Entanglement Growth as Transport Across Schmidt Scales

Shi-Xin Zhang; Shuo Liu; Yu-Qin Chen

REVIEW

Entanglement Growth as Transport Across Schmidt Scales

Shi-Xin Zhang, Shuo Liu, Yu-Qin Chen

Quantum Physics · Submitted 6 Sep 2026 · 40 references

Pith's verdict: The paper introduces Schmidt-scale coordinates that resolve where entanglement weight lives in rank space, then demonstrates a robust temporal hierarchy (entropy peak, then spectral roughness/magic, then transport of the dominant scale) with clean numerics and an analytic head-tail model.

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arXiv ID2609.06643 (version 1)

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CitationShi-Xin Zhang, Shuo Liu, Yu-Qin Chen. "Entanglement Growth as Transport Across Schmidt Scales." arXiv:2609.06643 (2026). https://pith.science/paper/D7V66ORM

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Editor's note

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You should know two things. First, this paper introduces a genuinely useful diagnostic for entanglement dynamics — the Schmidt-scale coordinates (η*, u*) — that tell you where the dominant probability mass sits in logarithmic rank space, not just how much entropy there is. Second, it uses these coordinates to establish a clear temporal hierarchy (entropy production peaks first, spectral roughness and nonlocal magic peak next, dominant-scale transport arrives later) and backs the hierarchy with a solvable head-tail model and controlled numerical experiments. The central claims are well supported. The numerics go to L=22 and tJ=10⁸ with exact diagonalization and Chebyshev propagation. The analytic head-tail model predicts the ordering ϵF < ϵD < ϵℓ independent of system size, and the active-bond explanation correctly accounts for the early clocks. The paper is transparent about where the independent-two-bond theory underestimates the transport delay and uses the g-interpolation to confirm that intra-subsystem many-body dressing is responsible for the extra delay. The stress-test is right: I don't see an unaddressed logical gap. Weaknesses are minor. The finite-size trends for transport are shown, but the disorder scan for tℓ,50 beyond W=0.5 becomes marginal or unresolved; the paper honestly marks these as lower bounds. The head-tail model assumes a flat tail, which is a strong idealization, but the authors use it as a path through spectrum space, not a quantitative fit. The protocol dependence (OBC vs PBC, Néel vs random) is documented in the supplement and actually strengthens the picture by showing how the active-bond count controls the early clocks. This is a serious, well-executed piece of work. It deserves a good referee and a quick route to publication. I'd accept peer review without hesitation. Hm, I'd bring it to reading group, too — the Schmidt-scale coordinates are likely to be adopted by others studying entanglement dynamics. Would cite if I were working on diagnostics of entanglement growth or MBL.

Referee report

Significance

If the results hold, the Schmidt-scale coordinates are a genuinely useful, cheap-to-compute addition to the entanglement diagnostic toolbox: they separate 'how much' entanglement from 'where in rank space' it resides, and they convert the qualitative notion of spectral reorganization into specific, falsifiable clock times. The strongest asset of the paper is its combination of solid numerics with hypothesis testing: exact diagonalization and Chebyshev propagation with reported error controls (norm deviation 3.6×10^{-15} at early times; agreement with fixed-sector ED to 1.14×10^{-11} for the octave profile), trajectory-bootstrap uncertainty intervals, and a parameter-free two-bond prediction that matches the g = 0 limit of the numerics to three decimal places ((0.58, 0.65, 0.79) vs (0.585282, 0.652620, 0.785398)).

Minor objections (5)

