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Superposing random states adds exactly ln(m) entanglement — or nothing at all

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2026-07-08 04:33 UTC pith:ULIDZC47

load-bearing objection Universal ln(m) entanglement enhancement for sub-maximally entangled superpositions, with a gap between proven examples and the general claim the 2 major comments →

arxiv 2607.06474 v1 pith:ULIDZC47 submitted 2026-07-07 quant-ph

Typical Entanglement of Superpositions

Damien Quinn , Joshuah T. Heath , Graham Kells This is my paper
classification quant-ph
keywords entanglementstatessuperpositionsentangledregimesub-maximallysuperpositiondensity
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.

The paper proves that when you superpose m independently drawn random quantum states, the entanglement gained across a bipartition depends cleanly on a single number: the 2nd Rényi entanglement density s₂ of the individual components. If s₂ is below the maximal value ln(2) — as it is for random Gaussian states and random matrix-product states — the reduced density matrices of distinct components become orthogonal in the thermodynamic limit. This orthogonality collapses the purity tensor onto its diagonal, and the entanglement of the superposition gains a universal, parameter-free increment of exactly ln(m). If instead s₂ sits at the Haar-typical maximum ln(2) — as for fully random states and random stabilizer states — the components are not orthogonal, and superposition produces only an intensive, system-size-independent relaxation toward the maximal value. The paper computes s₂ in closed form for both canonical sub-maximal examples: random fermionic Gaussian states yield s₂ ≈ 0.317 (46% of maximum, from a Wachter/arcsine law), and random MPS of bond dimension χ yield s₂ = ln(dχ/(d+1)) (from a Marchenko–Pastur law). In both cases, numerical simulations confirm the predicted ln(m) growth. An exponential number of superposed components is therefore required to bridge the gap from sub-maximal to maximal entanglement, which has direct consequences for classical simulation costs and quantum chemistry expansions.

Core claim

The central object is the purity tensor T_{ijkℓ}, a four-index object whose diagonal entries measure the self-overlap of each component's reduced density matrix and whose off-diagonal entries measure cross-overlaps between distinct components. The paper's key mechanical insight is that sub-maximal s₂ forces the cross-to-self overlap ratio to vanish exponentially in subsystem size (at rate c⊥ = ln(2) − s₂ > 0), which makes the purity tensor diagonally dominant. For an equal-weight m-component superposition, the purity then reduces to Tr(ρ²_A) ≈ (1/m)⟨e^{−S₂}⟩, giving the universal enhancement ΔS₂(m) = ln(m). This is a concrete realization of the Linden–Popescu–Smolin bound on entanglement of超

What carries the argument

The argument runs through three linked objects: (1) the 2nd Rényi entanglement entropy density s₂, which classifies ensembles into sub-maximal (s₂ < ln 2) or maximal (s₂ ~ ln 2); (2) the purity tensor T_{ijkℓ}, whose diagonal dominance (or lack thereof) controls whether superposition adds ln(m) or only O(1) entanglement; and (3) the cross-to-self overlap ratio C₀ e^{−c⊥|A|}, whose exponential decay — guaranteed by Schur's lemma fixing the cross-overlap at 2^{−|A|} while the self-overlap sits at e^{−s₂|A|} — is the mechanism that enforces orthogonality. For Gaussian states, s₂ is computed via the Wachter law for the eigenvalue distribution of covariance subblocks; for MPS, s₂ comes from the第四

Load-bearing premise

The argument relies on Schur's lemma to fix the ensemble-averaged reduced density matrix as maximally mixed, which pins the cross-overlap between two independent components at 2^{−|A|}. This holds when the relevant symmetry group acts irreducibly on the Fock sector, but for ensembles with additional structure — conserved charges, restricted symmetry sectors, or cases where concentration of measure fails — the cross-to-self ratio may not vanish and the ln(m) law could break.

What would settle it

Construct a random ensemble whose components have sub-maximal s₂ < ln(2) but whose symmetry group does not act irreducibly on the relevant Fock sector, such that the ensemble-averaged RDM is not maximally mixed. If the cross-to-self overlap ratio does not decay exponentially in |A| for such an ensemble, the purity tensor would not become diagonally dominant and the ln(m) enhancement would fail, contradicting the claimed universality.

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

If this is right

  • Classical simulation of fermionic systems using Gaussian superpositions is bounded: the entanglement of a state with Gaussian extent ξ = m is at most ⟨S_Gaussian⟩ + ln(ξ), so reaching Haar-typical entanglement requires exponentially large ξ, matching known polynomial-cost simulation thresholds.
  • Truncated configuration-interaction methods in quantum chemistry, which expand a state as m Slater determinants, are predicted to gain at most ~ln(m) excess entanglement over the single-determinant baseline, implying that strongly correlated regimes with extensive entanglement cannot be captured without exponentially many determinants.
  • The single axis s₂/ln(2) ∈ [0,1] collapses the hierarchy of state families — MPS at low bond dimension, Gaussians at intermediate density, Haar and stabilizer states at the top — into a unified classification with quantitative endpoint behaviors.
  • For the maximal class, the paper derives a parameter-free 1/m relaxation law (one-bit collapse for maximally entangled components) and a 1/m³ tail (Jensen gap for generic stabilizers), both independent of system size.

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 / 6 minor

Summary. This manuscript studies the entanglement of m-component equal-weight superpositions of random states drawn from structured ensembles. The central claim is a classification into two regimes: (i) a sub-maximally entangled class (s2 < ln 2), where the component reduced density matrices (RDMs) become two-sided orthogonal in the thermodynamic limit, yielding a universal logarithmic enhancement ΔS2(m) = ln(m); and (ii) a maximally entangled class (s2 ~ ln 2), where superposition produces only intensive O(1) corrections. The authors compute s2 analytically for fermionic Gaussian states (via the Wachter law) and random MPS (via the Marchenko–Pastur law), derive N-independent relaxation laws for stabilizer superpositions, and provide numerical confirmation. The logical chain from sub-maximal s2 to orthogonality to diagonal dominance of the purity tensor to ln(m) is clearly laid out.

Significance. The paper provides a clean and physically transparent framework that unifies several ensemble-specific results under a single axis s2/ln(2). The parameter-free derivation of ΔS2 = ln(m) from the orthogonality criterion (Eq. 5) is a strength, as are the closed-form s2 computations from random matrix laws (Wachter, Marchenko–Pastur) and the falsifiable numerical predictions confirmed in Figs. 2–4. The stabilizer analysis (Appendices D–E), including the 1/m^3 Jensen gap law and the exact deficit variance computation, is technically substantial. The connection to the fermionic Gaussian extent (Eq. 7) and the CI expansion bound gives the results practical bite beyond the random-state setting.

