 (2013); open in new tab
  62. B. Damski and M. M. Rams, Exact Results for Fidelity Susceptibility of the Quantum Ising Model: The Interplay between Parity, System Size, and Magnetic Field, J. Phys. A 47, 025303 (2014). open in new tab
  63. S. Knysh, Zero-Temperature Quantum Annealing Bottle- necks in the Spin-Glass Phase, Nat. Commun. 7, 12370 (2016). open in new tab
  64. W. H. Zurek, U. Dorner, and P. Zoller, Dynamics of a Quantum Phase Transition, Phys. Rev. Lett. 95, 105701 (2005). open in new tab
  65. J. Dziarmaga, Dynamics of a Quantum Phase Transition and Relaxation to a Steady State, Adv. Phys. 59, 1063 (2010). open in new tab
  66. A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, Colloquium: Nonequilibrium Dynamics of Closed Interacting Quantum Systems, Rev. Mod. Phys. 83, 863 (2011). open in new tab
  67. J. Cardy, Scaling and Renormalization in Statistical Physics (Cambridge University Press, Cambridge, England, 1996), Vol. 5.
  68. W.-L. You, Y.-W. Li, and S.-J. Gu, Fidelity, Dynamic Structure Factor, and Susceptibility in Critical Phenom- ena, Phys. Rev. E 76, 022101 (2007). open in new tab
  69. S. Deffner and S. Campbell, Quantum Speed Limits: From Heisenberg's Uncertainty Principle to Optimal Quantum Control, J. Phys. A 50, 453001 (2017). open in new tab
  70. Following Ref. [70] consider the cluster-Ising spin chain such a system z ¼ 2 and ½h ¼ 0. open in new tab
  71. M. M. Wolf, G. Ortiz, F. Verstraete, and J. I. Cirac, Quantum Phase Transitions in Matrix Product Systems, Phys. Rev. Lett. 97, 110403 (2006). open in new tab
  72. See for instance Eqs. (7)-(9) in Ref. [14]. open in new tab
  73. In Eq. (8) we allow a small change of, say, λ 1 to depend on time. open in new tab
  74. I. Affleck and M. Oshikawa, Field-Induced Gap in Cu Benzoate and Other s ¼ 1 open in new tab
  75. Antiferromagnetic Chains, Phys. Rev. B 60, 1038 (1999). open in new tab
  76. L. Campos Venuti and P. Zanardi, Quantum Critical Scaling of the Geometric Tensors, Phys. Rev. Lett. 99, 095701 (2007). open in new tab
  77. J. Sirker, Finite-Temperature Fidelity Susceptibility for One-Dimensional Quantum Systems, Phys. Rev. Lett. 105, 117203 (2010). open in new tab
  78. G. Sun, A. K. Kolezhuk, and T. Vekua, Fidelity at Berezinskii-Kosterlitz-Thouless Quantum Phase Transi- tions, Phys. Rev. B 91, 014418 (2015). open in new tab
  79. The experienced reader may wonder that for odd N the ground state is exactly degenerate at the critical point λ c ¼ 0, which leads to singularity in χ F if one uses Eq. (9) directly. For simplicity of the discussion, we use even N in the numerics. open in new tab
  80. In the perturbative regime of t ≪ 1 it is easy to see that Δ δ λ ðĤ 1 ; λ c ; tÞ ∼ t −2 . Such a region is not shown in Fig. 2. open in new tab
  81. F. Verstraete, V. Murg, and J. I. Cirac, Matrix Product States, Projected Entangled Pair States, and Variational Renormalization Group Methods for Quantum Spin Systems, Adv. Phys. 57, 143 (2008). open in new tab
  82. U. Schollwöck, The Density-Matrix Renormalization Group in the Age of Matrix Product States, Ann. Phys. (Amsterdam) 326, 96 (2011). open in new tab
  83. J. Haegeman, J. I. Cirac, T. J. Osborne, I. Pižorn, H. Verschelde, and F. Verstraete, Time-Dependent Variational Principle for Quantum Lattices, Phys. Rev. Lett. 107, 070601 (2011); open in new tab
  84. J. Haegeman, C. Lubich, I. Oseledets, B. Vandereycken, and F. Verstraete, Unifying Time Evolution and Optimization with Matrix Product States, Phys. Rev. B 94, 165116 (2016). open in new tab
  85. M. Suzuki, General Decomposition Theory of Ordered Exponentials, Proc. Jpn. Acad. Ser. B 69, 161 (1993); open in new tab
  86. N. Hatano and M. Suzuki, Quantum Annealing and Other Optimization Methods (Springer, New York, 2005), pp. 37-68. open in new tab
  87. A. Eckardt, C. Weiss, and M. Holthaus, Superfluid- Insulator Transition in a Periodically Driven Optical Lattice, Phys. Rev. Lett. 95, 260404 (2005). open in new tab
  88. A. Eckardt, P. Hauke, P. Soltan-Panahi, C. Becker, K. Sengstock, and M. Lewenstein, Frustrated Quantum Antiferromagnetism with Ultracold Bosons in a Triangu- lar Lattice, Europhys. Lett. 89, 10010 (2010). open in new tab
