Enhanced Entanglement in the Measurement-Altered Quantum Ising Chain

Alessio Paviglianiti; Xhek Turkeshi; Marco Schirò; Alessandro Silva

doi:10.22331/q-2024-12-23-1576

Enhanced Entanglement in the Measurement-Altered Quantum Ising Chain

Alessio Paviglianiti¹, Xhek Turkeshi^2,3, Marco Schirò³, and Alessandro Silva¹

¹International School for Advanced Studies (SISSA), via Bonomea 265, 34136 Trieste, Italy
²Institut für Theoretische Physik, Universität zu Köln, Zülpicher Strasse 77, 50937 Köln, Germany
³JEIP, UAR 3573 CNRS, Collège de France, PSL Research University, 11 Place Marcelin Berthelot, 75321 Paris Cedex 05, France

Published:	2024-12-23, volume 8, page 1576
Eprint:	arXiv:2310.02686v3
Doi:	https://doi.org/10.22331/q-2024-12-23-1576
Citation:	Quantum 8, 1576 (2024).

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Abstract

Understanding the influence of measurements on the properties of many-body systems is a fundamental problem in quantum mechanics and for quantum technologies. This paper explores how a finite density of stochastic local measurement modifies a given state’s entanglement structure. Considering various measurement protocols, we explore the typical quantum correlations of their associated projected ensembles arising from the ground state of the quantum Ising model. Using large-scale numerical simulations, we demonstrate substantial differences among inequivalent measurement protocols. Surprisingly, we observe that forced on-site measurements can enhance both bipartite and multipartite entanglement. We present a phenomenological toy model and perturbative calculations to analytically support these results. Furthermore, we extend these considerations to the non-Hermitian Ising model, naturally arising in optically monitored systems, and we show that its qualitative entanglement features are not altered by a finite density of projective measurements. Overall, these results reveal a complex phenomenology where local quantum measurements do not simply disentangle degrees of freedom, but may actually strengthen the entanglement in the system.

Popular summary

Quantum entanglement links microscopic objects, like atoms or electrons, in deeply interconnected ways, allowing them to exist in states that are correlated. However, this remarkable phenomenon is fragile: observing a particle with a detector can instantly disrupt its entanglement, breaking its correlations with others. At the same time, this process also impacts the rest of the system, reshaping its correlations in surprising ways. In our work, we explore how observing some particles affects the entanglement structure of the unmeasured ones. Remarkably, we find that while measurements destroy certain correlations, their backaction can generate new ones, sometimes leading to a net increase in entanglement. Our study reveals how the action of an external observer transforms the properties of quantum states of matter, highlighting a mechanism that simultaneously destroys and rebuilds correlations.

► BibTeX data

@article{Paviglianiti2024enhanced,
  doi = {10.22331/q-2024-12-23-1576},
  url = {https://doi.org/10.22331/q-2024-12-23-1576},
  title = {Enhanced {E}ntanglement in the {M}easurement-{A}ltered {Q}uantum {I}sing {C}hain},
  author = {Paviglianiti, Alessio and Turkeshi, Xhek and Schir{\`{o}}, Marco and Silva, Alessandro},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1576},
  month = dec,
  year = {2024}
}

► References

[1] R. Shankar. ``Principles of quantum mechanics''. Springer New York, NY. (1994).
https://doi.org/10.1007/978-1-4757-0576-8

[2] N. Gisin and I. C. Percival. ``The quantum-state diffusion model applied to open systems''. J. Phys. A: Math. Theor. 25, 5677–5691 (1992).
https://doi.org/10.1088/0305-4470/25/21/023

[3] H.-P. Breuer and F. Petruccione. ``The theory of open quantum systems''. Oxford University Press. (2007).
https://doi.org/10.1093/acprof:oso/9780199213900.001.0001

[4] K. Jacobs. ``Quantum measurement theory and its applications''. Cambridge University Press. (2014).
https://doi.org/10.1017/CBO9781139179027

[5] A. J. Daley. ``Quantum trajectories and open many-body quantum systems''. Adv. Phys. 63, 77–149 (2014).
https://doi.org/10.1080/00018732.2014.933502

[6] H. Carmichael. ``Statistical methods in quantum optics 1''. Springer Berlin, Heidelberg. (1999).
https://doi.org/10.1007/978-3-662-03875-8

[7] C. Gardiner and P. Zoller. ``Quantum noise''. Springer Berlin, Heidelberg. (2004).

[8] H. M. Wiseman and G. J. Milburn. ``Quantum measurement and control''. Cambridge University Press. (2009).
https://doi.org/10.1017/CBO9780511813948

[9] M. P. A. Fisher, V. Khemani, A. Nahum, and S. Vijay. ``Random quantum circuits''. Annu. Rev. Condens. Matter Phys. 14, 335–379 (2023).
https://doi.org/10.1146/annurev-conmatphys-031720-030658

[10] X. Cao, A. Tilloy, and A. De Luca. ``Entanglement in a fermion chain under continuous monitoring''. SciPost Phys. 7, 024 (2019).
https://doi.org/10.21468/SciPostPhys.7.2.024

[11] Y. Li, X. Chen, and M. P. A. Fisher. ``Quantum Zeno effect and the many-body entanglement transition''. Phys. Rev. B 98, 205136 (2018).
https://doi.org/10.1103/PhysRevB.98.205136

[12] Y. Li, X. Chen, and M. P. A. Fisher. ``Measurement-driven entanglement transition in hybrid quantum circuits''. Phys. Rev. B 100, 134306 (2019).
https://doi.org/10.1103/PhysRevB.100.134306

[13] B. Skinner, J. Ruhman, and A. Nahum. ``Measurement-induced phase transitions in the dynamics of entanglement''. Phys. Rev. X 9, 031009 (2019).
https://doi.org/10.1103/PhysRevX.9.031009

[14] A. Zabalo, M. J. Gullans, J. H. Wilson, S. Gopalakrishnan, D. A. Huse, and J. H. Pixley. ``Critical properties of the measurement-induced transition in random quantum circuits''. Phys. Rev. B 101, 060301 (2020).
https://doi.org/10.1103/PhysRevB.101.060301

[15] R. Vasseur, A. C. Potter, Y.-Z. You, and A. W. W. Ludwig. ``Entanglement transitions from holographic random tensor networks''. Phys. Rev. B 100, 134203 (2019).
https://doi.org/10.1103/PhysRevB.100.134203

[16] F. Barratt, U. Agrawal, S. Gopalakrishnan, D. A. Huse, R. Vasseur, and A. C. Potter. ``Field Theory of Charge Sharpening in Symmetric Monitored Quantum Circuits''. Phys. Rev. Lett. 129, 120604 (2022).
https://doi.org/10.1103/PhysRevLett.129.120604

[17] S. Choi, Y. Bao, X.-L. Qi, and E. Altman. ``Quantum error correction in scrambling dynamics and measurement-induced phase transition''. Phys. Rev. Lett. 125, 030505 (2020).
https://doi.org/10.1103/PhysRevLett.125.030505

[18] Y. Bao, S. Choi, and E. Altman. ``Theory of the phase transition in random unitary circuits with measurements''. Phys. Rev. B 101, 104301 (2020).
https://doi.org/10.1103/PhysRevB.101.104301

[19] M. J. Gullans and D. A. Huse. ``Dynamical purification phase transition induced by quantum measurements''. Phys. Rev. X 10, 041020 (2020).
https://doi.org/10.1103/PhysRevX.10.041020

[20] M. J. Gullans and D. A. Huse. ``Scalable probes of measurement-induced criticality''. Phys. Rev. Lett. 125, 070606 (2020).
https://doi.org/10.1103/PhysRevLett.125.070606

[21] U. Agrawal, A. Zabalo, K. Chen, J. H. Wilson, A. C. Potter, J. H. Pixley, S. Gopalakrishnan, and R. Vasseur. ``Entanglement and Charge-Sharpening Transitions in U(1) Symmetric Monitored Quantum Circuits''. Phys. Rev. X 12, 041002 (2022).
https://doi.org/10.1103/PhysRevX.12.041002

[22] M. Block, Y. Bao, S. Choi, E. Altman, and N. Y. Yao. ``Measurement-induced transition in long-range interacting quantum circuits''. Phys. Rev. Lett. 128, 010604 (2022).
https://doi.org/10.1103/PhysRevLett.128.010604

[23] S. Sharma, X. Turkeshi, R. Fazio, and M. Dalmonte. ``Measurement-induced criticality in extended and long-range unitary circuits''. SciPost Phys. Core 5, 023 (2022).
https://doi.org/10.21468/SciPostPhysCore.5.2.023

[24] T. Müller, S. Diehl, and M. Buchhold. ``Measurement-induced dark state phase transitions in long-ranged fermion systems''. Phys. Rev. Lett. 128, 010605 (2022).
https://doi.org/10.1103/PhysRevLett.128.010605

[25] T. Minato, K. Sugimoto, T. Kuwahara, and K. Saito. ``Fate of measurement-induced phase transition in long-range interactions''. Phys. Rev. Lett. 128, 010603 (2022).
https://doi.org/10.1103/PhysRevLett.128.010603

[26] P. Sierant, G. Chiriacò, F. M. Surace, S. Sharma, X. Turkeshi, M. Dalmonte, R. Fazio, and G. Pagano. ``Dissipative Floquet Dynamics: from Steady State to Measurement Induced Criticality in Trapped-ion Chains''. Quantum 6, 638 (2022).
https://doi.org/10.22331/q-2022-02-02-638

[27] G. Passarelli, X. Turkeshi, A. Russomanno, P. Lucignano, M. Schirò, and R. Fazio. ``Many-Body Dynamics in Monitored Atomic Gases without Postselection Barrier''. Phys. Rev. Lett. 132, 163401 (2024).
https://doi.org/10.1103/PhysRevLett.132.163401

[28] P. Sierant, M. Schirò, M. Lewenstein, and X. Turkeshi. ``Measurement-induced phase transitions in $(d+1)$-dimensional stabilizer circuits''. Phys. Rev. B 106, 214316 (2022).
https://doi.org/10.1103/PhysRevB.106.214316

[29] K. Klocke and M. Buchhold. ``Majorana Loop Models for Measurement-Only Quantum Circuits''. Phys. Rev. X 13, 041028 (2023).
https://doi.org/10.1103/PhysRevX.13.041028

[30] O. Lunt, M. Szyniszewski, and A. Pal. ``Measurement-induced criticality and entanglement clusters: A study of one-dimensional and two-dimensional clifford circuits''. Phys. Rev. B 104, 155111 (2021).
https://doi.org/10.1103/PhysRevB.104.155111

[31] X. Turkeshi, R. Fazio, and M. Dalmonte. ``Measurement-induced criticality in $(2+1)$-dimensional hybrid quantum circuits''. Phys. Rev. B 102, 014315 (2020).
https://doi.org/10.1103/PhysRevB.102.014315

[32] P. Sierant and X. Turkeshi. ``Universal behavior beyond multifractality of wave functions at measurement-induced phase transitions''. Phys. Rev. Lett. 128, 130605 (2022).
https://doi.org/10.1103/PhysRevLett.128.130605

[33] A. Nahum, S. Roy, B. Skinner, and J. Ruhman. ``Measurement and entanglement phase transitions in all-to-all quantum circuits, on quantum trees, and in landau-ginsburg theory''. PRX Quantum 2, 010352 (2021).
https://doi.org/10.1103/PRXQuantum.2.010352

[34] M. Szyniszewski, A. Romito, and H. Schomerus. ``Universality of Entanglement Transitions from Stroboscopic to Continuous Measurements''. Phys. Rev. Lett. 125, 210602 (2020).
https://doi.org/10.1103/PhysRevLett.125.210602

[35] A. C. Potter and R. Vasseur. ``Entanglement dynamics in hybrid quantum circuits''. Pages 211–249. Springer International Publishing. Cham (2022).
https://doi.org/10.1007/978-3-031-03998-0_9

[36] O. Lunt, J. Richter, and A. Pal. ``Quantum simulation using noisy unitary circuits and measurements''. Pages 251–284. Springer International Publishing. Cham (2022).
https://doi.org/10.1007/978-3-031-03998-0_10

[37] A. Russomanno, G. Piccitto, and D. Rossini. ``Entanglement transitions and quantum bifurcations under continuous long-range monitoring''. Phys. Rev. B 108, 104313 (2023).
https://doi.org/10.1103/physrevb.108.104313

[38] J. Preskill. ``Quantum Computing in the NISQ era and beyond''. Quantum 2, 79 (2018).
https://doi.org/10.22331/q-2018-08-06-79

[39] J. Fraxanet, T. Salamon, and M. Lewenstein. ``The Coming Decades of Quantum Simulation''. Pages 85––125. Springer International Publishing. (2023).
https://doi.org/10.1007/978-3-031-32469-7_4

[40] K. J. Ferris, A. J. Rasmusson, N. T. Bronn, and O. Lanes. ``Quantum Simulation on Noisy Superconducting Quantum Computers'' (2022). arXiv:2209.02795.
arXiv:2209.02795

[41] S. J. Garratt, Z. Weinstein, and E. Altman. ``Measurements Conspire Nonlocally to Restructure Critical Quantum States''. Phys. Rev. X 13, 021026 (2023).
https://doi.org/10.1103/PhysRevX.13.021026

[42] Z. Weinstein, R. Sajith, E. Altman, and S. J. Garratt. ``Nonlocality and entanglement in measured critical quantum Ising chains''. Phys. Rev. B 107, 245132 (2023).
https://doi.org/10.1103/PhysRevB.107.245132

[43] J. Y. Lee, C.-M. Jian, and C. Xu. ``Quantum Criticality Under Decoherence or Weak Measurement''. PRX Quantum 4, 030317 (2023).
https://doi.org/10.1103/PRXQuantum.4.030317

[44] G.-Y. Zhu, N. Tantivasadakarn, A. Vishwanath, S. Trebst, and R. Verresen. ``Nishimori's Cat: Stable Long-Range Entanglement from Finite-Depth Unitaries and Weak Measurements''. Phys. Rev. Lett. 131, 200201 (2023).
https://doi.org/10.1103/physrevlett.131.200201

[45] Z. Yang, D. Mao, and C.-M. Jian. ``Entanglement in a one-dimensional critical state after measurements''. Phys. Rev. B 108, 165120 (2023).
https://doi.org/10.1103/PhysRevB.108.165120

[46] X. Sun, H. Yao, and S.-K. Jian. ``New critical states induced by measurement'' (2023). arXiv:2301.11337.
arXiv:2301.11337

[47] K. Su, N. Myerson-Jain, C. Wang, C.-M. Jian, and C. Xu. ``Higher-Form Symmetries under Weak Measurement''. Phys. Rev. Lett. 132, 200402 (2024).
https://doi.org/10.1103/PhysRevLett.132.200402

[48] S. Murciano, P. Sala, Y. Liu, R. S. K. Mong, and J. Alicea. ``Measurement-Altered Ising Quantum Criticality''. Phys. Rev. X 13, 041042 (2023).
https://doi.org/10.1103/PhysRevX.13.041042

[49] O. Alberton, M. Buchhold, and S. Diehl. ``Entanglement transition in a monitored free-fermion chain: From extended criticality to area law''. Phys. Rev. Lett. 126, 170602 (2021).
https://doi.org/10.1103/PhysRevLett.126.170602

[50] M. Buchhold, Y. Minoguchi, A. Altland, and S. Diehl. ``Effective Theory for the Measurement-Induced Phase Transition of Dirac Fermions''. Phys. Rev. X 11, 041004 (2021).
https://doi.org/10.1103/PhysRevX.11.041004

[51] B. Ladewig, S. Diehl, and M. Buchhold. ``Monitored open fermion dynamics: Exploring the interplay of measurement, decoherence, and free hamiltonian evolution''. Phys. Rev. Res. 4, 033001 (2022).
https://doi.org/10.1103/PhysRevResearch.4.033001

[52] X. Turkeshi, L. Piroli, and M. Schirò. ``Density and current statistics in boundary-driven monitored fermionic chains''. Phys. Rev. B 109, 144306 (2024).
https://doi.org/10.1103/PhysRevB.109.144306

[53] X. Turkeshi, L. Piroli, and M. Schiró. ``Enhanced entanglement negativity in boundary-driven monitored fermionic chains''. Phys. Rev. B 106, 024304 (2022).
https://doi.org/10.1103/PhysRevB.106.024304

[54] X. Turkeshi, A. Biella, R. Fazio, M. Dalmonte, and M. Schiró. ``Measurement-induced entanglement transitions in the quantum Ising chain: From infinite to zero clicks''. Phys. Rev. B 103, 224210 (2021).
https://doi.org/10.1103/PhysRevB.103.224210

[55] X. Turkeshi and M. Schiró. ``Entanglement and correlation spreading in non-Hermitian spin chains''. Phys. Rev. B 107, L020403 (2023).
https://doi.org/10.1103/PhysRevB.107.L020403

[56] X. Turkeshi, M. Dalmonte, R. Fazio, and M. Schirò. ``Entanglement transitions from stochastic resetting of non-Hermitian quasiparticles''. Phys. Rev. B 105, L241114 (2022).
https://doi.org/10.1103/PhysRevB.105.L241114

[57] A. Paviglianiti and A. Silva. ``Multipartite entanglement in the measurement-induced phase transition of the quantum Ising chain''. Phys. Rev. B 108, 184302 (2023).
https://doi.org/10.1103/PhysRevB.108.184302

[58] C. Zerba and A. Silva. ``Measurement phase transitions in the no-click limit as quantum phase transitions of a non-hermitean vacuum''. SciPost Phys. Core 6, 051 (2023).
https://doi.org/10.21468/SciPostPhysCore.6.3.051

[59] G. Piccitto, A. Russomanno, and D. Rossini. ``Entanglement transitions in the quantum Ising chain: A comparison between different unravelings of the same Lindbladian''. Phys. Rev. B 105, 064305 (2022).
https://doi.org/10.1103/PhysRevB.105.064305

[60] E. Tirrito, A. Santini, R. Fazio, and M. Collura. ``Full counting statistics as probe of measurement-induced transitions in the quantum Ising chain''. SciPost Phys. 15, 096 (2023).
https://doi.org/10.21468/SciPostPhys.15.3.096

[61] M. Coppola, E. Tirrito, D. Karevski, and M. Collura. ``Growth of entanglement entropy under local projective measurements''. Phys. Rev. B 105, 094303 (2022).
https://doi.org/10.1103/PhysRevB.105.094303

[62] M. Fava, L. Piroli, T. Swann, D. Bernard, and A. Nahum. ``Nonlinear Sigma Models for Monitored Dynamics of Free Fermions''. Phys. Rev. X 13, 041045 (2023).
https://doi.org/10.1103/PhysRevX.13.041045

[63] C.-M. Jian, H. Shapourian, B. Bauer, and A. W. W. Ludwig. ``Measurement-induced entanglement transitions in quantum circuits of non-interacting fermions: Born-rule versus forced measurements'' (2023). arXiv:2302.09094.
arXiv:2302.09094

[64] S.-K. Jian, C. Liu, X. Chen, B. Swingle, and P. Zhang. ``Measurement-Induced Phase Transition in the Monitored Sachdev-Ye-Kitaev Model''. Phys. Rev. Lett. 127, 140601 (2021).
https://doi.org/10.1103/PhysRevLett.127.140601

[65] I. Poboiko, P. Pöpperl, I. V. Gornyi, and A. D. Mirlin. ``Theory of Free Fermions under Random Projective Measurements''. Phys. Rev. X 13, 041046 (2023).
https://doi.org/10.1103/PhysRevX.13.041046

[66] H. Lóio, A. De Luca, J. De Nardis, and X. Turkeshi. ``Purification timescales in monitored fermions''. Phys. Rev. B 108, L020306 (2023).
https://doi.org/10.1103/PhysRevB.108.L020306

[67] A. Biella and M. Schiró. ``Many-Body Quantum Zeno Effect and Measurement-Induced Subradiance Transition''. Quantum 5, 528 (2021).
https://doi.org/10.22331/q-2021-08-19-528

[68] J. A. Maki, A. Berti, I. Carusotto, and A. Biella. ``Monte Carlo matrix-product-state approach to the false vacuum decay in the monitored quantum Ising chain''. SciPost Phys. 15, 152 (2023).
https://doi.org/10.21468/SciPostPhys.15.4.152

[69] W. W. Ho and S. Choi. ``Exact Emergent Quantum State Designs from Quantum Chaotic Dynamics''. Phys. Rev. Lett. 128, 060601 (2022).
https://doi.org/10.1103/PhysRevLett.128.060601

[70] P. W. Claeys and A. Lamacraft. ``Emergent quantum state designs and biunitarity in dual-unitary circuit dynamics''. Quantum 6, 738 (2022).
https://doi.org/10.22331/q-2022-06-15-738

[71] M. Ippoliti and W. W. Ho. ``Solvable model of deep thermalization with distinct design times''. Quantum 6, 886 (2022).
https://doi.org/10.22331/q-2022-12-29-886

[72] M. Ippoliti and W. W. Ho. ``Dynamical Purification and the Emergence of Quantum State Designs from the Projected Ensemble''. PRX Quantum 4, 030322 (2023).
https://doi.org/10.1103/PRXQuantum.4.030322

[73] M. Lucas, L. Piroli, J. De Nardis, and A. De Luca. ``Generalized deep thermalization for free fermions''. Phys. Rev. A 107, 032215 (2023).
https://doi.org/10.1103/PhysRevA.107.032215

[74] P. Łydżba, M. Mierzejewski, M. Rigol, and L. Vidmar. ``Generalized Thermalization in Quantum-Chaotic Quadratic Hamiltonians''. Phys. Rev. Lett. 131, 060401 (2023).
https://doi.org/10.1103/PhysRevLett.131.060401

[75] J. Choi, A. L. Shaw, I. S. Madjarov, X. Xie, R. Finkelstein, J. P. Covey, J. S. Cotler, D. K. Mark, H.-Y. Huang, A. Kale, H. Pichler, F. G. S. L. Brandão, S. Choi, and M. Endres. ``Preparing random states and benchmarking with many-body quantum chaos''. Nature 613, 468–473 (2023).
https://doi.org/10.1038/s41586-022-05442-1

[76] J. S. Cotler, D. K. Mark, H.-Y. Huang, F. Hernández, J. Choi, A. L. Shaw, M. Endres, and S. Choi. ``Emergent Quantum State Designs from Individual Many-Body Wave Functions''. PRX Quantum 4, 010311 (2023).
https://doi.org/10.1103/PRXQuantum.4.010311

[77] J. Surace and L. Tagliacozzo. ``Fermionic Gaussian states: an introduction to numerical approaches''. SciPost Phys. Lect. Notes 54 (2022).
https://doi.org/10.21468/SciPostPhysLectNotes.54

[78] N. Laflorencie. ``Quantum entanglement in condensed matter systems''. Phys. Rep. 646, 1–59 (2016).
https://doi.org/10.1016/j.physrep.2016.06.008

[79] L. Amico, R. Fazio, A. Osterloh, and V. Vedral. ``Entanglement in many-body systems''. Rev. Mod. Phys. 80, 517–576 (2008).
https://doi.org/10.1103/RevModPhys.80.517

[80] P. Calabrese and J. Cardy. ``Entanglement entropy and quantum field theory''. J. Stat. Mech.: Theory Exp. 2004, P06002 (2004).
https://doi.org/10.1088/1742-5468/2004/06/P06002

[81] P. Calabrese and J. Cardy. ``Entanglement entropy and conformal field theory''. J. Phys. A 42, 504005 (2009).
https://doi.org/10.1088/1751-8113/42/50/504005

[82] G. B. Mbeng, A. Russomanno, and G. E. Santoro. ``The quantum Ising chain for beginners''. SciPost Phys. Lect. Notes 82 (2024).
https://doi.org/10.21468/SciPostPhysLectNotes.82

[83] L. Pezzè, A. Smerzi, M. K. Oberthaler, R. Schmied, and P. Treutlein. ``Quantum metrology with nonclassical states of atomic ensembles''. Rev. Mod. Phys. 90, 035005 (2018).
https://doi.org/10.1103/RevModPhys.90.035005

[84] S. Pappalardi, A. Russomanno, A. Silva, and R. Fazio. ``Multipartite entanglement after a quantum quench''. J. Stat. Mech.: Theory Exp. 2017, 053104 (2017).
https://doi.org/10.1088/1742-5468/aa6809

[85] S. Pappalardi, A. Russomanno, B. Žunkovič, F. Iemini, A. Silva, and R. Fazio. ``Scrambling and entanglement spreading in long-range spin chains''. Phys. Rev. B 98, 134303 (2018).
https://doi.org/10.1103/PhysRevB.98.134303

[86] J.-Y. Desaules, F. Pietracaprina, Z. Papić, J. Goold, and S. Pappalardi. ``Extensive multipartite entanglement from su(2) quantum many-body scars''. Phys. Rev. Lett. 129, 020601 (2022).
https://doi.org/10.1103/PhysRevLett.129.020601

[87] M. Brenes, S. Pappalardi, J. Goold, and A. Silva. ``Multipartite Entanglement Structure in the Eigenstate Thermalization Hypothesis''. Phys. Rev. Lett. 124, 040605 (2020).
https://doi.org/10.1103/PhysRevLett.124.040605

[88] E. R. Caianiello and S. Fubini. ``On the Algorithm of Dirac spurs''. Il Nuovo Cimento 9, 1218–1226 (1952).
https://doi.org/10.1007/BF02782927

[89] E. Barouch and B. M. McCoy. ``Statistical Mechanics of the $XY$ Model. II. Spin-Correlation Functions''. Phys. Rev. A 3, 786–804 (1971).
https://doi.org/10.1103/PhysRevA.3.786

[90] H. Shapourian, K. Shiozaki, and S. Ryu. ``Partial time-reversal transformation and entanglement negativity in fermionic systems''. Phys. Rev. B 95, 165101 (2017).
https://doi.org/10.1103/PhysRevB.95.165101

[91] J. Eisert, V. Eisler, and Z. Zimborás. ``Entanglement negativity bounds for fermionic Gaussian states''. Phys. Rev. B 97, 165123 (2018).
https://doi.org/10.1103/PhysRevB.97.165123

[92] A. Neven, J. Carrasco, V. Vitale, C. Kokail, A. Elben, M. Dalmonte, P. Calabrese, P. Zoller, B. Vermersch, R. Kueng, and B. Kraus. ``Symmetry-resolved entanglement detection using partial transpose moments''. Inf. 7, 152 (2021).
https://doi.org/10.1038/s41534-021-00487-y

[93] V. Vitale, A. Elben, R. Kueng, A. Neven, J. Carrasco, B. Kraus, P. Zoller, P. Calabrese, B. Vermersch, and M. Dalmonte. ``Symmetry-resolved dynamical purification in synthetic quantum matter''. SciPost Phys. 12, 106 (2022).
https://doi.org/10.21468/SciPostPhys.12.3.106

[94] H. Shapourian, P. Ruggiero, S. Ryu, and P. Calabrese. ``Twisted and untwisted negativity spectrum of free fermions''. SciPost Phys. 7, 37 (2019).
https://doi.org/10.21468/SciPostPhys.7.3.037

[95] X. Turkeshi, P. Ruggiero, and P. Calabrese. ``Negativity spectrum in the random singlet phase''. Phys. Rev. B 101, 064207 (2020).
https://doi.org/10.1103/PhysRevB.101.064207

[96] P. Ruggiero and X. Turkeshi. ``Quantum information spreading in random spin chains''. Phys. Rev. B 106, 134205 (2022).
https://doi.org/10.1103/PhysRevB.106.134205

[97] S. Murciano, V. Vitale, M. Dalmonte, and P. Calabrese. ``Negativity Hamiltonian: An Operator Characterization of Mixed-State Entanglement''. Phys. Rev. Lett. 128, 140502 (2022).
https://doi.org/10.1103/PhysRevLett.128.140502

[98] S. Murciano, R. Bonsignori, and P. Calabrese. ``Symmetry decomposition of negativity of massless free fermions''. SciPost Phys. 10, 111 (2021).
https://doi.org/10.21468/SciPostPhys.10.5.111

[99] V. Alba and F. Carollo. ``Spreading of correlations in Markovian open quantum systems''. Phys. Rev. B 103, L020302 (2021).
https://doi.org/10.1103/PhysRevB.103.L020302

[100] V. Alba and F. Carollo. ``Logarithmic negativity in out-of-equilibrium open free-fermion chains: An exactly solvable case''. SciPost Phys. 15, 124 (2023).
https://doi.org/10.21468/SciPostPhys.15.3.124

[101] F. Rottoli, S. Murciano, E. Tonni, and P. Calabrese. ``Entanglement and negativity Hamiltonians for the massless Dirac field on the half line''. J. Stat. Mech.: Theory Exp. 2023, 013103 (2023).
https://doi.org/10.1088/1742-5468/acb262

[102] F. Rottoli, S. Murciano, and P. Calabrese. ``Finite temperature negativity Hamiltonians of the massless Dirac fermion''. J. High Energy Phys. 2023, 139 (2023).
https://doi.org/10.1007/jhep06(2023)139

[103] B. Groisman, S. Popescu, and A. Winter. ``Quantum, classical, and total amount of correlations in a quantum state''. Phys. Rev. A 72, 032317 (2005).
https://doi.org/10.1103/PhysRevA.72.032317

[104] C.-J. Lin, W. Ye, Y. Zou, S. Sang, and T. H. Hsieh. ``Probing sign structure using measurement-induced entanglement''. Quantum 7, 910 (2023).
https://doi.org/10.22331/q-2023-02-02-910

[105] Z. Cheng, R. Wen, S. Gopalakrishnan, R. Vasseur, and A. C. Potter. ``Universal structure of measurement-induced information in many-body ground states''. Phys. Rev. B 109, 195128 (2024).
https://doi.org/10.1103/PhysRevB.109.195128

[106] Y. Zhang and S. Gopalakrishnan. ``Nonlocal growth of quantum conditional mutual information under decoherence''. Phys. Rev. A 110, 032426 (2024).
https://doi.org/10.1103/PhysRevA.110.032426

[107] G. Vidal and R. F. Werner. ``Computable measure of entanglement''. Phys. Rev. A 65, 032314 (2002).
https://doi.org/10.1103/PhysRevA.65.032314

[108] M. Plenio and S. Virmani. ``An introduction to entanglement measures''. Quantum Information and Computation 7, 001–051 (2005).
https://doi.org/10.26421/QIC7.1-2-1

[109] X.-G. Wen. ``Quantum Field Theory of Many-Body Systems: From the Origin of Sound to an Origin of Light and Electrons''. Oxford University Press. (2007).
https://doi.org/10.1093/acprof:oso/9780199227259.001.0001

[110] F. Iglói and C. Monthus. ``Strong disorder RG approach of random systems''. Phys. Rep. 412, 277–431 (2005).
https://doi.org/10.1016/j.physrep.2005.02.006

[111] F. Iglói and C. Monthus. ``Strong disorder RG approach – a short review of recent developments''. J. B 91, 290 (2018).
https://doi.org/10.1140/epjb/e2018-90434-8

[112] A. Nahum and B. Skinner. ``Entanglement and dynamics of diffusion-annihilation processes with Majorana defects''. Phys. Rev. Res. 2, 023288 (2020).
https://doi.org/10.1103/PhysRevResearch.2.023288

[113] S. Gopalakrishnan and M. J. Gullans. ``Entanglement and Purification Transitions in Non-Hermitian Quantum Mechanics''. Phys. Rev. Lett. 126, 170503 (2021).
https://doi.org/10.1103/PhysRevLett.126.170503

[114] E. Granet, C. Zhang, and H. Dreyer. ``Volume-Law to Area-Law Entanglement Transition in a Nonunitary Periodic Gaussian Circuit''. Phys. Rev. Lett. 130, 230401 (2023).
https://doi.org/10.1103/PhysRevLett.130.230401

[115] B. Barch, N. Anand, J. Marshall, E. Rieffel, and P. Zanardi. ``Scrambling and operator entanglement in local non-Hermitian quantum systems''. Phys. Rev. B 108, 134305 (2023).
https://doi.org/10.1103/PhysRevB.108.134305

[116] X. Turkeshi, M. Dalmonte, R. Fazio, and M. Schirò. ``Erratum: Entanglement transitions from stochastic resetting of non-Hermitian quasiparticles [Phys. Rev. B 105, L241114 (2022)]''. Phys. Rev. B 107, 079901 (2023).
https://doi.org/10.1103/PhysRevB.107.079901

[117] Y. Le Gal, X. Turkeshi, and M. Schirò. ``Entanglement Dynamics in Monitored Systems and the Role of Quantum Jumps''. PRX Quantum 5, 030329 (2024).
https://doi.org/10.1103/PRXQuantum.5.030329

[118] C. Y. Leung, D. Meidan, and A. Romito. ``Theory of free fermions dynamics under partial post-selected monitoring'' (2023). arXiv:2312.14022.
arXiv:2312.14022

[119] B. Xing, X. Turkeshi, M. Schiró, R. Fazio, and D. Poletti. ``Interactions and integrability in weakly monitored Hamiltonian systems''. Phys. Rev. B 109, L060302 (2024).
https://doi.org/10.1103/PhysRevB.109.L060302

[120] H. M. Wiseman. ``Quantum trajectories and quantum measurement theory''. Quantum Semiclass. Opt. 8, 205–222 (1996).
https://doi.org/10.1088/1355-5111/8/1/015

[121] S. E. Ahnert and M. C. Payne. ``General implementation of all possible positive-operator-value measurements of single-photon polarization states''. Phys. Rev. A 71, 012330 (2005).
https://doi.org/10.1103/physreva.71.012330

[122] T. A. Brun. ``A simple model of quantum trajectories''. Am. J. of Phys. 70, 719–737 (2002).
https://doi.org/10.1119/1.1475328

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