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Many-Body Quantum Zeno Effect and Measurement-Induced Subradiance Transition

Alberto Biella, Marco Schiró

arXiv:2011.11620v3quant-phcond-mat.stat-mech

TL;DR

The paper examines how stochastic measurements competing with unitary many-body dynamics produce the quantum Zeno effect in a quantum Ising chain. It studies no-click evolution with a non-Hermitian Hamiltonian and finds a sharp transition associated with subradiance, although the transition is invisible in average dynamics.

  • Problem

    The paper asks how a sharp many-body quantum Zeno onset emerges from competing unitary dynamics and stochastic measurements, and whether it appears in average dynamics.

    Source: §1 · p. 2

  • Method

    The authors study a quantum Ising chain with local measurements, using no-click evolution, exact numerics, and fermionization of the resulting non-Hermitian transverse-field Ising model.

    Source: §2 · p. 2, §1 · p. 2, §5 · p. 7

  • Results

    The many-body Zeno effect appears sharply, with a non-analytic slowest decay mode and a closing gap in the quasiparticle spectrum.

    Source: §5 · p. 7

  • Takeaways & Limitations

    Rare fluctuations, captured by no-click evolution, encode the transition, whereas the average dynamics do not.

    Source: §3 · p. 3, §5 · p. 7

    For the considered Born-rule protocol, the mapping cannot be used to exactly solve the problem.

    Source: §1 · p. 2

Abstract

from arXiv · show

It is well known that by repeatedly measuring a quantum system it is possible to completely freeze its dynamics into a well defined state, a signature of the quantum Zeno effect. Here we show that for a many-body system evolving under competing unitary evolution and variable-strength measurements the onset of the Zeno effect takes the form of a sharp phase transition. Using the Quantum Ising chain with continuous monitoring of the transverse magnetization as paradigmatic example we show that for weak measurements the entanglement produced by the unitary dynamics remains protected, and actually enhanced by the monitoring, while only above a certain threshold the system is sharply brought into an uncorrelated Zeno state. We show that this transition is invisible to the average dynamics, but encoded in the rare fluctuations of the stochastic measurement process, which we show to be perfectly captured by a non-Hermitian Hamiltonian which takes the form of a Quantum Ising model in an imaginary valued transverse field. We provide analytical results based on the fermionization of the non-Hermitian Hamiltonian in supports of our exact numerical calculations.

1 Introduction

The introduction places the many-body quantum Zeno effect among measurement-driven phenomena and asks how it emerges when unitary dynamics competes with stochastic measurements. It presents a sharp transition in rare measurement histories and relates it to a subradiance transition in a non-Hermitian Ising model.

1 Introduction

  • Frequent projective measurements can localize a system through the quantum Zeno effect, while many-body measurement protocols have also been linked to entanglement phase transitions.Continuous monitoring and variable-strength weak measurements use POVMs and produce stochastic backaction.Source: §1 · pp. 1–2
  • For a quantum Ising chain with continuous monitoring of local transverse magnetization, the many-body Zeno effect appears sharply above a critical measurement strength.The competing unitary and measurement processes act on noncommuting spin components.Source: §1 · p. 2
  • The transition is encoded in rare stochastic realizations, not average dynamics, and the no-click evolution is described by a non-Hermitian transverse-field Ising model.The model’s subradiance transition is marked by a non-analytic change in the eigenstate with the smallest imaginary part.Source: §1 · p. 2
  • Fermionization provides analytical arguments supporting exact numerics, while the many-body behavior arises from competing entangling dynamics and local measurements.The unmeasured quantum Ising chain itself is integrable through free-fermion techniques.Source: §1 · p. 2
  • For the considered Born-rule measurement protocol, the mapping cannot be used to exactly solve the problem.The passage contrasts this with other stochastic settings that also show genuine many-body signatures.Source: §1 · p. 2

2 The model and the measurement protocol

The protocol combines quantum Ising unitary evolution with stochastic local measurements, beginning from an uncorrelated product state. It is designed to examine how unitary dynamics builds correlations while local measurements suppress them.

2 The model and the measurement protocol

  • The stochastic protocol combines unitary evolution generated by a quantum Ising model with simultaneous local measurements implemented by a normalized POVM Kraus operator.Measurements occur on each site with probability psite.Source: §2 · p. 2
  • The protocol starts from an uncorrelated product state with all spins aligned along the z-axis to study entanglement from nonlocal unitary evolution and correlation loss from local measurements.The authors note that nonlocal measurement processes are an alternative scenario expected to stabilize correlated entangled states.Source: §2 · p. 2
  • Implementing the alternative scenario with nonlocal measurement processes would be theoretically treatable but experimentally very demanding..Source: §2 · pp. 2–3
  • The state is normalized after each protocol step because the measurement operator is non-unitary and leaves the wavefunction unnormalized..Source: §2 · p. 3

3 Results

The results reveal a sharp measurement-induced Zeno transition: weak monitoring can enhance entanglement, while strong monitoring favors an uncorrelated product state. The transition is expressed in rare stochastic trajectories and captured by a non-Hermitian Ising description, rather than by the average dynamics.

3 Results

  • A sharp transition separates weak-measurement dynamics from the strong-measurement Zeno product state, appearing in rare trajectories while average dynamics shows only a smooth crossover.The dimensionless control parameter g compares the measurement frequency with the unitary dynamics.Source: §3 · p. 3

3.1 Measurement-induced entanglement

  • For small finite g, single stochastic trajectories can exceed the unitary entanglement entropy, while excess entanglement peaks near g∼1 and then rapidly declines.The reported measurement-induced entanglement ΘE is measured relative to the unitary case.Source: Figure 2 · p. 3, §3.1 · pp. 3–4

3.2 Onset of many-body quantum Zeno effect

  • For g≲1 the overlap distribution peaks at zero with fat tails; above g≈1 a second peak emerges and becomes dominant at larger g.The secondary peak shifts toward overlap values of order one in the deep Zeno regime.Source: Figure 3 · p. 4, §3.2 · p. 4

3.3 No-click dynamics and subradiance transition

  • For g>gc, f∼g^2 places the subradiant state essentially in the Zeno subspace; below gc, Osub,T scales as L^-β with β=0.049±0.003.At large g the overlap depends weakly on system size.Source: Figure 4 · p. 5, §3.3 · pp. 5–6
  • The no-click evolution is governed by a non-Hermitian TFIM with an imaginary transverse field, whose long-time state is subradiant and undergoes a transition at a critical g.For g>gc, the non-Hermitian overlap agrees with the secondary-peak overlap from the full stochastic dynamics.Source: §3.3 · p. 5
  • The real quasiparticle spectrum is gapped at k=±π/2 below gc and gapless above it, while the imaginary part is negative for all k above gc.The imaginary part has maxima at k=±π/2 and vanishes there below gc.Source: Figure 5 · p. 6

4 Analytical approach

The non-Hermitian TFIM is diagonalized by mapping it to fermions, revealing a quasiparticle-spectrum transition at the critical coupling. Its subradiant eigenstates and the limitations of average dynamics clarify how the transition appears in this analysis.

4 Analytical approach

  • The non-Hermitian TFIM is diagonalized exactly using a Jordan-Wigner representation and a generalized Bogoliubov transformation.The quasiparticle spectrum and many-body eigenstates follow from this fermionic mapping.Source: §4 · p. 6
  • At gc = 4, the quasiparticle real-part gap closes for g < gc, while for g > gc the real part stays zero and the imaginary part becomes nonzero.The gap closing is stated in the thermodynamic limit.Source: §4 · p. 6
  • The subradiant states’ structure factor is strongly peaked near k ≃ π/2 before the transition.Source: §4 · p. 6
  • At weak coupling the subradiant eigenvalue has a g-independent real part and a linearly growing imaginary part; at strong coupling it is purely imaginary and tends to zero.The reported asymptotic forms are εsub ∼ 1 − ig/gc for g/gc ≪ 1 and εsub ∼ −2igc/g for g/gc ≫ 1.Source: §4 · p. 6
  • The GKLS dynamics leads to a fully depolarized steady state that retains no statistical features of the studied observables, although its spectrum could reveal the Zeno-onset timescale.Source: §4 · p. 7

