- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Letter
  • Open Access

Geometric phases distinguish entangled states in wormhole quantum mechanics

Flavio S. Nogueira1, Souvik Banerjee2, Moritz Dorband2, René Meyer2, Jeroen van den Brink1,3, and Johanna Erdmenger2

  • 1Institute for Theoretical Solid State Physics, IFW Dresden, 01069 Dresden, Germany
  • 2Institute for Theoretical Physics and Würzburg-Dresden Cluster of Excellence ct.qmat, Julius-Maximilians-Universität Würzburg, 97074 Würzburg, Germany
  • 3Institute for Theoretical Physics and Würzburg-Dresden Cluster of Excellence ct.qmat, Technische Universität Dresden, 01069 Dresden, Germany

PDF

Phys. Rev. D 105, L081903 – Published 19 April, 2022

DOI: https://doi.org/10.1103/PhysRevD.105.L081903

Abstract

We establish a relation between entanglement in simple quantum mechanical qubit systems and in wormhole physics as considered in the context of the AdS/CFT correspondence. We show that in both cases, states with the same entanglement structure, indistinguishable by any local measurement, nevertheless are characterized by a different Berry phase. This feature is experimentally accessible in coupled qubit systems where states with different Berry phase are related by unitary transformations. In the wormhole case, these transformations are identified with a time evolution of one of the two throats.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (39)

  1. S. Ryu and T. Takayanagi, Holographic Derivation of Entanglement Entropy from AdS/CFT, Phys. Rev. Lett. 96, 181602 (2006).
  2. J. M. Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2, 231 (1998).
  3. J. M. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortschr. Phys. 61, 781 (2013).
  4. M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Relativ. Gravit. 42, 2323 (2010).
  5. J. Wheeler, Geons, Phys. Rev. 97, 511 (1955).
  6. D. Garfinkle and A. Strominger, Semiclassical Wheeler wormhole production, Phys. Lett. B 256, 146 (1991).
  7. S. Novikov, The Hamiltonian formalism and a many-valued analogue of Morse theory, Usp. Mat. Nauk 37, 3 (1982).
  8. H. Verlinde, Wormholes in quantum mechanics, arXiv:2105.02129.
  9. J. M. Maldacena, Eternal black holes in anti-de Sitter, J. High Energy Phys. 04 (2003) 021.
  10. J. B. Hartle and S. W. Hawking, Path integral derivation of black hole radiance, Phys. Rev. D 13, 2188 (1976).
  11. W. Israel, Thermo field dynamics of black holes, Phys. Lett. 57A, 107 (1976).
  12. D. Harlow, Jerusalem lectures on black holes and quantum information, Rev. Mod. Phys. 88, 015002 (2016).
  13. K. Papadodimas and S. Raju, Local Operators in the Eternal Black Hole, Phys. Rev. Lett. 115, 211601 (2015).
  14. H. Verlinde, ER=EPR revisited: On the entropy of an Einstein-Rosen Bridge, arXiv:2003.13117.
  15. R. Blatt and D. Wineland, Entangled states of trapped atomic ions, Nature (London) 453, 1008 (2008).
  16. I. Bengtsson and K. Życzkowski, Geometry of Quantum States: An Introduction to Quantum Entanglement (Cambridge University Press, Cambridge, England, 2017).
  17. M. Nakahara, Geometry, Topology and Physics (IOP Publishing Ltd., London, 2003).
  18. B. Simon, Holonomy, the Quantum Adiabatic Theorem, and Berry’s Phase, Phys. Rev. Lett. 51, 2167 (1983).
  19. H. Georgi, Lie Algebras In Particle Physics: From Isospin to Unified Theories (CRC Press, Boca Raton, 2019).
  20. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevD.105.L081903 for details about the bundle connection in CP3, time-shifted TFD states and their associated entanglement entropy.
  21. K. Papadodimas and S. Raju, Remarks on the necessity and implications of state-dependence in the black hole interior, Phys. Rev. D 93, 084049 (2016).