[Four spectral clocks (Fig. 2d)] The main-text sentence that the clock ordering 'persists across the disorder scan' reads more strongly than what Fig. 2(d) establishes: the fourth clock is resolved only at W = 0.5, and for W ≥ 1 the symbols are finite-window lower bounds that are consistent with, but do not verify, the ordering. The figure caption says this, but one clarifying clause in the text would remove any impression that t_{ℓ,50} has been measured across the scan.
[Four spectral clocks, two-bond prediction] The text states that the independent-two-bond model 'predicts the onset of dominant-Schmidt-scale transport at tℓJ = 1.231'; the numerics are actually compared with the disorder-averaged 50% crossing prediction of tJ = 1.25, while 1.231 = 2 arcsin(1/√3) is the clean resonant threshold (SM Fig. S4). Please label the quoted value as the clean-resonant threshold or quote the averaged 1.25 value, so the comparison with the numerical range t_{ℓ,50}J ≈ 1.73–1.79 is unambiguous.
[Abstract and Eq. (4)] The abstract's 'ordered hierarchy' does not name the protocol, but Eq. (4) is scoped to periodic boundary conditions with a Néel initial state; in the open-chain Néel protocol the middle clocks are nearly coincident and can exchange point-estimate order ((t*_F, t*_D) = (0.88, 0.87) vs (0.70, 0.87) for PBC, SM Sec. 7A). A protocol qualifier in the abstract or a one-sentence cross-reference to the SM would prevent readers from attributing the strict middle-pair ordering to open-chain dynamics.
[Fig. 2(c) and Fig. S4] The claimed size stability L = 14–22 rests on only 16 disorder realizations at L = 22 (and 32 at L = 20), for which the 50%-crossing times agree with the smaller sizes to within ~0.03. Please state the realization counts in the main-text panel caption, as done for Fig. S4, so the reader can weigh the statistical strength of the size-stability claim.
[Numerical methods, SM Sec. 2] The manuscript would benefit from a data-availability statement identifying which scripts and data files (TensorCircuit-NG implementations of the Chebyshev propagator and the clock-extraction pipeline) accompany the publication, since the clock definitions involve smoothing and peak-extraction conventions whose precise implementation is needed for independent reproduction.

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The paper

The paper claims

New Schmidt-scale coordinates reveal a hidden three-stage race in entanglement growth: entropy first, spectral reorganization second, dominant-weight transport last.

The reading

The paper introduces two new diagnostics—Schmidt-scale concentration and dominant Schmidt scale—that track where probability mass resides in the exponentially large Schmidt spectrum during quantum quenches. Applied to a random-field spin chain, these coordinates expose a robust temporal hierarchy: entropy production peaks first, spectral roughness and nonlocal magic peak next, and the transport of the dominant Schmidt weight to higher rank occurs only after a substantial delay. This hierarchy distinguishes entanglement production, entanglement-spectrum reorganization, and dominant-Schmidt-scale transport as distinct dynamical events that entropy alone cannot separate. The authors trace the hierarchy's origin to a solvable head-tail model and to the competition between local cross-cut exchange bonds and intra-subsystem many-body dressing. If correct, this work provides a unified framework for understanding how correlations spread in many-body systems, with implications for thermalization, many-body localization, and the design of quantum simulators and tensor-network algorithms.

What carries the argument

The central objects are the Schmidt-scale concentration η⋆ and the dominant Schmidt scale u⋆, defined via a sliding octave window in ordered Schmidt-rank space: η⋆ is the maximum probability contained in any factor-of-two window, and u⋆ = log₂(ℓ⋆+1) locates that window. Together (η⋆, u⋆) separate the erosion of the leading Schmidt weight from the outward migration of the dominant probability pile. The argument also relies on a head-plus-flat-tail analytic model that gives closed-form thresholds for the three events, and on an independent-two-bond theory that supplies the microscopic clock times from local exchange dynamics across the bipartition. A dressing parameter g interpolates between isolated cut bonds and the full many-body chain, isolating the role of intra-subsystem interactions in delaying transport.

Extended reading notes

Core claim

The paper discovers a universal ordering of three spectral events during entanglement growth: the entropy-production rate reaches its maximum first, followed by the maxima of spectral anti-flatness (roughness) and exact nonlocal magic, and finally the fraction of disorder realizations whose dominant Schmidt scale has left the spectral head crosses 50% only after a noticeable delay. This ordering holds across the thermal-to-MBL crossover and persists for different system sizes, boundary conditions, and initial states, though the delay for the last event grows indefinitely in the localized phase. The physical mechanism is captured by a minimal head-plus-flat-tail spectrum model: the thresholds for the three events occur at tail weights ϵ=1/4 (roughness), ϵ=1/2 (magic), and ϵ=2/3 (transport), and the time to reach each threshold is set first by local exchange bonds crossing the bipartition and then, for transport only, by additional many-body dressing within the two subsystems.

Load-bearing premise

The independent-two-bond theory neglects all intra-subsystem XXZ interactions, assuming the two exchange bonds crossing the bipartition evolve in isolation; this idealization underestimates the time for dominant-Schmidt-scale transport, and the head-tail model further assumes a flat spectral tail, which is a strong idealization of the real spectrum.

