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The half-chain waiting-time distribution of quantum jumps is not the trivial Poissonian one: its long-time tail is controlled by a single eigenvalue of the no-jump superoperator, and under strong measurement that tail survives as the system

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2026-08-02 16:58 UTC pith:SPS2HXY6

load-bearing objection New subsystem WTD result, real but needs a sharper thermodynamic-limit argument. the 2 major comments →

arxiv 2604.00358 v2 pith:SPS2HXY6 submitted 2026-04-01 cond-mat.stat-mech cond-mat.quant-gasquant-ph

Anomalous waiting-time distributions in postselection-free quantum many-body dynamics under continuous monitoring

Kazuki Yamamoto , Ryusuke Hamazaki This is my paper
classification cond-mat.stat-mech cond-mat.quant-gasquant-ph
keywords waiting-time distributionquantum jumpscontinuous monitoringLiouvillian spectrumsubsystem statisticsmeasurement-induced dynamicspostselection-freehard-core bosons
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper studies the statistics of waiting times between quantum jumps in a continuously monitored quantum many-body chain whose unconditional steady state is the maximally mixed, infinite-temperature state. While the whole-system waiting-time distribution is exactly Poissonian, the distribution for a half-chain subsystem develops an anomalously heavy tail. The authors trace this tail to the eigenvalue λ0 with the largest real part of the superoperator L0, defined by deleting the subsystem jump terms from the full Liouvillian. They show that λ0 scales with system size for weak measurements but saturates to a size-independent value for strong measurements, implying the anomalous tail persists in the thermodynamic limit only in the strong-measurement regime. Because the distribution is extracted directly from the spacetime record of jumps, it offers a postselection-free, experimentally accessible probe of many-body monitored dynamics.

Core claim

The central claim is that the half-chain WTD obeys Whalf(τ) ∼ e^{λ0τ}, where λ0 < 0 is the rightmost eigenvalue of L0 = L − Σ_{i∈M} Li, the superoperator that generates time evolution with no jumps in the half chain while allowing jumps elsewhere. Unlike the full Liouvillian, L0 has no zero eigenvalue, and its spectral decomposition gives the long-time decay of the WTD. The eigenvalue obeys λ0 Tr[ρR0] = −Σ_{i∈M} Tr[Li†Li ρR0], which fixes λ0 < 0 under a mild condition. Numerically, λ0 scales as −O(L) for γ ≪ 1 and as −O(1) for γ = O(1), so the anomalous tail remains robust in the thermodynamic limit only for strong measurement. At short times the WTD still shows the Poissonian slope γL/4 bef

What carries the argument

The central object is the superoperator L0 = L − Σ_{i∈M} Li, where L is the Lindblad Liouvillian and Li(ρ) = Li ρ Li† are the jump terms in the subsystem M; physically it describes evolution conditioned on no jumps in M. Its right-most eigenvalue λ0, assumed unique and with all real parts ordered as in Eq. (30), dominates the long-time WTD through the spectral decomposition e^{L0 t} = Σ e^{λα t}|ρRα)(ρLα|. The identity λ0 Tr[ρR0] = −Σ_{i∈M} Tr[Li† Li ρR0] proves λ0 < 0 and gives a direct link between the eigenvalue and the decay rate of the no-jump probability.

Load-bearing premise

The central claim relies on L0 being diagonalizable with a unique eigenvalue λ0 of largest real part (Eq. 30); the paper states this assumption explicitly and notes that exceptional points, where eigenvalues coalesce, could invalidate the pure exponential tail but does not rule them out.

What would settle it

Measure or compute the half-chain WTD at strong measurement γ = O(1) for increasing system sizes (e.g., L = 8, 10, 12, 14) and extract the long-time decay rate. If the rate approaches a nonzero constant as L grows, the persistence claim holds; if the rate decays like 1/L or the tail becomes non-exponential, the λ0-dominated picture fails. An exact-diagonalization scan of L0's spectrum to locate exceptional points in this parameter regime would also settle the question.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The half-chain WTD is a new, postselection-free observable that carries nontrivial many-body information even when the unconditional steady state is featureless.
  • For weak measurement the anomalous tail is suppressed exponentially with system size, while for strong measurement the tail rate saturates, so the crossover in λ0 scaling acts as a sharp diagnostic of monitored dynamics.
  • The whole-system WTD remains exactly Poissonian, so a comparison between full-chain and half-chain waiting times directly reveals the subsystem effect.
  • The framework gives a spectral route to computing WTDs in other monitored models by diagonalizing the corresponding L0 operator.
  • The short-time Poissonian slope followed by a λ0-dominated tail means the full statistics are not single-exponential, which is relevant for interpreting continuous-monitoring experiments.

Where Pith is reading between the lines

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

  • A natural extension the paper leaves implicit is to vary the subsystem fraction M/L; the same L0 formalism should produce an M-dependent tail rate, and plotting λ0 against subsystem fraction could expose where the Poissonian whole-chain behavior crosses into the anomalous regime.
  • The sudden change in λ0 scaling with γ resembles a nonequilibrium crossover; a systematic scaling collapse of λ0/L versus γ, or a derivative of λ0, could locate the crossover measurement strength and test whether it sharpens with system size.
  • The exceptional points mentioned in Sec. 5 could alter the exponential tail into polynomial corrections; a direct numerical search for eigenvalue coalescence in L0 in the reported parameter regime would show whether the e^{λ0τ} prediction is exact or only approximate.
  • Because the WTD is extracted from jump records, the same analysis could be applied to other jump observables like factorial cumulants of subsystem jump counts, potentially yielding a family of postselection-free probes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies waiting-time distributions (WTDs) of quantum jumps in a continuously monitored hard-core boson chain with particle-number detection, whose unconditional steady state is the infinite-temperature state. The authors define a superoperator L0 obtained by removing the jump terms in a subsystem M from the full Liouvillian, and show that the half-chain WTD has a long-time tail governed by the eigenvalue λ0 of L0 with the largest real part, in contrast to the Poissonian WTD of the whole system. Based on exact diagonalization of L0 for small system sizes, they report a qualitative change in the system-size scaling of λ0: for weak measurement λ0 ∝ L, while for strong measurement λ0 is independent of L, implying that the anomalous tail persists in the thermodynamic limit. The WTD is extracted from the spacetime record of jumps and thus does not require postselection.

Significance. If the thermodynamic-limit claim holds, the paper provides a novel, experimentally accessible observable—subsystem waiting-time statistics—that detects many-body effects in monitored quantum dynamics without postselection. The spectral framework connecting the WTD tail to L0 is natural and the exact derivation of the whole-system Poissonian is a clear strength. The numerical simulations (L up to 14) and the explicit comparison of the tail slope with λ0 provide initial support. The paper is also careful to state its main diagonalizability assumption. However, the headline prediction—the O(1) decay rate for strong measurement in the thermodynamic limit—rests on finite-size numerics without an analytic scaling argument or larger-system data, which limits the confidence in the central claim.

major comments (2)
  1. [§4.2, Eq. (1), Fig. 2(c)] The claim that for γ=O(1) the half-chain WTD has a tail ∼e^{−O(1)τ} in the thermodynamic limit is supported only by exact diagonalization of L0 for small systems (Fig. 2 shows spectra for L=6; Appendix B presents WTDs only up to L=14). No analytic scaling argument is given for why λ0 becomes L-independent at strong measurement, and the finite-size data in Fig. 2(c) do not include error bars or a systematic extrapolation. If the apparent constancy of λ0 is a finite-size crossover and λ0 eventually grows with L beyond the accessible sizes, the central prediction of Eq. (1) would not survive. The authors should either provide an analytic argument (e.g., a strong-measurement effective theory showing a finite gap) or present significantly larger-L data using sparse or tensor-network methods.
  2. [§3.2, Eq. (30), §5] The spectral decomposition in Eq. (50), which yields the tail W_half(τ)∼e^{λ0τ}, explicitly assumes that L0 is diagonalizable and that the eigenvalue λ0 with the largest real part is unique. The paper acknowledges the possible emergence of exceptional points in Sec. 5, but does not verify that these do not occur in the parameter regimes used for the central claim. If exceptional points are present, Eq. (50) must be replaced by a Jordan-form expression, and the tail would acquire a polynomial prefactor. The authors should either check the absence of such degeneracies for the studied γ and L, or clearly state in the main text that the result is conditional on this assumption and discuss the expected modification.
minor comments (4)
  1. [§2.2, Eq. (13)] The normalization proof assumes p_no(∞)=0, but this is only established later in §3.2 via the negativity of λ0. The logical ordering should be adjusted or this decay should be stated as an assumption at this point.
  2. [§4.2, Eq. (51)] The approximation leading to the short-time Poissonian decay is heuristic: 'the local operator n_i can be regarded as approximately conserved' and the replacement e^{L0τ}∼e^{Lτ}e^{−∑_{i∈M}L_iτ} are not rigorously justified. It is acceptable as a physical argument, but the limitations should be stated.
  3. [§4.1] Typo: 'Poisonian' should be 'Poissonian' (also in §4.2 and the figure captions). In §2.1, 'trance preserving' should be 'trace preserving'.
  4. [Appendix B] For L=14 the number of trajectories is 10^7, and the long-time tail at γ=0.05 appears to be sampled over a narrow time window. Showing error bars or a comparison of the tail over a wider dynamic range would strengthen the claim that the data are consistent with the λ0 slope.

Circularity Check

0 steps flagged

No significant circularity: the λ0 tail is derived from a spectral decomposition and verified by independent trajectory simulations, not fitted.

full rationale

The central claimed result is the long-time tail W_half(τ) ~ e^{λ0τ}. This is not a fitted input: it follows analytically from Eq. (8), W(τ,j;i)=∫0∞ Tr[Lj e^{L0τ} Li e^{L0t} ρss] dt, by inserting the spectral decomposition of L0 (Eqs. (33)-(35)) to obtain Eq. (50). The eigenvalue λ0 is defined as the rightmost eigenvalue of L0 and is computed independently by exact diagonalization of the superoperator, not by fitting the WTD. The quantum-trajectory histograms in Figs. 3 and 4 are separate dynamical simulations, and the overlay −λ0 exp(λ0τ) uses the independently computed λ0 with no adjustable amplitude. Thus the numerical agreement is a genuine test of the spectral prediction rather than a tautology. The system-size scaling of λ0 (weak: −O(L), strong: −O(1)) is presented as a numerical observation from Fig. 2(c); the thermodynamic-limit conclusion inherits the usual finite-size extrapolation caveat, but that is a missing-support/robustness concern, not a circular reduction. The self-citations [30,70] are used only as background for the stochastic Schrödinger equation, whole-system Poissonian WTD, and current fluctuations; the whole-system Poissonian result is re-derived analytically in Eq. (45), so the citations are not load-bearing. Assumptions such as diagonalizability and uniqueness of λ0 in Eq. (30) and the possible exceptional points mentioned in Sec. 5 are explicitly stated limitations rather than hidden circular inputs. Overall, the derivation chain is self-contained and no step reduces to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The central claim relies on the spectral decomposition of L0, which requires the diagonalizability and uniqueness assumption. The normalization of the WTD depends on vanishing no-jump probabilities at infinite time. No free parameters are fitted; γ is a control parameter. No new physical entities are postulated.

axioms (5)
  • ad hoc to paper The superoperator L0 is diagonalizable with a unique eigenvalue λ0 of largest real part.
    Used for spectral decomposition in Eq. (35) and the conclusion that the long-time tail is governed by λ0. Not proven for all parameter regimes; the paper itself notes possible exceptional points in Sec. 5.
  • domain assumption The probabilities of no jumps in M after infinite time vanish: p_no(∞)=0 and p'_no(∞)=0.
    Assumed in Sec. 2.2 to prove normalization of the WTD. Without this, the WTD would not be normalized.
  • domain assumption The unconditional steady state is the maximally mixed state ρ_ss = I/D0.
    Used throughout to compute WTD. Standard for particle-number monitoring in a symmetry sector, but it is an input.
  • standard math Tr[Lρ]=0 for any ρ for the Liouvillian L.
    Trace-preservation property of the Lindblad master equation, used in Eqs. (37) and (10).
  • standard math The quantum trajectory method accurately simulates the Lindblad dynamics.
    Used for generating numerical WTDs. Standard in the field.

pith-pipeline@v1.3.0-alltime-deepseek · 17227 in / 13152 out tokens · 113750 ms · 2026-08-02T16:58:25.959035+00:00 · methodology

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read the original abstract

We investigate waiting-time distributions (WTDs) of quantum jumps in continuously monitored quantum many-body systems, whose unconditional dynamics lead to the trivial infinite-temperature state. We demonstrate that the WTD of a half-chain subsystem exhibits an anomalous tail, markedly deviating from the Poissonian distribution in stark contrast to that of the whole system. By analyzing the spectral properties of the superoperator $\mathscr L_0$, which is defined by removing the jump terms associated with the half-chain subsystem from the full Liouvillian, we find that the long-time behavior with the anomalous tail of the half-chain WTD is governed by the eigenvalue $\lambda_0\:(<0)$ with the largest real part. We further reveal a qualitative change in the system-size dependence of $\lambda_0$ as a function of the measurement strength: for sufficiently weak measurement, $\lambda_0$ decreases proportionally to the system size, while for strong measurement, $\lambda_0$ scales independently of the system size, signaling the persistence of the anomalous half-chain WTD in the thermodynamic limit. The WTD is extracted solely from the spacetime record of quantum jumps $\{t_i,x_i\}$ and can be experimentally accessed without postselection. Our work establishes a spectral framework for understanding nontrivial WTDs in subsystems of monitored quantum dynamics and provides a novel diagnostics to assess many-body effects on WTDs.

Figures

Figures reproduced from arXiv: 2604.00358 by Kazuki Yamamoto, Ryusuke Hamazaki.

Figure 1
Figure 1. Figure 1: Schematic figure of our setup. We focus on the first and the second jumps (red crosses) in a half chain after the system reaches the steady state (see text) and calculate the probability distribution of the waiting time τ ≡ t2 − t1 along trajectory realizations. where ⟨·⟩ denotes a quantum expectation value for the state |ψ(t)⟩. Here, a discrete random variable dNi = 0, 1 that counts the increment of a jum… view at source ↗
Figure 2
Figure 2. Figure 2: Eigenspectrum of the superoperator L0 for the Heisenberg model under continuous monitoring for (a) L = 6, γ = 0.05 and (b) L = 6, γ = 1. The eigenvalue with the largest real part, λ0, is negative in contrast to the case of the Liouvillian. (c) System-size dependence of λ0. For weak measurement strength, λ0 decreases proportional to the system size, but for strong measurement, λ0 is independent of the syste… view at source ↗
Figure 3
Figure 3. Figure 3: Numerical results of the WTD for the Heisenberg model under continuous monitoring with L = 8 for γ = 0.05 [(a)], 0.5 [(b)], and 1 [(c)], demonstrating the emergence of the anomalous tail in the half-chain subsystem characterized by λ0. Left panel shows the WTD of the half-chain subsystem (blue histogram). The red solid line denotes the Poissonian distribution, while the green solid line corresponds to an e… view at source ↗
Figure 4
Figure 4. Figure 4: Numerical results of the WTD for the Heisenberg model under continuous monitoring with L = 14 for γ = 0.05 [(a)], 0.5 [(b)], and 1 [(c)], demonstrating the emergence of the anomalous tail in the half-chain subsystem characterized by λ0. Left panel shows the WTD of the half-chain subsystem (blue histogram). The red solid line denotes the Poissonian distribution, while the green solid line corresponds to an … view at source ↗

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