  22. In differential geometry, a nontrivial holonomy arises when the considered manifold has to be described by more than one coordinate patch, each of which patch has its own connection. Therefore, one cannot define a symplectic form as an exterior derivative of a single connection. Hence, the symplectic form cannot be globally exact. However, the connections on individual patches are related to each other by U(1) gauge transformations and yields a nonzero Berry phase. In the case of gravity with a horizon, the identifications δ=tL and δ=tR only work at the left and right boundaries and can at most be analytically continued in the near-boundary regions, leading to two different coordinate patches. The connections on the coordinate patches are related by a unitary of the same form as (10). This can be interpreted as a holonomy of a U(1) bundle, as in the quantum mechanical case, and results in a nonvanishing Berry phase [18]. For the TFD state, this nonvanishing Berry connection can also be argued in terms of the holonomy associated to the modular Hamiltonian [23, 24, 25].

  23. B. Czech, L. Lamprou, S. Mccandlish, and J. Sully, Modular Berry Connection for Entangled Subregions in AdS/CFT, Phys. Rev. Lett. 120, 091601 (2018).
  24. B. Czech, L. Lamprou, and L. Susskind, Entanglement holonomies, arXiv:1807.04276.
  25. B. Czech, J. De Boer, D. Ge, and L. Lamprou, A modular sewing kit for entanglement wedges, J. High Energy Phys. 11 (2019) 094.

  26. This follows from the fact that the bulk Killing vector switches sign at the horizon. This is accounted for by relating the CFT times at the left and right boundaries to the Schwarzschild times in the respective bulk wedges with a relative flip of sign. Note that this is independent of any additional phases in the TFD state (7).

  27. Roman Jackiw, Lower dimensional gravity, Nucl. Phys. B252, 343 (1985).
  28. Claudio Teitelboim, Gravitation and Hamiltonian structure in two spacetime dimensions, Phys. Lett. 126B, 41 (1983).
  29. G. Sárosi, AdS2 holography and the SYK model, Proc. Sci., Modave2017 (2018) 001 [arXiv:1711.08482].
  30. D. Harlow and D. Jafferis, The factorization problem in Jackiw-Teitelboim gravity, J. High Energy Phys. 02 (2020) 177.

  31. Note that the wormhole structure is supported by a nonvanishing ϕh only; setting ϕh=0 collapses the geometry to pure AdS2.

  32. M. Van Raamsdonk, Comments on quantum gravity and entanglement, arXiv:0907.2939.
  33. B. Czech, J. Karczmarek, F. Nogueira, and M. Van Raamsdonk, The gravity dual of a density matrix, Classical Quantum Gravity 29, 155009 (2012).
  34. C. A. Ryan, M. Laforest, and R. Laflamme, Randomized benchmarking of single- and multi-qubit control in liquid-state NMR quantum information processing, New J. Phys. 11, 013034 (2009).
  35. A. Imamoglu, D. D. Awschalom, G. Burkard, D. P. DiVincenzo, D. Loss, M. Sherwin, and A. Small, Quantum Information Processing Using Quantum Dot Spins and Cavity QED, Phys. Rev. Lett. 83, 4204 (1999).
  36. M. Steffen, M. Ansmann, R. C. Bialczak, N. Katz, E. Lucero, R. McDermott, M. Neeley, E. M. Weig, A. N. Cleland, and J. M. Martinis, Measurement of the entanglement of two superconducting qubits via state tomography, Science 313, 1423 (2006).
  37. M. H. Devoret and R. J. Schoelkopf, Superconducting circuits for quantum information: An outlook, Science 339, 1169 (2013).
  38. C. Song, K. Xu, W. Liu, C.-P. Yang, S.-B. Zheng, H. Deng, Q. Xie, K. Huang, Q. Guo, L. Zhang, P. Zhang, D. Xu, D. Zheng, X. Zhu, H. Wang, Y.-A. Chen, C.-Y. Lu, S. Han, and J.-W. Pan, 10-Qubit Entanglement and Parallel Logic Operations with a Superconducting Circuit, Phys. Rev. Lett. 119, 180511 (2017).
  39. D. Zhu, S. Johri, N. M. Linke, K. A. Landsman, N. H. Nguyen, C. H. Alderete, A. Y. Matsuura, T. H. Hsieh, and C. Monroe, Generation of thermofield double states and critical ground states with a quantum computer, Proc. Natl. Acad. Sci. U.S.A. 117, 25402 (2020).

Outline

Information

Phys. Rev. D 105, L081903– Published 19 April, 2022
  • Received 21 September 2021
  • Accepted 22 March 2022
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation