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Entanglement transition and heterogeneity in long-range quadratic Lindbladians

Alejandro Cros Carrillo de Albornoz, Dominic C. Rose, and Arijeet Pal
Phys. Rev. B 109, 214204 – Published 11 June 2024
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  • INTRODUCTION
  • MODEL
  • GAUSSIANITY AND PURE DISSIPATION…
  • MUTUAL INFORMATION
  • FERMIONIC NEGATIVITY
  • BATH STRUCTURE
  • STATIONARY-STATE HETEROGENEITY
  • EFFECT OF COHERENT HOPPING
  • CONCLUSIONS
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    The generation of entanglement in mixed states is relevant to quantum systems coupled to an environment. The dissipative and mixing properties of the environment are unavoidable in physical platforms for quantum simulation and information processing, where entanglement can be a vital resource. In this work, we explore entanglement and heterogeneity in random Lindbladian dynamics describing open quantum systems. We propose a model of a one-dimensional chain of noninteracting, spinless fermions coupled to a local ensemble of baths. The jump operator mediating the interaction with the bath linked to each site has a power-law tail with an exponent p. We show that the system undergoes volume-to-area law entanglement phase transition in the mixed steady state by tuning p which remains stable in the presence of coherent hopping. Unlike the entanglement transition in the pure-state quantum trajectories of open systems, this transition is exhibited by the averaged steady-state density matrix of the Lindbladian. The steady state in the area-law phase is characterized by a spatial heterogeneity in local population imbalance, while the jump operators exhibit a constant participation ratio of the sites they affect. Our work provides a theoretical description of an entanglement transition realized in long-ranged open quantum systems and provides an avenue to stabilize quantum correlations in mixed states.

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    • Received 28 March 2023
    • Revised 16 May 2024
    • Accepted 16 May 2024

    DOI:https://doi.org/10.1103/PhysRevB.109.214204

    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.

    Published by the American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    FermionsLocalizationOpen quantum systems & decoherenceQuantum criticalityQuantum entanglementSecond order phase transitionsThermal properties
    1. Techniques
    Lattice models in condensed matter
    Quantum Information, Science & TechnologyCondensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Alejandro Cros Carrillo de Albornoz, Dominic C. Rose, and Arijeet Pal

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    Vol. 109, Iss. 21 — 1 June 2024

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    • Figure 1
      Figure 1

      (a) Schematic of the model: The spatial power-law profile of each jump operator for the heterogeneous baths shows the coupling to a system of spinless fermions hopping in a 1D lattice. We consider the mutual information between fermions in regions A and B (I). (b) Disorder-averaged mutual information between A and B in the steady state (I¯∞) as a function of system size L, with LA=LB=LC=L/3. For each value of p (shown in the legend) the curves are fitted to the form ln(I¯∞)=Δln(L)+c. The fitting parameter Δ is shown in the legend. (c) I¯∞ between A and B in the steady state as a function of subsystem size LA with total system size L=103. LC is fixed at L/3. Curves were fitted according to I¯∞=a(LA)c+blog(LA). Note c≈1/2 for all p. For both (b), (c): Curves are color coded to a given p according to the legend in panel (b). Error bars are not visible. Details on disorder realizations are included in Appendix pp8-s1.

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    • Figure 2
      Figure 2

      Mutual information between A and B as a function of p, with LA=LB=LC=L/3, for various L shown, averaged over disorder. Inset shows a finite-size scaling collapse with pc=0.98±0.01 and ν=6.3±0.7. Error bars are not visible. Details on disorder realizations and fitting procedure are included in Appendix pp8.

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    • Figure 3
      Figure 3

      (a) Disorder-averaged fermionic negativity (EF) between A and B (cf. Fig. 1) in the steady state as a function of system size L, with LA=LB=LC=L/3. (b) Steady state EF between A and B as a function of subsystem size LA with LC=L/3 fixed at L=103. For each value of p the curves are fitted to ln[f(x)]=Δln(x)+c. The corresponding Δ is shown in the legend. Details on disorder realizations are included in Appendix pp8-s1.

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    • Figure 4
      Figure 4

      The disorder-averaged participation ratio PR¯ against p for sizes L = [400, 800, 1200, 1600, 2000], increasing along the arrow. Inset: Power a as a function of p found when fitting PR¯(L,p)=La(p)/c(p) to PR¯(L) curves. Data for c are shown in Appendix pp4.

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    • Figure 5
      Figure 5

      Distribution of the standard deviation of the bias ñ calculated for each realization using the central L/5 sites in the bulk in the area (blue, p=4)) and volume law (orange, p=0.4) phases for system size L=103. Inset: Average standard deviation as a function of L; error bars are not visible.

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    • Figure 6
      Figure 6

      (a) Dynamics of the bipartite mutual information I¯t between A and B where LA=LB=L/2 and LC=0, of an initially uncorrelated state, i.e., Ω(0)=diag{a1,...,aL}, for random real {an} with system size L=100 and λ=5. Dashed lines here denote their associated I¯∞. (b) Nonequilibrium steady state bipartite mutual information I¯∞ against hopping strength λ for different dissipation and system sizes. All curves were averaged over disorder realizations, and error bars are not visible.

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    • Figure 7
      Figure 7

      (a)–(c) Generalized fractal dimensions as a function of system size for p=[0.95,1.0,1.05,1.1], increasing along the arrow.

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    • Figure 8
      Figure 8

      Coefficients as a function of p found when fitting PR¯(L,p)=La(p)/c(p) to PR¯(L) curves.

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    • Figure 9
      Figure 9

      Eigenvalues of X†X (markers) and their average against L (lines).

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    • Figure 10
      Figure 10

      Log-log plot of the decay of the matrix elements of Γ as a distance d away from the diagonal. For large enough d, the decay is given by a power law with an exponent approximately equal to p. In the legend, glog is the linear gradient of the black lines fitted to the log-log data. Note how p∼glog, suggesting a power-law decay with power p for large enough d.

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    • Figure 11
      Figure 11

      (a), (b) Five sample eigenstates with a maximum magnitude value within the middle 25th of the system at (a) p=0.5 and (b) p=1.1. (c) Log-log plot of the averaged positional probability as a function of distance from the most probable site, for p=[0.5,0.7,0.9,1.0,1.2,1.4,1.7,2.0]. Averaged over eigenstates with most probable sites located in the middle 5th of the system, and averaged over 1000 disorder realizations. Linear fits of the probabilities for distances of 100 to 1000 are shown by the dashed lines. Inset: (solid) gradients of the linear fits are shown vs p, (dashed) the curve y=p.

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    • Figure 12
      Figure 12

      Steady-state mutual information between A and B including their boundary term ∂AB.

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    • Figure 13
      Figure 13

      Relative height of the bump in I¯t in (b) with respect to its I¯∞ scaled by 1/I¯∞, denoted by ΔB/I¯∞, against system size, L.

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