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Entanglement in one-dimensional critical state after measurements (2301.08255v2)

Published 19 Jan 2023 in quant-ph, cond-mat.stat-mech, and cond-mat.str-el

Abstract: The entanglement entropy (EE) of the ground state of a one-dimensional Hamiltonian at criticality has a universal logarithmic scaling with a prefactor given by the central charge $c$ of the underlying 1+1d conformal field theory. When the system is probed by measurements, the entanglement in the critical ground state is inevitably affected due to wavefunction collapse. In this paper, we study the effect of weak measurements on the entanglement scaling in the ground state of the one-dimensional critical transverse-field Ising model. For the measurements of the spins along their transverse spin axis, we identify interesting post-measurement states associated with spatially uniform measurement outcomes. The EE in these states still satisfies the logarithmic scaling but with an alternative prefactor given by the effective central charge $c_{\text{eff}}$. We derive the analytical expression of $c_{\text{eff}}$ as a function of the measurement strength. Using numerical simulations, we show that for the EE averaged over all post-measurement states based on their Born-rule probabilities, the numerically extracted effective central charge appears to be independent of the measurement strength, contrary to the usual expectation that local and non-overlapping measurements reduce the entanglement in the system. We also examine the behavior of the average EE under (biased) forced measurements where the measurement outcomes are sampled with a pre-determined probability distribution without inter-site correlations. In particular, we find an optimal probability distribution that can serve as a mean-field approximation to the Born-rule probabilities and lead to the same $c_{\text{eff}}$ behavior. The effects of the measurements along the longitudinal spin axis and the post-measurement correlation functions are also discussed.

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References (34)
  1. A. Y. Kitaev, Fault-tolerant quantum computation by anyons, Annals of Physics 303, 2 (2003), arXiv:quant-ph/9707021 [quant-ph] .
  2. M. A. Levin and X.-G. Wen, String-net condensation: A physical mechanism for topological phases, Phys. Rev. B 71, 045110 (2005), arXiv:cond-mat/0404617 [cond-mat.str-el] .
  3. B. Zeng and X.-G. Wen, Gapped quantum liquids and topological order, stochastic local transformations and emergence of unitarity, Phys. Rev. B 91, 125121 (2015).
  4. R. Raussendorf and H. J. Briegel, A one-way quantum computer, Phys. Rev. Lett. 86, 5188 (2001).
  5. D. Gottesman and I. L. Chuang, Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations, Nature (London) 402, 390 (1999), arXiv:quant-ph/9908010 [quant-ph] .
  6. R. Raussendorf, D. E. Browne, and H. J. Briegel, Measurement-based quantum computation on cluster states, Phys. Rev. A 68, 022312 (2003).
  7. H. J. Briegel and R. Raussendorf, Persistent entanglement in arrays of interacting particles, Phys. Rev. Lett. 86, 910 (2001).
  8. J. Cardy, Boundary Conformal Field Theory, arXiv e-prints , hep-th/0411189 (2004), arXiv:hep-th/0411189 [hep-th] .
  9. M. Levin and X.-G. Wen, Detecting Topological Order in a Ground State Wave Function, Phys. Rev. Lett.  96, 110405 (2006), arXiv:cond-mat/0510613 [cond-mat.str-el] .
  10. R. C. Myers and A. Sinha, Seeing a c-theorem with holography, Phys. Rev. D 82, 046006 (2010), arXiv:1006.1263 [hep-th] .
  11. R. C. Myers and A. Sinha, Holographic c-theorems in arbitrary dimensions, Journal of High Energy Physics 2011, 125 (2011), arXiv:1011.5819 [hep-th] .
  12. H. Casini and M. Huerta, A c-theorem for entanglement entropy, Journal of Physics A Mathematical General 40, 7031 (2007), arXiv:cond-mat/0610375 [cond-mat.stat-mech] .
  13. H. Casini and M. Huerta, Renormalization group running of the entanglement entropy of a circle, Phys. Rev. D 85, 125016 (2012), arXiv:1202.5650 [hep-th] .
  14. H. Li and F. D. M. Haldane, Entanglement spectrum as a generalization of entanglement entropy: Identification of topological order in non-abelian fractional quantum hall effect states, Phys. Rev. Lett. 101, 010504 (2008).
  15. M. A. Metlitski and T. Grover, Entanglement Entropy of Systems with Spontaneously Broken Continuous Symmetry, arXiv e-prints , arXiv:1112.5166 (2011), arXiv:1112.5166 [cond-mat.str-el] .
  16. S. J. Garratt, Z. Weinstein, and E. Altman, Measurements conspire nonlocally to restructure critical quantum states, arXiv e-prints , arXiv:2207.09476 (2022), arXiv:2207.09476 [cond-mat.stat-mech] .
  17. I. Affleck, A current algebra approach to the kondo effect, Nuclear Physics B 336, 517 (1990).
  18. I. Affleck and A. W. Ludwig, Critical theory of overscreened kondo fixed points, Nuclear Physics B 360, 641 (1991).
  19. C. L. Kane and M. P. A. Fisher, Transmission through barriers and resonant tunneling in an interacting one-dimensional electron gas, Phys. Rev. B 46, 15233 (1992).
  20. J. Y. Lee, C.-M. Jian, and C. Xu, Quantum criticality under decoherence or weak measurement, arXiv e-prints , arXiv:2301.05238 (2023), arXiv:2301.05238 [cond-mat.stat-mech] .
  21. Y. Bao, S. Choi, and E. Altman, Theory of the phase transition in random unitary circuits with measurements, Phys. Rev. B 101, 104301 (2020), arXiv:1908.04305 [cond-mat.stat-mech] .
  22. M. A. Nielsen and I. L. Chuang, Computation and Quantum Information (Cambridge University Press, Cambridge, UK, 2010).
  23. P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, Journal of physics a: mathematical and theoretical 42, 504005 (2009).
  24. V. Eisler and I. Peschel, Entanglement in fermionic chains with interface defects, Annalen der Physik 522, 679 (2010).
  25. J. Preskill, Quantum Computing in the NISQ era and beyond, Quantum 2, 79 (2018), arXiv:1801.00862 [quant-ph] .
  26. T.-C. Lu and T. Grover, Spacetime duality between localization transitions and measurement-induced transitions, PRX Quantum 2, 040319 (2021), arXiv:2103.06356 [quant-ph] .
  27. X. Turkeshi and M. Schiró, Entanglement and correlation spreading in non-Hermitian spin chains, Phys. Rev. B 107, L020403 (2023), arXiv:2201.09895 [cond-mat.stat-mech] .
  28. The expectation value ⟨Ω|σjx|Ω⟩=2/πquantum-operator-productΩsubscriptsuperscript𝜎𝑥𝑗Ω2𝜋\langle\Omega|\sigma^{x}_{j}|\Omega\rangle=2/\pi⟨ roman_Ω | italic_σ start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT | roman_Ω ⟩ = 2 / italic_π can be straightforwardly calculated using the Majorana-fermion representation of cTFIM.
  29. S. Kullback and R. A. Leibler, On Information and Sufficiency, The Annals of Mathematical Statistics 22, 79 (1951).
  30. Y. Zou, S. Sang, and T. H. Hsieh, Channeling quantum criticality, arXiv e-prints , arXiv:2301.07141 (2023), arXiv:2301.07141 [quant-ph] .
  31. F. Iglói, I. Peschel, and L. Turban, Inhomogeneous systems with unusual critical behaviour, Advances in Physics 42, 683 (1993).
  32. R. Bariev, Effect of linear defects on the local magnetization of a plane ising lattice, Soviet Physics-JETP 50, 613 (1979).
  33. B. M. McCoy and J. H. Perk, Two-spin correlation functions of an ising model with continuous exponents, Physical Review Letters 44, 840 (1980).
  34. S. Bravyi, Lagrangian representation for fermionic linear optics, arXiv e-prints , quant-ph/0404180 (2004), arXiv:quant-ph/0404180 [quant-ph] .
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