major comments (2)
  1. The central universality claim — that ANY ensemble with s2 < ln 2 exhibits ΔS2(m) = ln(m) — rests on the step from ensemble-averaged cross-overlap to typical (per-instance) diagonal dominance of the purity tensor. Eq. (5) and Appendix A2 show that E[Tr(ρ_A^ψ ρ_A^ϕ)] / E[Tr((ρ_A^ψ)^2)] → 0, but diagonal dominance of T_{ijkℓ} (Eq. 4) requires that the off-diagonal elements are negligible for typical instances, not merely that the ratio of ensemble averages vanishes. The paper acknowledges this gap implicitly by rigorously verifying concentration for the Gaussian case (Appendix B, via self-averaging of linear statistics) and the MPS case (via the MP law), but the general claim stated in the abstract and main text ('for any ensemble in the sub-maximally entangled class') does not establish that the cross-overlap concentrates around its mean. An ensemble could have s2 < ln 2 while cross-overl
  2. Appendix A2, paragraph on the one-sided-orthogonality criterion: the statement that 'a vanishing ratio makes [the cross-overlap] negligible beside the self-purity' conflates the ratio of expectations with the expectation of the ratio. The text says E[cross]/E[self] → 0 implies Tr(ρ_A^ψ ρ_A^ϕ) ≪ Tr((ρ_A^ψ)^2) for typical pairs, but this implication requires concentration of both numerator and denominator. For the Gaussian and MPS cases this is established (Appendix B, MP law), but the general criterion as stated in Eq. (5) and Appendix A2 does not include this as a hypothesis. The authors should either (a) restrict the general claim to ensembles where concentration is provable, or (b) add an explicit assumption that the cross-overlap concentrates, and note that this is verified for the canonical examples but is an additional condition beyond s2 < ln 2. This is load-bearing because the ln(
minor comments (6)
  1. The abstract states ΔS(m) = ln(m) while Eq. (1) and the main text use ΔS2(m) = ln(m). Consistency would help.
  2. Fig. 2 caption: the inset describes MPS superpositions with a bond-dimension cutoff, but the main panel for Gaussian states does not mention whether a similar cutoff or normalization is applied. Clarification would aid readability.
  3. The typesetting of Fig. 5 (Appendix E) appears corrupted in the manuscript, with garbled font characters rendering the axis labels and legends unreadable. This should be fixed in the final version.
  4. Eq. (6): the closed-form s2 = ln(2) - ln((3+2√2)/4) is stated as ≈ 0.317, but the text also mentions von Neumann density ≈ 0.386 without derivation. A brief reference to where this comes from (presumably a similar integral) would be helpful.
  5. The reference to Ref. [14] (Świętek et al., arXiv:2607.01326) appears to be a very recent preprint. The authors note it reached a 'similar conclusion' for fermionic Gaussian superpositions. A sentence clarifying the relationship — whether the present work subsumes, complements, or independently derives overlapping results — would help readers assess the novel contribution here.
  6. In Appendix B5, the comparison with full Haar-random states states c_Haar_⊥ = ln 2/|A|, which 'vanishes as |A| → ∞.' This is correct, but the subsequent statement that 'the cross-to-self ratio sits at 1/2 regardless of |A|' could be misread; a brief clarification that this 1/2 is the finite-size ratio (not the asymptotic limit) would improve the summary of the two classes.

Simulated Author's Rebuttal

2 responses · 0 unresolved

The referee raises a single, well-defined concern (stated as two related major comments): the gap between ensemble-averaged cross-overlap vanishing (Eq. 5) and per-instance concentration of that cross-overlap, which is needed for diagonal dominance of the purity tensor in typical instances. We agree this gap exists in the general statement and will revise the manuscript to make the concentration assumption explicit, restricting the universal claim accordingly.

read point-by-point responses

  1. Referee: The central universality claim — that ANY ensemble with s2 < ln 2 exhibits ΔS2(m) = ln(m) — rests on the step from ensemble-averaged cross-overlap to typical (per-instance) diagonal dominance of the purity tensor. Eq. (5) and Appendix A2 show that E[Tr(ρ_A^ψ ρ_A^ϕ)] / E[Tr((ρ_A^ψ)^2)] → 0, but diagonal dominance of T_{ijkℓ} (Eq. 4) requires that the off-diagonal elements are negligible for typical instances, not merely that the ratio of ensemble averages vanishes. The paper acknowledges this gap implicitly by rigorously verifying concentration for the Gaussian case (Appendix B, via self-averaging of linear statistics) and the MPS case (via the MP law), but the general claim stated in the abstract and main text ('for any ensemble in the sub-maximally entangled class') does not establish that the cross-overlap concentrates around its mean. An ensemble could have s2 < ln 2 while cross-overl

    Authors: The referee is correct that the step from vanishing ratio of expectations to per-instance diagonal dominance requires concentration, and that this is not established for a completely arbitrary ensemble with s2 < ln 2. We acknowledge this gap in the general claim and will revise the manuscript to address it. Specifically, we will: (1) add an explicit hypothesis in the statement of the general result (around Eq. 5 and in Appendix A2) that the cross-overlap concentrates around its mean — i.e., that Tr(ρ_A^ψ ρ_A^ϕ) / Tr((ρ_A^ψ)^2) → 0 holds not only in expectation but for typical instances; (2) restate the abstract and main-text claims to say that the ln(m) enhancement holds for any ensemble in the sub-maximally entangled class *that satisfies this concentration condition*, rather than unconditionally; (3) clarify that concentration is verified, not merely assumed, for the two canonical examples — fermionic Gaussian states (Appendix B, via self-averaging of the linear statistic with variance O((ln|A|)^2/|A|^2) from Johansson's fluctuation results) and random MPS (via the Marchenko–Pastur law). We agree that a pathological ensemble with s2 < ln 2 but non-concentrating cross-overlaps could evade the ln(m) law, and the revised text will say so. The core physical results of the paper — the parameter-free ln(m) derivation for the canonical ensembles, the closed-form s2 computations, and the stabilizer relaxation laws — are unaffected, since each rests on verified concentration rather than the general criterion alone. revision: partial

  2. Referee: Appendix A2, paragraph on the one-sided-orthogonality criterion: the statement that 'a vanishing ratio makes [the cross-overlap] negligible beside the self-purity' conflates the ratio of expectations with the expectation of the ratio. The text says E[cross]/E[self] → 0 implies Tr(ρ_A^ψ ρ_A^ϕ) ≪ Tr((ρ_A^ψ)^2) for typical pairs, but this implication requires concentration of both numerator and denominator. For the Gaussian and MPS cases this is established (Appendix B, MP law), but the general criterion as stated in Eq. (5) and Appendix A2 does not include this as a hypothesis. The authors should either (a) restrict the general claim to ensembles where concentration is provable, or (b) add an explicit assumption that the cross-overlap concentrates, and note that this is verified for the canonical examples but is an additional condition beyond s2 < ln 2. This is load-bearing because the ln(

    Authors: This is the same concern as the first major comment, focused on the specific wording in Appendix A2. The referee is right that the text as written conflates E[cross]/E[self] → 0 with E[cross/self] → 0, and we will fix this. We adopt the referee's option (b): we will add an explicit assumption that the cross-overlap concentrates around its mean (and that the self-purity concentrates around its typical value), state this as an additional condition beyond s2 < ln 2, and note that it is verified for the canonical examples (Gaussian states via self-averaging of linear statistics in Appendix B; MPS via the Marchenko–Pastur law). The wording in Appendix A2 — 'a vanishing ratio makes it negligible beside the self-purity' — will be revised to state that the vanishing ratio of expectations, *combined with concentration of both quantities*, yields per-instance orthogonality. We will also add a brief remark that the von Neumann result (Appendix A4) does not require this concentration step, as noted in the manuscript, since it follows from the exact decomposition S(ρ) = H({p_i}) + Σ p_i S(ρ_i) under orthogonal support. revision: partial

Circularity Check

0 steps flagged

No significant circularity; the ln(m) result follows from external mathematical inputs (Schur's lemma, Wachter/Marchenko–Pastur laws) and the definition of s₂, not from fitted parameters or self-cited premises.

full rationale

The central derivation chain is: (1) Schur's lemma (external, standard representation theory) fixes the ensemble-averaged cross-overlap at 2^{-|A|}; (2) the self-overlap e^{-s₂|A|} is the definition of s₂; (3) their ratio vanishes when s₂ < ln 2 (Eq. 5), which is arithmetic, not a fit; (4) vanishing ratio → orthogonality → diagonal purity tensor → ln(m) enhancement (Eq. 4, Appendix A4), a standard calculation for orthogonal mixtures. The s₂ values for Gaussian states and MPS are computed from external random-matrix laws (Wachter, Marchenko–Pastur), not fitted to superposition data. The numerical simulations in Fig. 2 confirm the predicted ln(m) scaling rather than defining it. The LPS bound [19–21] is cited as the origin of the orthogonality concept but is by external authors. One minor self-citation exists (Ref. [47], Heath as co-author) in the matchgate discussion, but it is not load-bearing for the central ln(m) result. The gap between ensemble-averaged and typical orthogonality—flagged by the skeptic—is a correctness/completeness concern, not a circularity issue: the paper's equations do not reduce to their own inputs by construction. Score 1 reflects the minor non-load-bearing self-citation.