  89. M. Łącki and J. Zakrzewski, Fast Dynamics for Atoms in Optical Lattices, Phys. Rev. Lett. 110, 065301 (2013). open in new tab
  90. N. Goldman and J. Dalibard, Periodically Driven Quan- tum Systems: Effective Hamiltonians and Engineered Gauge Fields, Phys. Rev. X 4, 031027 (2014). open in new tab
  91. M. Bukov, L. D'Alessio, and A. Polkovnikov, Universal High-Frequency Behavior of Periodically Driven Systems: From Dynamical Stabilization to Floquet Engineering, Adv. Phys. 64, 139 (2015). open in new tab
  92. A. Eckardt and E. Anisimovas, High-Frequency Approxi- mation for Periodically Driven Quantum Systems from a Floquet-Space Perspective, New J. Phys. 17, 093039 (2015). open in new tab
  93. A. Dutta, G. Aeppli, B. K. Chakrabarti, U. Divakaran, T. F. Rosenbaum, and D. Sen, Quantum Phase Transitions in Transverse Field Spin Models: From Statistical Physics to Quantum Information (Cambridge University Press, Cambridge, England, 2015). open in new tab
  94. H. T. Quan, Z. Song, X. F. Liu, P. Zanardi, and C. P. Sun, Decay of Loschmidt Echo Enhanced by Quantum Criticality, Phys. Rev. Lett. 96, 140604 (2006). open in new tab
  95. A. Peres, Stability of Quantum Motion in Chaotic and Regular Systems, Phys. Rev. A 30, 1610 (1984). open in new tab
  96. Note that here the scaling remains tight also beyond the perturbative regime of very short times. open in new tab
  97. L. Campos Venuti and P. Zanardi, Unitary Equilibrations: Probability Distribution of the Loschmidt Echo, Phys. Rev. A 81, 022113 (2010). open in new tab
  98. V. Mukherjee, S. Sharma, and A. Dutta, Loschmidt Echo with a Nonequilibrium Initial State: Early-Time Scaling and Enhanced Decoherence, Phys. Rev. B 86, 020301 (2012). open in new tab
  99. C. Sträter and A. Eckardt, Orbital-Driven Melting of a Bosonic Mott Insulator in a Shaken Optical Lattice, Phys. Rev. A 91, 053602 (2015). open in new tab
  100. A. Przysiężna, O. Dutta, and J. Zakrzewski, Rice-Mele Model with Topological Solitons in an Optical Lattice, New J. Phys. 17, 013018 (2015). open in new tab
  101. O. Dutta, A. Przysiężna, and J. Zakrzewski, Spontaneous Magnetization and Anomalous Hall Effect in an Emergent Dice Lattice, Sci. Rep. 5, 11060 (2015). open in new tab
  102. A. Eckardt, Colloquium: Atomic Quantum Gases in Periodically Driven Optical Lattices, Rev. Mod. Phys. 89, 011004 (2017). open in new tab
  103. P. Sierant, O. Dutta, and J. Zakrzewski, Effective Spin Models from Cold Bosons in Optical Shaken Potentials (to be published). open in new tab
  104. A. K. Ekert, C. M. Alves, D. K. L. Oi, M. Horodecki, P. Horodecki, and L. C. Kwek, Direct Estimations of Linear and Nonlinear Functionals of a Quantum State, Phys. Rev. Lett. 88, 217901 (2002). open in new tab
  105. P. Horodecki and A. Ekert, Method for Direct Detection of Quantum Entanglement, Phys. Rev. Lett. 89, 127902 (2002). open in new tab
  106. J. A. Miszczak, Z. Puchała, P. Horodecki, A. Uhlmann, and K. Życzkowski, Sub-and Super-Fidelity as Bounds for Quantum Fidelity, Quantum Inf. Comput. 9, 0103 (2009).
  107. F. A. Bovino, G. Castagnoli, A. Ekert, P. Horodecki, C. M. Alves, and A. V. Sergienko, Direct Measurement of Nonlinear Properties of Bipartite Quantum States, Phys. Rev. Lett. 95, 240407 (2005). open in new tab
  108. A. J. Daley, H. Pichler, J. Schachenmayer, and P. Zoller, Measuring Entanglement Growth in Quench Dynamics of Bosons in an Optical Lattice, Phys. Rev. Lett. 109, 020505 (2012). open in new tab
  109. H. Pichler, L. Bonnes, A. J. Daley, A. M. Läuchli, and P. Zoller, Thermal versus Entanglement Entropy: A Meas- urement Protocol for Fermionic Atoms with a Quantum Gas Microscope, New J. Phys. 15, 063003 (2013). open in new tab
  110. A. Elben, B. Vermersch, M. Dalmonte, J. I. Cirac, and P. Zoller, Rényi Entropies from Random Quenches in Atomic Hubbard and Spin Models, Phys. Rev. Lett. 120, 050406 (2018). open in new tab
  111. B. Vermersch, A. Elben, M. Dalmonte, J. I. Cirac, and P. Zoller, Unitary n-Designs via Random Quenches in Atomic Hubbard and Spin Models: Application to the Measurement of Rényi Entropies, Phys. Rev. A 97, 023604 (2018). open in new tab
Sources of funding:
Verified by:
Gdańsk University of Technology

seen 141 times

Recommended for you

Meta Tags