Editorial extensions

If this is right

The temporal hierarchy provides a benchmark for numerical and experimental tests of entanglement dynamics: one can now distinguish whether a system is merely producing entanglement or also reorganizing and transporting its Schmidt weight.
The Schmidt-scale coordinates can serve as discriminative probes for detecting the many-body localization transition, where transport of the dominant scale is suppressed even as entropy grows logarithmically.
The framework extends beyond Hamiltonian dynamics to random circuits and l-bit models, suggesting a common language for entanglement growth across platforms.
The results inform adaptive bond-dimension truncation in tensor-network simulations by indicating when the dominant weight is still concentrated in the spectral head versus when it has spread to higher rank.
Experimental routes via classical shadows or tomography may directly measure η⋆ and u⋆, enabling direct observation of dominant-Schmidt-scale transport in quantum simulators.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper leaves implicit that the same temporal hierarchy may appear in the dynamics of other resource theories, such as coherence or discord, when analyzed through a scale-resolved lens; one could test whether their spectral counterparts also exhibit delayed transport.
The sliding-octave construction could be applied to entanglement spectra of ground states and finite-temperature states, not just quench dynamics, to probe how spectral weight reorganizes across quantum phase transitions.
A natural extension is to investigate whether the hierarchy holds for higher-dimensional systems or long-range interactions, where the active-bond picture may need revision.
The 'many-body dressing' that delays transport may be linked to the buildup of mutual information between distant sites; a direct calculation of the mutual information between the two subsystems could quantify this dressing and possibly predict the delay time.

Original abstract

Quantum entanglement growth is commonly summarized by a single entropy, obscuring where correlations reside in the exponentially large Schmidt spectrum and how they form. Here, we introduce Schmidt-scale concentration and dominant Schmidt scale, two coordinates that locate the probability maximum across logarithmic windows in ordered Schmidt-rank space. Applied to quenches of a random-field spin chain, these coordinates distinguish rapid transport of the dominant scale to higher Schmidt rank at weak disorder from strongly suppressed transport despite continued logarithmic entropy growth at strong disorder. The disorder-averaged dynamics exhibit an ordered hierarchy: entropy production peaks first, spectral roughness and exact nonlocal magic peak next, and dominant-Schmidt-scale transport becomes typical only after a substantial delay. Moreover, a solvable head--tail model and controlled numerical experiments reveal the physical origin of this hierarchy: the spectral path determines the order of events, local dynamics on active exchange bonds set their early timing, and intra-subsystem many-body dressing further delays dominant-Schmidt-scale transport. These results establish the Schmidt-scale coordinates as powerful dynamical probes for uncovering fine-grained entanglement structures distinguishing entanglement production, entanglement-spectrum reorganization, and dominant-Schmidt-scale transport beyond entropy alone.

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The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Quantum entanglement growth is commonly summarized by a single entropy, obscuring where correlations reside in the exponentially large Schmidt spectrum and how they form. Here, we introduce Schmidt-scale concentration and dominant Schmidt scale, two coordinates that locate the probability maximum across logarithmic windows in ordered Schmidt-rank space. Applied to quenches of a random-field spin chain, these coordinates distinguish rapid transport of the dominant scale to higher Schmidt rank at weak disorder from strongly suppressed transport despite continued logarithmic entropy growth at strong disorder. The disorder-averaged dynamics exhibit an ordered hierarchy: entropy production peaks first, spectral roughness and exact nonlocal magic peak next, and dominant-Schmidt-scale transport becomes typical only after a substantial delay. Moreover, a solvable head--tail model and controlled numerical experiments reveal the physical origin of this hierarchy: the spectral path determines the order of events, local dynamics on active exchange bonds set their early timing, and intra-subsystem many-body dressing further delays dominant-Schmidt-scale transport. These results establish the Schmidt-scale coordinates as powerful dynamical probes for uncovering fine-grained entanglement structures distinguishing entanglement production, entanglement-spectrum reorganization, and dominant-Schmidt-scale transport beyond entropy alone.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Quantum entanglement growth is commonly summarized by a single entropy, obscuring where correlations reside in the exponentially large Schmidt spectrum and how they form.	gap 0.60	0.05	0.04	asserted 1.00
s1	Here, we introduce Schmidt-scale concentration and dominant Schmidt scale, two coordinates that locate the probability maximum across logarithmic windows in ordered Schmidt-rank space.	contribution 0.85	0.92	0.07	asserted 0.99
s2	Applied to quenches of a random-field spin chain, these coordinates distinguish rapid transport of the dominant scale to higher Schmidt rank at weak disorder from strongly suppressed transport despite continued logarithmic entropy growth at strong disorder.	result 0.99	0.94	0.23	asserted 1.00
s3	The disorder-averaged dynamics exhibit an ordered hierarchy: entropy production peaks first, spectral roughness and exact nonlocal magic peak next, and dominant-Schmidt-scale transport becomes typical only after a substantial delay.	result 1.00	0.91	0.16	asserted 1.00
s4	Moreover, a solvable head--tail model and controlled numerical experiments reveal the physical origin of this hierarchy: the spectral path determines the order of events, local dynamics on active exchange bonds set their early timing, and intra-subsystem many-body dressing further delays dominant-Schmidt-scale transport.	interpretation 0.47	0.94	0.12	asserted 1.00
s5	These results establish the Schmidt-scale coordinates as powerful dynamical probes for uncovering fine-grained entanglement structures distinguishing entanglement production, entanglement-spectrum reorganization, and dominant-Schmidt-scale transport beyond entropy alone.	interpretation 0.95	0.88	0.87	asserted 1.00