Axiom & Free-Parameter Ledger

1 free parameters · 5 axioms · 0 invented entities

The paper introduces no new physical entities, particles, forces, or postulated objects. All mathematical objects (purity tensor, orthogonality rate c_perp, rank deficit delta) are defined in terms of standard quantum information quantities. The single free parameter C0 is determined by symmetry, not fitted. The axioms are standard results from representation theory and random matrix theory. No ad hoc constructions are introduced.

free parameters (1)
  • C0 = 1 for BdG; 2/sqrt(3) for number-conserving at half-filling
    O(1) prefactor in the orthogonality ratio (Eq. 5) accounting for symmetry sectors. Not fitted to data but determined by the ensemble's symmetry structure.
axioms (5)
  • standard math Schur's lemma: the ensemble-averaged RDM is maximally mixed when the symmetry group acts irreducibly on the relevant Fock sector.
    Invoked in Eq. (5), Appendix A2, and Appendix B1 to fix the cross-overlap at 2^{-|A|}. Standard representation theory result.
  • domain assumption Concentration of measure / self-averaging of linear statistics of random matrix eigenvalues.
    Invoked in Appendix B2 (Eq. B5) to assert that the per-mode orthogonality rate c_perp self-averages. Standard in random matrix theory but a load-bearing assumption for the thermodynamic limit.
  • domain assumption The Wachter law describes the limiting spectral distribution of the Jacobi (MANOVA) ensemble formed by Gaussian covariance subblocks.
    Invoked in the Gaussian s2 computation (main text and Appendix B4). Established result in random matrix theory (Refs. 32-33).
  • domain assumption Random stabilizer states form an exact 3-design (Refs. 51-52).
    Invoked in Appendix E3 to pin the mean purity to the Haar value for all m. Established result.
  • domain assumption The Schmidt spectrum of a random MPS converges to the Marchenko-Pastur distribution with aspect ratio gamma=1/d.
    Invoked in the MPS s2 computation (main text, Eq. 9, Fig. 4). Supported by Ref. 34 and numerical confirmation, but the specific gamma=1/d identification appears to be a claim of this paper.

pith-pipeline@v1.1.0-glm · 25038 in / 3384 out tokens · 596252 ms · 2026-07-08T04:33:09.540230+00:00 · methodology

0 comments
read the original abstract

We investigate universal entanglement properties inherent to superpositions of randomized states. We find that an $m$-fold superposition of typical states may be classified into two distinct entanglement classes via the 2nd R\'enyi entropy density $s_2$. The maximally entangled regime is defined by $s_2 \sim \ln (2)$, for which superposition adds no additional entanglement. The sub-maximally entangled regime, $s_2<\ln 2$, instead constrains the reduced density matrices of independent components to be orthogonal in the thermodynamic limit, which fixes the entanglement of the superposition to a logarithmic enhancement $\Delta S(m)=\ln (m)$. As a consequence, an exponentially large number of superpositions is required to transition from the sub-maximally entangled class to maximal entanglement. We explicitly calculate $s_2$ and the logarithmic enhancement, and demonstrate orthogonality for two canonical examples of the sub-maximally entangled regime (superpositions of pure Gaussian states and of random matrix-product states). We also examine the entanglement of superpositions of random stabilizer states, and discuss their relaxation to the Haar limit.

Figures

Figures reproduced from arXiv: 2607.06474 by Damien Quinn, Graham Kells, Joshuah T. Heath.

