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.
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- Articles
- Type:
- artykuł w czasopiśmie wyróżnionym w JCR
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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
-
- C. M. Caves, Quantum-Mechanical Noise in an Interfer- ometer, Phys. Rev. D 23, 1693 (1981). open in new tab
- 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
- 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
- V. Giovannetti, S. Lloyd, and L. Maccone, Quantum- Enhanced Measurements: Beating the Standard Quantum Limit, Science 306, 1330 (2004). open in new tab
- A. Uhlmann, The "Transition Probability" in the State Space of a *-Algebra, Rep. Math. Phys. 9, 273 (1976). open in new tab
- 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
- V. Giovannetti, S. Lloyd, and L. Maccone, Advances in Quantum Metrology, Nat. Photonics 5, 222 (2011). open in new tab
- G. Tóth and I. Apellaniz, Quantum Metrology from a Quantum Information Science Perspective, J. Phys. A 47, 424006 (2014). open in new tab
- R. Demkowicz-Dobrzanski, M. Jarzyna, and J. Kolodynski, Quantum Limits in Optical Interferometry, Prog. Opt. 60, 345 (2015). open in new tab
- S. M. Roy and S. L. Braunstein, Exponentially Enhanced Quantum Metrology, Phys. Rev. Lett. 100, 220501 (2008). open in new tab
- 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
- V. Giovannetti, S. Lloyd, and L. Maccone, Quantum Metrology, Phys. Rev. Lett. 96, 010401 (2006). open in new tab
- 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
- 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
- S. Pang and T. A. Brun, Quantum Metrology for a General Hamiltonian Parameter, Phys. Rev. A 90, 022117 (2014). open in new tab
- J. M. E. Fraisse and D. Braun, Hamiltonian Extensions in Quantum Metrology, Quantum Meas. Quantum Metrol. 4, 8 (2017). open in new tab
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- G. Tóth, Multipartite Entanglement and High-Precision Metrology, Phys. Rev. A 85, 022322 (2012). open in new tab
- 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
- 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
- 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
- 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
- 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
- D. Clément, N. Fabbri, L. Fallani, C. Fort, and M. Inguscio, J. Low Temp. Phys. 158, 5 (2010). open in new tab
- M. Tsang, Quantum Transition-Edge Detectors, Phys. Rev. A 88, 021801 (2013). open in new tab
- T. Macrì, A. Smerzi, and L. Pezzè, Loschmidt Echo for Quantum Metrology, Phys. Rev. A 94, 010102 (2016). open in new tab
- 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
- 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
- M. A. Rajabpour, Multipartite Entanglement and Quantum Fisher Information in Conformal Field Theories, Phys. Rev. D 96, 126007 (2017). open in new tab
- 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
- P. Zanardi and N. Paunković, Ground State Overlap and Quantum Phase Transitions, Phys. Rev. E 74, 031123 (2006). open in new tab
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- I. Frérot and T. Roscilde, Quantum Critical Metrology, arXiv:1707.08804. open in new tab
- 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
- 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
- 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
- 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
- H.-Q. Zhou and J. P. Barjaktarevič, Fidelity and Quantum Phase Transitions, J. Phys. A 41, 412001 (2008). open in new tab
- 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
- L. Campos Venuti and P. Zanardi, Quantum Critical Scaling of the Geometric Tensors, Phys. Rev. Lett. 99, 095701 (2007). open in new tab
- 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
- 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
- A. Polkovnikov and V. Gritsev, Universal Dynamics Near Quantum Critical Points, Understanding Quantum Phase Transitions, arXiv:0910.3692. open in new tab
- S Sachdev, Quantum Phase Transitions (Cambridge University Press, Cambridge, England, 1999).
- 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
- M. A. Continentino, Quantum Scaling in Many-Body Systems (World Scientific Publishing, Singapore, 2001). open in new tab
- 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
- 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
- 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
- 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
- S. Knysh, Zero-Temperature Quantum Annealing Bottle- necks in the Spin-Glass Phase, Nat. Commun. 7, 12370 (2016). open in new tab
- W. H. Zurek, U. Dorner, and P. Zoller, Dynamics of a Quantum Phase Transition, Phys. Rev. Lett. 95, 105701 (2005). open in new tab
- J. Dziarmaga, Dynamics of a Quantum Phase Transition and Relaxation to a Steady State, Adv. Phys. 59, 1063 (2010). open in new tab
- 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
- J. Cardy, Scaling and Renormalization in Statistical Physics (Cambridge University Press, Cambridge, England, 1996), Vol. 5.
- 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
- 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
- Following Ref. [70] consider the cluster-Ising spin chain such a system z ¼ 2 and ½h ¼ 0. open in new tab
- 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
- See for instance Eqs. (7)-(9) in Ref. [14]. open in new tab
- In Eq. (8) we allow a small change of, say, λ 1 to depend on time. open in new tab
- I. Affleck and M. Oshikawa, Field-Induced Gap in Cu Benzoate and Other s ¼ 1 open in new tab
- Antiferromagnetic Chains, Phys. Rev. B 60, 1038 (1999). open in new tab
- L. Campos Venuti and P. Zanardi, Quantum Critical Scaling of the Geometric Tensors, Phys. Rev. Lett. 99, 095701 (2007). open in new tab
- J. Sirker, Finite-Temperature Fidelity Susceptibility for One-Dimensional Quantum Systems, Phys. Rev. Lett. 105, 117203 (2010). open in new tab
- 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
- 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
- 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
- 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
- U. Schollwöck, The Density-Matrix Renormalization Group in the Age of Matrix Product States, Ann. Phys. (Amsterdam) 326, 96 (2011). open in new tab
- 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
- 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
- M. Suzuki, General Decomposition Theory of Ordered Exponentials, Proc. Jpn. Acad. Ser. B 69, 161 (1993); open in new tab
- N. Hatano and M. Suzuki, Quantum Annealing and Other Optimization Methods (Springer, New York, 2005), pp. 37-68. open in new tab
- 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
- 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
- M. Łącki and J. Zakrzewski, Fast Dynamics for Atoms in Optical Lattices, Phys. Rev. Lett. 110, 065301 (2013). open in new tab
- 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
- 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
- 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
- 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
- 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
- A. Peres, Stability of Quantum Motion in Chaotic and Regular Systems, Phys. Rev. A 30, 1610 (1984). open in new tab
- Note that here the scaling remains tight also beyond the perturbative regime of very short times. open in new tab
- L. Campos Venuti and P. Zanardi, Unitary Equilibrations: Probability Distribution of the Loschmidt Echo, Phys. Rev. A 81, 022113 (2010). open in new tab
- 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
- 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
- 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
- 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
- A. Eckardt, Colloquium: Atomic Quantum Gases in Periodically Driven Optical Lattices, Rev. Mod. Phys. 89, 011004 (2017). open in new tab
- P. Sierant, O. Dutta, and J. Zakrzewski, Effective Spin Models from Cold Bosons in Optical Shaken Potentials (to be published). open in new tab
- 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
- P. Horodecki and A. Ekert, Method for Direct Detection of Quantum Entanglement, Phys. Rev. Lett. 89, 127902 (2002). open in new tab
- 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).
- 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
- 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
- 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
- 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
- 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
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