What kind of paper the classifier reports this is: method or tool 0.94 · evidence from simulation or computation 1.00.

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sources read

arXiv listing abstract v1

not examined

full text
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references and citing sentences
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 3.** (b) tests whether the early-time dynamical pattern persists beyond Hamiltonian dynamics. Starting from a half-filled product state, a random circuit with weakly entangling U(1)-conserving gates again produces a roughness maximum before dominant-Schmidtscale transport. A separate SWAP-only circuit exchanges Bell pairs prepared within the two halves and raises u ⋆ while keeping F = DNL min = 0, demonstrating dominantSchmidt-scale transport without roughness or magic. We further examine noninteracting Anderson localization and an interacting l-bit effective model for MBL; their…

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References

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MODELS, TIME EVOLUTION, AND ST A TISTICAL ANAL YSIS This section specifies the physical protocols, propagation algorithms, spectrum extraction, and statistical postpro- cessing used in every figure. The random-field XXZ model is H = X i Sx iSx i+1 +Sy iSy i+1 +Sz iSz i+1 +hiSz i , h i Uniform[W,W ], (S7) 3 with Sα i = σα i/2 and exchange scale J = 1 . The...

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HEAD EROSION AND THE ORDERING OF SPECTRAL EVENTS This section gives the minimal analytic head–tail model for the clocks. It separates two microscopic ways to increase entropy and derives the order of the roughness, magic, and octave thresholds. Write any ordered spectrum as λ0 = 1 ϵ andλi =ϵqi fori 1, with P iqi = 1. Here ϵ is the total tail weight and q ...

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(S30) To locate the roughness maximum, set a = 1 + 1/R and write F =ϵ(1 ϵ)(1 aϵ)2. Its derivative factorizes as dF dϵ = (aϵ 1) 4aϵ2 (3a + 2)ϵ + 1 . (S31) The smaller root of the quadratic is the first interior maximum reached from the product state, ϵF (R) = 2 + 3a p 9a2 4a + 4 8a , a = 1 + 1 R, (S32) which tends to 1/4 as R ! 1; the larger root is the la...

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= 1.230959. (S44) Before this crossing, the dominant concentration is the decreasing head branch η0 = (1 q)2; afterward, the first transported window η1 = 2q(1 q) is larger. Their upper envelope η⋆ = max fη0,η 1g therefore has a cusp-shaped minimumη⋆ = 4/9 at the clean crossing. On later coherent branches, each binary factor must be reordered by replacing...

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INTERACTING L-BIT DEPHASING AND DOMINANT-SCHMIDT-SCALE TRANSPOR T This section analyzes the dynamics of Anderson localization and an l-bit effective model of MBL. The analysis proceeds from the terms that change the Schmidt spectrum, to one reduced l-bit mode, to the many-pair spectrum, and finally to the logarithmic activation of increasingly distant pai...

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The two fidelity branches in Eq. ( S52) are equal at the same value, proving that the roughness maximum and the nonlocal-magic optimizer switch coincide for one dephasing mode. Equation ( S51) therefore fixes the common local roughness–magic threshold and shows why additional l-bit pairs are necessary for dominant-Schmidt-scale transport. 11 This binary m...

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NUMERICAL EVIDENCE AND MECHANISM CONTROLS This section gathers the extended data behind the three main figures. A. Early-time protocols and dominant-Schmidt-scale transport The first control determines why changing from the two active cut bonds of the main-text PBC Néel protocol to a single active OBC cut bond nearly merges the two middle clocks, and why ...

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SYMMETR Y-RESOL VED RANDOM-ST A TE BENCHMARKS The late-time thermal reference must respect the conserved total magnetization of the XXZ quench. This section derives the corresponding concentration and location baselines and quantifies the shift from an unconstrained Haar state. For an unconstrained Haar-random state on Cd Cd, where d = dim HA = dim HB is ...

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(S74) The normalizations Q0(0) = Q1(0) = 1 check the count and probability measures. The threshold y(α) is defined by Q0[y(α)] = α. Exactly as above, the octave mass is Q1[y(2α)] Q 1[y(α)], and its stationary point satisfies y(α) = 2y(2α). Numerical quadrature of these one-dimensional integrals gives α⋆ = 0.1792176 and η⋆ = 0.2582366. For nonlocal magic, ...

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(S23) To recover the dimension-independent constant in the last inequality, also use q =P2 p2. For p 1/2, unconstrained maximization over q gives q(p q) p2/4 1/16. For p 1/2, the parabola is decreasing throughout the allowed intervalq p2 p/2, so F p2(p p2) = p3(1 p) 27 256, (S24) where the final supremum occurs at p = 3/4. As established in Ref. [ 13], th...

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Forward citations

Cited by 4 Pith papers

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

Observing the magic Mpemba effect in localized dynamics on a digital quantum computer
quant-ph 2026-10 conditional novelty 7.0 of 10

In localized interacting spin systems, interaction-induced dephasing makes initially less magical states saturate at higher magic than more magical states, establishing a magic Mpemba effect in the absence of transport.
Universal entanglement embezzlement and divergent nonlocal magic from generic local chaotic quantum evolution
quant-ph 2026-09 conditional novelty 7.0 of 10

Local chaotic quantum evolution generically creates intermediate states that are universal entanglement embezzlers, with nonlocal magic diverging logarithmically with system size.
Nonlocal Magic across the Many-Body Localization Crossover
cond-mat.dis-nn 2026-09 accept novelty 7.0 of 10

Nonlocal magic evolves nonmonotonically with disorder and exposes a dome-shaped peak at intermediate disorder, providing a new probe of the ergodic-to-MBL crossover.
Entanglement Embezzlement from Diffusive Hydrodynamics
quant-ph 2026-09 conditional novelty 6.0 of 10

Conserved charge fluctuations convert into borrowable entanglement at a rate fixed by the charge bias, growing diffusively as the fourth root of time.

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

W. van Dam and P. Hayden, Universal entanglement transformations without communication, Physical Review A 67, 060302(R) (2003)

View on Pith 2003

[2] [2]

MODELS, TIME EVOLUTION, AND ST A TISTICAL ANAL YSIS This section specifies the physical protocols, propagation algorithms, spectrum extraction, and statistical postpro- cessing used in every figure. The random-field XXZ model is H = X i Sx iSx i+1 +Sy iSy i+1 +Sz iSz i+1 +hiSz i , h i Uniform[W,W ], (S7) 3 with Sα i = σα i/2 and exchange scale J = 1 . The...

View on Pith

[3] [3]

GENERAL GEOMETR Y OF SCHMIDT-SCALE CONCENTRA TION The entanglement-related observables obey useful bounds before any dynamical assumption is imposed. These bounds define the allowed region for the numerical trajectories and clarify how concentration constrains nonlocal magic. Let M = dlog2(r + 1) e and write the fixed shells introduced in Sec. 1 in zero-b...

View on Pith

[4] [4]

DNL min log2η⋆ minflog2M,S 2,S g. (S17) The maximum possible nonlocal magic therefore grows at most as log2 log2r even for spectra whose entropy grows as log2r. Ordering also constrains the position of a concentrated octave. Fix ℓ 1 and abbreviate its mass by m =ηℓ. Since theℓ + 1 entries inside the window satisfy λi λℓ, their sum obeys m (ℓ + 1)λℓ, or λℓ...

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[5] [5]

HEAD EROSION AND THE ORDERING OF SPECTRAL EVENTS This section gives the minimal analytic head–tail model for the clocks. It separates two microscopic ways to increase entropy and derives the order of the roughness, magic, and octave thresholds. Write any ordered spectrum as λ0 = 1 ϵ andλi =ϵqi fori 1, with P iqi = 1. Here ϵ is the total tail weight and q ...

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[6] [6]

(S30) To locate the roughness maximum, set a = 1 + 1/R and write F =ϵ(1 ϵ)(1 aϵ)2. Its derivative factorizes as dF dϵ = (aϵ 1) 4aϵ2 (3a + 2)ϵ + 1 . (S31) The smaller root of the quadratic is the first interior maximum reached from the product state, ϵF (R) = 2 + 3a p 9a2 4a + 4 8a , a = 1 + 1 R, (S32) which tends to 1/4 as R ! 1; the larger root is the la...

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[7] [7]

(S35) For the R = 63 spectrum plotted in Fig. 3(a) of the Letter, the three thresholds are ϵF = 0.24738, ϵD = 7/16, and ϵℓ = 63/95; in particular, ϵD(R) = 1 /2 1/(2 p R) + O(R−1) approaches its limit relatively slowly. The dominant Schmidt scale moves when the second branch of Eq. ( S29) exceeds 1 ϵ, giving the finite- R threshold above and ϵℓ ! 2/3. The ...

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[8] [8]

ACTIVE EXCHANGE BONDS ACROSS AN ENT ANGLEMENT CUT The head–tail model specifies thresholds in spectrum space without assigning them times. This section supplies a microscopic trajectory near the beginning of an XXZ quench, including the entropy-rate peak that spectral geometry alone cannot fix, and explains the dependence on bipartition cuts and initial s...

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[9] [9]

= 1.230959. (S44) Before this crossing, the dominant concentration is the decreasing head branch η0 = (1 q)2; afterward, the first transported window η1 = 2q(1 q) is larger. Their upper envelope η⋆ = max fη0,η 1g therefore has a cusp-shaped minimumη⋆ = 4/9 at the clean crossing. On later coherent branches, each binary factor must be reordered by replacing...

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[10] [10]

INTERACTING L-BIT DEPHASING AND DOMINANT-SCHMIDT-SCALE TRANSPOR T This section analyzes the dynamics of Anderson localization and an l-bit effective model of MBL. The analysis proceeds from the terms that change the Schmidt spectrum, to one reduced l-bit mode, to the many-pair spectrum, and finally to the logarithmic activation of increasingly distant pai...

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The two fidelity branches in Eq. ( S52) are equal at the same value, proving that the roughness maximum and the nonlocal-magic optimizer switch coincide for one dephasing mode. Equation ( S51) therefore fixes the common local roughness–magic threshold and shows why additional l-bit pairs are necessary for dominant-Schmidt-scale transport. 11 This binary m...

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[12] [12]

NUMERICAL EVIDENCE AND MECHANISM CONTROLS This section gathers the extended data behind the three main figures. A. Early-time protocols and dominant-Schmidt-scale transport The first control determines why changing from the two active cut bonds of the main-text PBC Néel protocol to a single active OBC cut bond nearly merges the two middle clocks, and why ...

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[13] [13]

SYMMETR Y-RESOL VED RANDOM-ST A TE BENCHMARKS The late-time thermal reference must respect the conserved total magnetization of the XXZ quench. This section derives the corresponding concentration and location baselines and quantifies the shift from an unconstrained Haar state. For an unconstrained Haar-random state on Cd Cd, where d = dim HA = dim HB is ...

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(S74) The normalizations Q0(0) = Q1(0) = 1 check the count and probability measures. The threshold y(α) is defined by Q0[y(α)] = α. Exactly as above, the octave mass is Q1[y(2α)] Q 1[y(α)], and its stationary point satisfies y(α) = 2y(2α). Numerical quadrature of these one-dimensional integrals gives α⋆ = 0.1792176 and η⋆ = 0.2582366. For nonlocal magic, ...

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(S23) To recover the dimension-independent constant in the last inequality, also use q =P2 p2. For p 1/2, unconstrained maximization over q gives q(p q) p2/4 1/16. For p 1/2, the parabola is decreasing throughout the allowed intervalq p2 p/2, so F p2(p p2) = p3(1 p) 27 256, (S24) where the final supremum occurs at p = 3/4. As established in Ref. [ 13], th...

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