Figure 1
Figure 1. Figure 1: Schematic of the two entanglement classes for typ [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The 2nd Rényi entanglement entropy ⟨S2(m)⟩ ver￾sus number of Gaussian components m for a system size of N = 100 at equal bipartition. The dashed baseline is the single-component average ⟨S2(1)⟩. The ln(m) growth pre￾dicted by Eq. (1) is confirmed up to m = 70. (Inset) S2 entan￾glement entropy of superpositions of m random MPS (d = 2, N = 100, χ = 10) with total bond-dimension cutoff χmax = 100. The ln(m) s… view at source ↗
Figure 3
Figure 3. Figure 3: Entanglement profile across all partitions for su [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Schmidt spectrum for a random MPS for a few [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (Top) Numerical calculations of the maximally entangled [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

79 extracted references · 79 canonical work pages · 8 internal anchors

  1. [1]

    Beyond this regime, the entropy plateaus atS2 ≈ln(dχ max/(d+1))

    Theln(m)scaling holds whilemχ≤χ max. Beyond this regime, the entropy plateaus atS2 ≈ln(dχ max/(d+1)). Orthogonality and entanglement scaling for typical superpositions.—Theln(m)scaling law rests on a logical chain of equivalent conditions on the components of the ensemble; namely, a sub-maximal entanglement density (i.e.,s 2 <ln(2)) forces the component R...

    work page

  2. [2]

    Superposition removes the single-component gapJ(1) and moves the entanglementdownonto the Haar-typical s2, essentially complete bym= 2with a1/m 3 tail. Conclusions.– We have shown that the enhancement of entanglement via superpositions of random states de- pends on whether the components of the reduced density matrices have sub-maximal 2nd Rényi entanglem...

    work page 2026

  3. [3]

    (A2) reduces to a diagonal under either of two averages

    Surviving index patterns The quadruple sum in Eq. (A2) reduces to a diagonal under either of two averages. Over a random ensemble of independent componentsE|ψ a⟩= 0, soE[T ijkℓ]vanishes unless every component enters once as a ket and once as a bra; the only such pairings are thedirect(i=j, k=ℓ) and theswap(i=ℓ, k=j). These are the two elements of the two-...

    work page

  4. [4]

    (A4) collapse to the univer- sal cross-overlap2 −|A|: by independenceE[Tr(ρ A ψ ρA ϕ )] = Tr((¯ρA)2) = 2−|A|, fixed by the Hilbert-space dimension alone

    The one-sided-orthogonality criterion For an ensemble whose average RDM is maximally mixed,¯ρA =I/2 |A| (Schur’s lemma, whenever a group 6 acts irreducibly on the symmetry-restricted sector), both off-diagonal terms of Eq. (A4) collapse to the univer- sal cross-overlap2 −|A|: by independenceE[Tr(ρ A ψ ρA ϕ )] = Tr((¯ρA)2) = 2−|A|, fixed by the Hilbert-spa...

    work page

  5. [5]

    The direct and swap pairings ofT(Eq

    The two sides of orthogonality The reduced state of the superposition splits into an incoherent part and a coherent one, ρA = X i pi ρ(i) A + X i̸=k λiλ∗ k TrB |ψi⟩⟨ψk|,(A5) withp i =|λ i|2; the first sum is the incoherent mixture, the second the coherences. The direct and swap pairings ofT(Eq. (A4)) control these two parts, and orthogonal- ity on the two...

    work page

  6. [6]

    This is the two-sided-orthogonal mixture of Sec

    Entropy of an orthogonal mixture Consider an incoherent mixtureρ= P i piρi whose components satisfyTr(ρ iρj) = 0fori̸=j. This is the two-sided-orthogonal mixture of Sec. A3. Von Neumann entropy.Using the concavity ofS and the orthogonality condition,S(ρ) =H({p i}) +P i piS(ρi), whereHis the Shannon entropy of the weights. For equal weights:S= ln(m) +⟨S(ρ ...

    work page

  7. [7]

    A natural question is whether theln(m)scaling persists for general coefficient distribu- tions