5 Conclusion

The conclusion identifies a sharp many-body Zeno transition in a monitored quantum Ising chain and relates it to a subradiance transition in a non-Hermitian model. It points to spatially inhomogeneous monitoring, different initial conditions, higher dimensions, and interacting unmeasured dynamics as future directions.

5 Conclusion

  • Strong continuous monitoring of transverse magnetization sharply freezes the quantum Ising chain as the measurement-to-interaction ratio increases.Source: §5 · p. 7
  • The transition is controlled by rare stochastic fluctuations and is captured by the no-click evolution, which reduces to a non-Hermitian TFIM in the continuous-measurement limit.Source: §5 · p. 7
  • Exact numerics and fermion mapping identify a sharp subradiance transition marked by non-analyticity of the slowest decay mode and a quasiparticle gap closing.Source: §5 · p. 7
  • Future directions include spatially inhomogeneous measurements, different initial conditions, higher dimensions, and models with interacting unmeasured dynamics.Source: §5 · p. 7

APPENDICES

The appendix maps the non-Hermitian TFIM to fermions and diagonalizes it in a fixed parity sector. This yields the quasiparticle spectrum and identifies the strong-measurement subradiant state.

A Diagonalization of the non-Hermitian quantum Ising model

B Excess entanglement

Excess entanglement curves for different measurement densities collapse when plotted against g psite, showing that density and measurement strength play equivalent roles for this quantity.

B Excess entanglement

  • The excess-entanglement curves collapse when plotted against g psite, making that product the control parameter for this quantity.The curves are shown for several values of psite.Source: B Excess entanglement · pp. 8–9, Figure 6 · p. 8
  • Measurement density and measurement strength play equivalent roles in generating excess entanglement.The product is ptot = psite pmeas.Source: B Excess entanglement · p. 9

C Validity of the non-Hermitian effective Hamiltonian

The effective non-Hermitian Hamiltonian reproduces the full no-click entanglement-entropy evolution exactly for g ≲ 1; above that threshold, the agreement remains qualitative but is not quantitatively exact.

C Validity of the non-Hermitian effective Hamiltonian

D Onset of the Zeno effect

The local-magnetization distribution changes sharply with measurement strength: it broadens across possible values at weak strength, then concentrates near the measurement eigenstate as g increases.

D Onset of the Zeno effect

E Spectral signatures of the superradiance transition

The section characterizes the subradiance transition through the subradiant width and finite-size scaling of its critical point.

E Spectral signatures of the superradiance transition

F k-mode occupancy

Across the transition, the subradiant-state k-mode distribution becomes sharply depleted, while the anti-subradiant state remains concentrated around k = π/2. The section defines these occupancies using modes fixed by open boundary conditions.

F k-mode occupancy

  • The k-mode occupancies use k = nπ/(L + 1), n = 1, . . . , L, as imposed by open boundary conditions.The spin operators are Fourier transformed to define the occupancies.Source: F k-mode occupancy · p. 10
  • After the transition, the subradiant-state k-mode distribution is sharply depleted as g/gc → ∞ and |sub⟩ → |T⟩.Figure 10 shows the k-space populations as γ varies at Jx = 1.Source: F k-mode occupancy · p. 10, Figure 10 · p. 10
  • The anti-subradiant state remains mainly populated at k = π/2 as g varies across the transition.It is the many-body state that becomes degenerate with the subradiant state at the transition.Source: F k-mode occupancy · p. 10
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