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arxiv: 2506.22755 · v2 · submitted 2025-06-28 · 🪐 quant-ph · cond-mat.dis-nn· cond-mat.stat-mech

Scaling Laws of Quantum Information Lifetime in Monitored Quantum Dynamics

Bingzhi Zhang , Fangjun Hu , Runzhe Mo , Tianyang Chen , Hakan E. T\"ureci , Quntao Zhuang This is my paper

Pith reviewed 2026-05-19 07:48 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.dis-nncond-mat.stat-mech
keywords quantum information lifetimemonitored quantum dynamicsmid-circuit measurementsscaling lawsHaar random unitariesquantum mutual informationrandom unitary circuitschaotic Hamiltonians
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The pith

Continuous monitoring of the environment lets quantum information lifetime grow exponentially with system size, independent of bath size.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that quantum information, starting from a maximally entangled state with a reference, survives for times that increase exponentially with the number of system qubits when the bath undergoes continuous mid-circuit measurements. This exponential protection holds for typical random unitary dynamics and persists across both circuit models and chaotic Hamiltonians, even when the bath itself is large. In the absence of any bath monitoring the lifetime instead grows at most linearly with system size and shrinks as the bath grows. The authors also identify a two-scale decay regime under partial monitoring, where information loss is logarithmic at short times and linear at long times. These scaling laws are verified numerically and on current quantum hardware.

Core claim

Starting from a maximally entangled state, the lifetime of quantum mutual information between system and reference scales exponentially with system size under continuous bath monitoring via mid-circuit measurements, for typical Haar random unitaries; without monitoring the lifetime grows at most linearly with system size and inversely with bath size.

What carries the argument

The scaling of quantum mutual information lifetime under continuous mid-circuit monitoring of the bath, proven analytically for Haar-random unitaries and checked in random circuits and chaotic Hamiltonians.

If this is right

  • Quantum diffusion models and reservoir computing can retain input information for times that grow with system size rather than saturating.
  • Quantum communication protocols gain a route to protect entanglement even when the environment is large.
  • Monitored circuits in the weak-measurement limit inherit the same exponential lifetime scaling.
  • Hardware experiments on IBM devices already confirm a measurable gap between monitored and unmonitored information persistence.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Monitoring strategies could be combined with existing error-correction codes to extend coherence times without increasing code size.
  • The two-scale transition may appear in other monitored many-body systems beyond circuits, offering a diagnostic for measurement strength.
  • Larger-scale simulations or future hardware could test whether the exponential scaling survives when the unitaries are only approximately random.

Load-bearing premise

The analytic proof requires the dynamics to be typical of Haar-random unitaries, which assumes sufficient randomness that may not hold for every possible Hamiltonian.

What would settle it

A direct numerical or experimental measurement showing that monitored lifetime remains linear or sub-linear with system size in a chaotic Hamiltonian system would contradict the central scaling claim.

Figures

Figures reproduced from arXiv: 2506.22755 by Bingzhi Zhang, Fangjun Hu, Hakan E. T\"ureci, Quntao Zhuang, Runzhe Mo, Tianyang Chen.

Figure 1
Figure 1. Figure 1: Schematic plot of quantum dynamics with mid-circuit measurement. In (a), initial input [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: a shows numerical simulations of the measurement-conditioned QMI dynamics in the model where each step we apply a random unitary sampled from Haar ensemble (dark blue dots), and its decay behavior versus time steps is well characterized by our theoretical lower bound (red dashed) in Eq. (9) as long as the bound remains positive. The gap between theoretical lower bound and numerical simulation arises from t… view at source ↗
Figure 3
Figure 3. Figure 3: Measurement-unconditioned QMI I(R : At) in unmonitored dynamics with reset. In (a) we plot numerical simulation results of I(R : At) of a Bell initial state with random Haar unitary (dark blue dots) and fixed Ising Hamiltonian evolution (light blue dots) in a system of NA = 5, NB = 1 qubits. Errorbars represent sample fluctuations of Haar unitary implementations. Red and orange lines represent the exact an… view at source ↗
Figure 4
Figure 4. Figure 4: Lifetime of measurement-unconitioned QMI [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Measurement-unconditioned QMI in unmonitored [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Reduction from the reservoir computing paradigm [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Equivalence of the monitored quantum circuit in the [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Connection from quantum dynamics with mid￾circuit measurements in unmonitored and monitored cases to entanglement-assisted communication without (top) and with (bottom) environment assistance. sender A0 and environment E = B0 · · · Bt−1 which is a joint system of expanded baths initialized in |0 ⊗NBt ⟩E . A reference system R to purify the input A0 is sent through the identity channel as the entangled anci… view at source ↗
Figure 9
Figure 9. Figure 9: QMI dynamics in monitored / unmonitored dy [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Dynamics of measurement-conditioned and uncon [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Comparisons on dynamics of QMI and the corre [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Measurement-conditioned QMI I(R : At|z) in monitored residual dynamics. The initial state |ψ⟩RA0 is chosen to be the late-time conditional state in dynamics with Bell initial state. In (a), we plot the normalized QMI dynamics for NA = 1 to 6 (light to dark). In (b), blue dots correspond to the estimated lifetime from numerical simulation in (a) for ϵ = 1/4 and red dots represent the scaling of 1/v where v… view at source ↗
Figure 13
Figure 13. Figure 13: Equivalent relation on the quantum dynamics with mid-circuit measurements and zero state reset via expanding the [PITH_FULL_IMAGE:figures/full_fig_p019_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Tensor network representation of Haar-averaged pseudo measurement-conditioned purity of [PITH_FULL_IMAGE:figures/full_fig_p021_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Equivalent relation on the quantum dynamics with mid-circuit measurements and [PITH_FULL_IMAGE:figures/full_fig_p024_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Tensor network representation of Haar-averaged purity for reduced states (a) [PITH_FULL_IMAGE:figures/full_fig_p026_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Equivalent relation on the quantum dynamics with mid-circuit measurements and diagonal-mixed-state reset via [PITH_FULL_IMAGE:figures/full_fig_p029_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Tensor network representation of Haar-averaged purity for reduced states (a) [PITH_FULL_IMAGE:figures/full_fig_p031_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Dynamics of normalized measurement-conditioned QMI [PITH_FULL_IMAGE:figures/full_fig_p034_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Dynamics of normalized measurement-conditioned QMI [PITH_FULL_IMAGE:figures/full_fig_p034_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Dynamics of normalized measurement-unconditioned QMI [PITH_FULL_IMAGE:figures/full_fig_p035_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: (a1)-(a5) Eigenspectrum of P0. Blue, green and orange dots indicate the eigenstates ρfix, σ1, σmid to be considered. (b1)-(b5) Normalized bipartite entanglement entropy of Hamiltonian H eigenstates versus normalized eigenenergies. (c1)-(c5) Normalized norm of overlap between eigenstates of P0 and eigenstates of Hamiltonian versus normalized eigenenergies. In all cases, we have bath NB = 2. scrambling mode… view at source ↗
Figure 23
Figure 23. Figure 23: Purity of post-measurement conditional state at different time steps in noisy simulation. Here we perform simulation [PITH_FULL_IMAGE:figures/full_fig_p036_23.png] view at source ↗
read the original abstract

Quantum information is typically fragile under measurements and environmental coupling. Remarkably, we find that its lifetime can scale exponentially with system size when the environment is continuously monitored via mid-circuit measurements -- regardless of bath size. Starting from a maximally entangled state with a reference, we analytically prove this exponential scaling for typical Haar random unitaries and confirm it through numerical simulations in both random unitary circuits and chaotic Hamiltonian systems. In the absence of bath monitoring, the lifetime exhibits a markedly different scaling: it grows at most linearly -- or remains constant -- with system size and decays inversely with the bath size. We further extend our findings numerically to a broad class of initial states. In the intermediate regime of partial monitoring, we identify and prove a two-scale transition, where the QMI decays logarithmically at microscopic time scales but linearly at macroscopic time scales.} We discuss implications for {monitored quantum circuits in the weak measurement limit, quantum algorithms such as quantum diffusion models and quantum reservoir computing, and quantum communication. Finally, we experimentally verify the gap of persisted information on IBM Quantum hardwares.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that continuous mid-circuit monitoring of the environment in quantum dynamics protects quantum information, yielding a lifetime (via quantum mutual information with a reference) that scales exponentially with system size independent of bath size. This is analytically proven for typical Haar-random unitaries starting from maximally entangled states, numerically confirmed in random unitary circuits and chaotic Hamiltonians, extended to other initial states, and shown to exhibit a two-scale (logarithmic then linear) transition under partial monitoring. Without monitoring the scaling is at most linear/constant and inversely dependent on bath size; experimental verification on IBM Quantum hardware is also reported.

Significance. If the central claims hold, the work would be significant for monitored quantum dynamics and quantum information science. The analytical proof for Haar-random unitaries supplies a rigorous, parameter-free foundation, while the numerical checks across circuit and Hamiltonian models plus hardware experiments add reproducibility and falsifiability. The reported independence from bath size and implications for quantum algorithms (diffusion models, reservoir computing) and communication would be noteworthy contributions if the scope beyond maximal scrambling is clarified.

major comments (2)
  1. [Abstract / Analytical Proof] Abstract and analytical proof section: the headline claim is for monitored quantum dynamics in general, yet the analytic result is restricted to typical Haar-random unitaries (maximal scrambling). The manuscript must supply either an extension argument or explicit caveats showing why the protection mechanism survives for non-random or weakly chaotic Hamiltonians, as the numerical evidence alone cannot establish this.
  2. [Numerical Simulations] Numerical simulations for chaotic Hamiltonians: system sizes are modest by necessity; the manuscript should include a finite-size scaling collapse or explicit comparison of exponential versus power-law fits (with R² or similar metrics) to demonstrate that the observed growth is not consistent with sub-exponential alternatives.
minor comments (2)
  1. [Introduction] Define QMI and all acronyms at first use; clarify the precise definition of 'lifetime' (e.g., time to reach a fixed QMI threshold) in the main text.
  2. [Experimental Verification] Experimental section: specify the IBM Quantum device names, circuit depths, number of shots, and any error-mitigation protocol used to support the hardware verification claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We address each major comment below and will revise the manuscript to improve clarity on the scope of the analytic results and to strengthen the numerical evidence.

read point-by-point responses

  1. Referee: [Abstract / Analytical Proof] Abstract and analytical proof section: the headline claim is for monitored quantum dynamics in general, yet the analytic result is restricted to typical Haar-random unitaries (maximal scrambling). The manuscript must supply either an extension argument or explicit caveats showing why the protection mechanism survives for non-random or weakly chaotic Hamiltonians, as the numerical evidence alone cannot establish this.

    Authors: We agree that the rigorous analytic proof applies specifically to typical Haar-random unitaries. In the revision we will add explicit caveats in the abstract, introduction, and discussion sections clarifying this scope. We will also include a short argument that the protection arises from the scrambling properties shared by both the Haar ensemble and sufficiently chaotic Hamiltonians, as evidenced by the consistent numerical behavior across models; a complete analytic extension to arbitrary Hamiltonians is beyond the present scope but will be noted as an open direction. revision: yes

  2. Referee: [Numerical Simulations] Numerical simulations for chaotic Hamiltonians: system sizes are modest by necessity; the manuscript should include a finite-size scaling collapse or explicit comparison of exponential versus power-law fits (with R² or similar metrics) to demonstrate that the observed growth is not consistent with sub-exponential alternatives.

    Authors: We acknowledge the limitation imposed by modest system sizes in the Hamiltonian simulations. In the revised manuscript we will add explicit comparisons of exponential versus power-law fits to the quantum mutual information lifetime data, including R² or equivalent goodness-of-fit metrics. Where computationally feasible we will also present a finite-size scaling analysis to further support the exponential scaling over sub-exponential alternatives. revision: yes

Circularity Check

0 steps flagged

No circularity: analytical proof for Haar-random unitaries stands independently of numerics and uses standard averaging.

full rationale

The derivation begins from a maximally entangled reference-system state and applies standard Haar averaging over the unitary group to prove exponential lifetime scaling under continuous bath monitoring. This averaging step is a conventional technique in quantum information and does not presuppose the target scaling law or reduce to a fitted parameter. Numerical simulations on random circuits and chaotic Hamiltonians are presented as separate confirmation rather than inputs to the analytic result. The contrast with the unmonitored case (linear or constant scaling) follows directly from the absence of the monitoring protocol in the same averaging framework. No self-citation chains, ansatz smuggling, or uniqueness theorems imported from prior author work are invoked as load-bearing steps for the central claim. The result is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no explicit free parameters, axioms, or invented entities are identifiable. The proof relies on standard properties of Haar random unitaries from quantum information theory.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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    quant-ph 2025-11 conditional novelty 8.0

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Reference graph

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