    Non-equal weights The results in the main text focus on equal-weight su- perpositions|λ k|2 = 1/m. A natural question is whether theln(m)scaling persists for general coefficient distribu- tions. For arbitrary weights{pk =|λ k|2}summing to unity, the purity of the reduced state under RDM orthogonality is Tr(ρ2 A)≈ X k p2 k e−S(k) 2 ,(A6) and the2nd Rényi e...

    work page

  8. [8]

    The maximally mixed average RDM The average RDM¯ρA =E[ρ A ψ]is fixed entirely by symmetry. Haar-random Gaussian states form the orbit { ˆUO |0⟩:O∈O(2N)}of the vacuum under Bogoliubov transformations, and the representation ofSpin(2N)on the even-parity Fock spaceH+ (dimension2 N−1) is ir- reducible [35]. Schur’s lemma therefore pins the ensem- ble average ...

    work page

  9. [9]

    The typical purity The self-overlap follows from Peschel’s factoriza- tion [41]. In the eigenbasis of the covariance subblock ΓA, the Gaussian RDM is a product over modes, ρA = |A|O k=1 1 +ϵ k 2 |0k⟩⟨0k|+ 1−ϵ k 2 |1k⟩⟨1k| ,(B3) where±iϵ k are the eigenvalues ofΓ A andϵ k ∈[0,1]. Each mode contributes independently, so Tr((ρA)2) = |A|Y k=1 " 1+ϵk 2 2 + 1−ϵ...

    work page

  10. [10]

    TheB-side orthogonality follows by the identical argument, since ∥TrB(|ψ⟩⟨ϕ|)∥2 HS = Tr(ρ B ψ ρB ϕ )obeys the same two es- timates

    Exponential decay of the ratio Dividing the cross-overlap by the self-purity gives the one-sided-orthogonality ratio for the Gaussian class, E[Tr(ρA ψ ρA ϕ )] E[Tr((ρA ψ)2)] =C 0 e−c⊥|A|+o(|A|) →0,(B6) withC 0 = 1for BdG states andC0 →2/ √ 3as|A| → ∞ for number-conserving states at half filling. TheB-side orthogonality follows by the identical argument, s...

    work page

  11. [11]

    Explicitc ⊥ at equal bipartition Atα= 1/2, for both BdG and number-conserving states at half filling, the Wachter distribution reduces to the arcsine formρλ(λ) = 2/[π √ 1−λ 2]on[0,1][11, 43]. Substitutingλ= sinθ: c⊥ = 2 π Z π/2 0 ln(1 + sin2 θ)dθ.(B7) Writing1+sin 2 θ= cos 2 θ+2 sin2 θandapplyingthestan- dard identity R π/2 0 ln(a2 cos2 θ+b 2 sin2 θ)dθ=πl...

    work page

  12. [12]

    The denominator, however, is barely above the floor: using the 2-replica method, the typical purity is (dA +d B)/(dAdB + 1)≈2 −|A| + 2−(N−|A|)

    Comparison with full Haar-random states For states drawn from the full Fock space, the av- erage RDM is alsoI/2 |A|, so the numerator is the same. The denominator, however, is barely above the floor: using the 2-replica method, the typical purity is (dA +d B)/(dAdB + 1)≈2 −|A| + 2−(N−|A|). At equal bipartition this gives⟨S 2⟩=|A|ln 2−ln 2and hence cHaar ⊥...

    work page

  13. [13]

    Us- ing the ricochet identity(M⊗I B)|Ω⟩= (I A ⊗M T)|Ω⟩, theA-side unitary can be moved to theBside,|ψ⟩= (IA ⊗V)|Ω⟩withV=U BU T A

    Connecting-unitary reduction Any maximally entangled state has flat, full-rank Schmidt form and can be written as|ψ⟩= (UA ⊗U B)|Ω⟩, where|Ω⟩=d −1/2P k |k⟩A |k⟩B is a fixed reference. Us- ing the ricochet identity(M⊗I B)|Ω⟩= (I A ⊗M T)|Ω⟩, theA-side unitary can be moved to theBside,|ψ⟩= (IA ