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REVIEW 4 major objections 5 minor 38 references

Deep thermalisation of two causally disconnected regions in a generic local quantum system is bounded by measurement-induced entanglement teleportation, and both timescales grow only logarithmically with the separation.

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2026-08-01 23:37 UTC pith:SQWUVMXT

load-bearing objection A clean inequality plus suggestive numerics, but the log-L scaling for generic circuits rests on an RTN argument that computes the wrong purity average. the 4 major comments →

arxiv 2607.15276 v1 pith:SQWUVMXT submitted 2026-07-16 quant-ph cond-mat.dis-nncond-mat.stat-mechhep-th

Locality of deep thermalisation through the lens of entanglement teleportation

Saptarshi Mandal , Alan Sherry , Sthitadhi Roy This is my paper
classification quant-ph cond-mat.dis-nncond-mat.stat-mechhep-th
keywords deep thermalisationprojected ensemblequantum state designsentanglement teleportationrandom tensor networksdual-unitary circuitsscramblinglocality
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 asks whether deep thermalisation — the emergence of a universal (Haar-like) ensemble of states on a subsystem after projective measurements on its complement — can be nonlocal. The authors consider a subsystem split into two regions that are never causally connected by the unitary dynamics, so any correlations between them must be generated by the measurements. They prove a general inequality: the distance of the projected ensemble from the Haar ensemble is bounded below by the deviation of the averaged bipartite entanglement (purity) between the two regions from its Haar value. For generic locally interacting circuits, numerics plus a random-tensor-network argument show that the time at which this purity starts to move toward the Page value grows as the logarithm of the separation, so deep thermalisation inherits that logarithmic timescale. The only exceptions are special circuits that perfectly transmit measurement randomness, such as dual-unitary or infinite-local-dimension gates, where deep thermalisation happens in O(1) time and is genuinely nonlocal.

Core claim

Deep thermalisation of two disjoint, causally disconnected regions is bounded by entanglement teleportation between them. For every k, the k-th moment distance Δ^(k) from Haar is at least D_R^{-k/2} δP^(k), where δP^(k) is the purity deviation from Page (Eq. 9). In generic local circuits, numerics (L_R ≤ 24) and a Gaussian random-tensor-network argument show δP^(k) stays flat until t*(L_R) ∼ ln L_R. The RTN picture, with bond dimension χ_i = exp[min(v_E t, ln D_i)], gives finite purity only for χ_max = L (depth ∝ ln L), not for finite depth. Exceptions are circuits that perfectly transmit measurement randomness (dual-unitary or infinite local dimension), where the ensemble is exactly Haar.

What carries the argument

The two central objects are (i) the bound Δ^(k) ≥ D_R^{-k/2} δP^(k), derived by applying Cauchy-Schwarz to the purity difference expressed in replica space, which converts a statement about the whole projected ensemble into a statement about the bipartite purity between the disjoint regions; and (ii) the Gaussian random-tensor-network (RTN) proxy, in which the state produced by a scrambling local circuit at time t is represented by a chain of Gaussian random tensors with bond dimension χ_i = exp[ min(v_E t, ln D_i) ]. The RTN yields an analytic annealed purity P_ann = (9 − Q)/(9 + Q) with Q = ((χ−1)/(χ+1))^(L+1), which shows that a finite purity at large L requires χ ∝ L, i.e., circuit depth

Load-bearing premise

The whole argument rests on modelling the state produced by a scrambling local circuit as a Gaussian random tensor network with bond dimension equal to the entanglement entropy profile; if that proxy misrepresents the higher moments of the projected ensemble, the logarithmic timescale for generic circuits is supported only by the L_R ≤ 24 numerics.

What would settle it

Compute Δ^(k) and δP^(k) for the brickwork circuit of Eqs. (14)-(16) at L_R = 30 or larger (or measure the purity between the two regions in a quantum simulator with more than 30 qubits between them). If the onset of the decay of δP^(k) does not grow with L_R, or if the projected ensemble reaches a k-design at a time that does not grow as ln L_R, the central claim is wrong.

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

If this is right

  • Deep thermalisation in generic local systems cannot occur at O(1) times: the projected ensemble is prevented from approaching Haar until the logarithmic teleportation time has elapsed.
  • The inequality Δ^(k) ≥ D_R^{-k/2} δP^(k) means that any protocol aiming to generate a quantum state design by measuring a local circuit must wait at least t*(L_R) ∼ ln L_R if the regions are separated by distance L_R.
  • Observation of deep thermalisation at O(1) times in a local system is a diagnostic of perfect randomness transmission (dual-unitary dynamics or effectively infinite local Hilbert space).
  • For distances L_R that are not too large, the logarithmic timescale may be mistaken for an O(1) thermalisation time, so finite-size scaling in L_R is necessary to identify the asymptotic behaviour.

Where Pith is reading between the lines

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

  • A direct check of the RTN proxy: extract the actual Schmidt spectrum of the brickwork-circuit state at various times and compare with the assumed χ_i = exp[min(v_E t, ln D_i)]; deviations would affect the prefactor of the logarithmic timescale but probably not the scaling.
  • The log timescale implies that local circuits of depth ∝ ln L are precisely at the threshold for a projected ensemble to form a state design; this resonates with the teleportation phase transition seen in random circuits at finite depth.
  • If the bound is tight, then the design time for generic local systems is set by the teleportation time; any k-dependence in design times must come from the k-dependence of Δ^(k), not from the purity.
  • An experimental signature is the second Rényi entropy between the two regions: its deviation from 1 should onset at a time that grows as the logarithm of the number of sites separating them, which could be measured in current Rydberg or trapped-ion simulators.

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

4 major / 5 minor

Summary. The paper studies deep thermalisation of a subsystem R = R1 ∪ R2 whose two parts are causally disconnected under unitary dynamics. Measurements on the complement generate an ensemble of conditional states on R; deep thermalisation occurs when this ensemble approaches the Haar ensemble (or a generalized Scrooge ensemble). The authors prove an inequality (Eq. 9) showing that the distance Δ(k) between the k-th moment of the projected ensemble and the Haar ensemble is bounded below by D_R^{-k/2} δP(k), where δP(k) measures the deficit of average bipartite purity between R1 and R2 relative to the Haar value. Thus entanglement teleportation between the disconnected parts must precede deep thermalisation. They demonstrate O(1) teleportation timescales in a minimal toy model with a perfect bath, and for a brickwork random circuit with local gates they present numerical evidence that both δP(k) and Δ(k) collapse when time is rescaled by ln L_R, implying t* ∼ ln L_R. They then propose a Gaussian random tensor network (RTN) proxy in which the bond dimensions are set by the entanglement entropy profile S_i(t) = min[v_E t, ln D_i], and use an annealed-purity calculation to argue that a depth ∝ ln L is necessary for finite teleported entanglement. Exceptions are discussed where perfectly transmitting gates yield O(1) deep thermalisation.

Significance. If the central claim holds, the paper establishes an emergent locality of deep thermalisation in generic locally interacting systems, tying it to measurement-induced entanglement teleportation. This is a conceptually interesting bridge between projected ensembles, quantum state designs, and information propagation. The proof of the inequality (Eq. 9) is clean and appears correct. The minimal model is exactly solvable and provides a clear ``perfect randomness transfer'' example. The RTN analytic calculation is explicit and compact. However, the main generic claim—t* ∼ ln L_R for local circuits—rests on two pieces of evidence: (i) brickwork numerics over a narrow range of L_R (16–24) without error bars, and (ii) an RTN proxy whose mapping from actual circuits is heuristic rather than derived. The analytic RTN result (Eq. 20) is for an annealed average, not the Born-weighted projected-ensemble quantity appearing in Eq. (9). These issues are load-bearing for the paper's main conclusion, so the manuscript requires revision rather than acceptance in its present form.

major comments (4)
  1. [Section 'Locally interacting circuits' (Eq. 17)] The mapping from scrambling local circuits to Gaussian RTNs with χ_i = exp[S_i(t)] is a heuristic assumption, not a controlled approximation. It is the only analytic bridge connecting circuit depth to the logarithmic timescale. No test is given that this proxy reproduces the projected ensemble (beyond k=2 purity) of the actual brickwork circuit. Since the brickwork numerics are limited to L_R=16–24 (ln L varies by only a factor ~1.15), the logarithmic scaling is not independently established over a wide range. Please state explicitly that this is a conjecture, test the RTN proxy against small brickwork circuits (e.g., compare δP(k) distributions and higher moments), or provide a derivation of the mapping.
  2. [Eqs. (19)–(20) and Eq. (9)] P_ann in Eq. (19) is the annealed purity E[Tr((ΦΦ†)^2)]/E[Tr(Φ†Φ)]^2. The quantity entering the bound Eq. (9) is instead the Born-weighted projected-ensemble purity P_Born = E[Tr((ΦΦ†)^2)/Tr(Φ†Φ)] / E[Tr(Φ†Φ)] (for k=2). These are equal only if the normalization Tr(Φ†Φ) is self-averaging, which is not shown. If D fluctuates strongly, P_ann can be <1 even when every typical conditional state has near-zero bipartite entanglement. The analytical conclusion that χ∝L is needed relies entirely on this annealed average. Please compute the quenched/Born-weighted purity directly (analytically or numerically) for the same RTN ensemble and show whether the χ∝L criterion survives.
  3. [Fig. 2(c) and surrounding text] The data collapse supporting t* ∼ ln L_R uses only L_R=16,18,20,22,24, and no error bars or sample sizes are reported. With such a narrow range, a ln L rescaling is not strongly distinguished from, e.g., a power law L^ε with small ε or a logarithmic-plus-constant fit. The claim that this identifies a universal curve and a well-defined t*(L_R) needs either a wider range of L_R, error bars, a quantitative collapse metric, or a scaling collapse over at least a decade of ln L. Without this, the numerical support for the logarithmic scaling is qualitative.
  4. [RTN analysis: k-dependence and non-uniform bond dimensions] The analytic RTN result Eq. (20) is only for k=2 and uniform bond dimension χ, while the paper's main claim concerns all k and the actual non-uniform profile χ_i=min[L,D_i] (the text in the second bullet of the RTN section writes min[L,ln D_i], presumably a typo). The inequality Eq. (9) applies for arbitrary k, and the numerical brickwork results in Fig. S1 show that Δ(k) has a k-dependent decay, so the k=2 annealed RTN calculation cannot by itself justify the statement that all moments thermalise on the same logarithmic timescale. Please either extend the RTN computation to higher k and non-uniform χ, or clearly state that the analytic argument is only for the second moment and that higher moments are extrapolated.
minor comments (5)
  1. [RTN section, second bullet] The text says ``χ_i = min[L,ln D_i]'' but the surrounding discussion and Fig. 3 indicate it should be ``min[L,D_i]'' (or min[L,2^{i+1}]). Please correct.
  2. [Fig. 1 and Fig. 2] No error bars or numbers of disorder/initial-state samples are reported. Please specify the averaging procedure and typical statistical error; this matters particularly for the small δP(k) values at late times.
  3. [Eq. (5)] The Hilbert-Schmidt norm is used without explicitly normalizing by the norm of the identity; this is fine, but it may help to state that Δ(k) is dimensionless and bounded because both density matrices are normalized.
  4. [General] The reference to the supplementary material is given as ``[URL]''; since the SM is part of the arXiv posting, please include a working link or clear pointer.
  5. [Introduction, Eq. (2)] The construction ``W_i |φ_Ri φ_Si> to be the even-parity Bell state'' is slightly terse. Clarify that W_i is a unitary preparing a Bell state on R_i ∪ S_i, so that ρ_Ri ∝ I, and that this choice makes the deep-thermal ensemble the Haar ensemble rather than a Scrooge ensemble.

Circularity Check

0 steps flagged

No significant circularity: the central bound is a Cauchy-Schwarz inequality, the RTN threshold is an internal model-derived condition, and the brickwork numerics provide an independent check.

full rationale

We walked the derivation chain. Equation (9) is obtained from Eq. (7) by Cauchy-Schwarz; it is a genuine inequality relating the moment distance to the teleported purity and does not define one quantity in terms of the other. The minimal-model results are self-contained solutions of the projected-ensemble moments. The local-circuit claim has two legs. (i) The brickwork simulations in Fig. 2 compute Delta^(k) and deltaP^(k) directly from the circuit; the collapse under t/ln L_R is an empirical inference, not a parameter fitted from a subset and then renamed as a prediction. (ii) The RTN argument is explicitly introduced as a 'premise' and a 'heuristic picture': 'states generated by scrambling local circuits, such as the model discussed above, can be proxied effectively by Gaussian RTNs'. Within that model, Eq. (20) is derived from Gaussian averaging; the condition chi proportional to L is a threshold consequence of that formula, and the conversion to t* proportional to ln L uses the stated scrambling relation chi_i = exp[S_i(t)] with S_i(t)=min[v_E t, ln D_i]. This is a physical mapping, not a restatement of the conclusion: the derivation does not assume P_ann < 1 at chi proportional to L, it computes it. The same-author citations ([20] and [22]) are topical references and are not load-bearing; no uniqueness theorem is imported. The annealed-versus-Born and small-L_R-range concerns raised by a skeptical reader concern whether the RTN proxy and the log fit are quantitatively reliable, not whether the derivation reduces to its own inputs. Therefore no circular step meets the quoted-evidence bar.

Axiom & Free-Parameter Ledger

4 free parameters · 6 axioms · 0 invented entities

The central claim rests on standard projected-ensemble definitions plus a heuristic RTN-to-circuit mapping; no new physical entities are introduced. The main free parameters are simulation/proxy choices rather than fitted constants, but the positive depth-∝lnL result is partly built into the RTN proxy.

free parameters (4)
  • Brickwork gate parameters (τ and H coefficients) = τ = 0.28; H = 0.3 X X + 0.2 X + 0.4 Z Z + 0.5 Z (site-local terms)
    Chosen by hand for the brickwork circuit (Eqs. 15-16). Not fit to the target scaling, but the claim of genericity implies the log-L scaling should be insensitive to these parameters, which is not demonstrated over a parameter range.
  • RTN finite-depth proxy χ_max = e^{v_E t} = assumed ~ O(1)
    The finite-depth case is represented by choosing χ_max = O(1) by construction; the conclusion that no teleportation occurs at finite depth follows from this modeling choice rather than from an independent derivation.
  • RTN depth-∝lnL proxy χ_max = L = χ_max = L
    The positive result for depth ∝ ln L is built into the proxy by setting χ_max = L. The analytic P_ann criterion independently shows χ ∝ L is needed, but the time-to-bond-dimension mapping is assumed.
  • Uniform bond dimension χ in analytic RTN = symbolic χ
    Simplifies the annealed purity computation to a 2×2 transfer matrix; the authors argue the qualitative threshold is unaffected by the uniformity approximation.
axioms (6)
  • standard math Projected-ensemble k-th moments and Haar-design definitions (Eqs. 3-4)
    Framework inherited from prior deep-thermalization works [12-16].
  • ad hoc to paper Gaussian random tensor network proxy for scrambling circuits
    The premise that states from scrambling local circuits can be proxied by Gaussian RTNs with bond dimensions set by entanglement growth is unproved; the RTN section states it as a premise.
  • domain assumption Operator-scrambling survival count giving λ = 2/5 per side in the minimal model
    The rate γ = ln(25/4) is argued from a Pauli-string counting argument; supported by numerics but not a rigorous derivation.
  • domain assumption Infinite Hilbert-space dimension D_E3 → ∞ in the minimal model
    The perfect-bath limit removes finite-size corrections and produces O(1) timescales; it is an idealization, not a generic local-system regime.
  • domain assumption No conserved quantities in the circuits
    The deep-thermal target is the Haar ensemble; adding conservation laws would change the target ensemble and possibly the timescales, as the outlook acknowledges.
  • standard math Weingarten calculus and replica tricks
    Used in the Supplementary Material for dual-unitary and Haar-averaged computations.

pith-pipeline@v1.3.0-alltime-deepseek · 12316 in / 11681 out tokens · 114041 ms · 2026-08-01T23:37:50.108341+00:00 · methodology

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read the original abstract

Deep thermalisation characterises the emergence of universal quantum state ensembles on subsystems due to projective measurements on their complement. We study the notion of locality, or lack thereof, in this phenomenon by considering a subsystem partitioned into two disjoint subregions which remain causally disconnected at all times under unitary dynamics. We show that the onset of deep thermalisation in this geometry is fundamentally bounded by measurement-induced entanglement teleportation between the subregions. While measurements on the environment generate entanglement across the disconnected partitions -- suggesting an apparent non-locality -- we demonstrate that generic locally interacting systems exhibit an emergent locality. Specifically, the timescales for both deep thermalisation and entanglement teleportation scale logarithmically with the distance separating the subregions. Exceptions to this include special circuits where the randomness of the measurement outcomes is perfectly transmitted to the ensemble of states of the subsystem, conditioned on the outcomes; in such cases the timescale for deep thermalisation is finite leading to genuine non-locality.

Figures

Figures reproduced from arXiv: 2607.15276 by Alan Sherry, Saptarshi Mandal, Sthitadhi Roy.

Figure 1
Figure 1. Figure 1: FIG. 1. Numerical results for the minimal model in Eq. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Numerical results for (a) ∆ [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Results for the RTN states of the form in Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

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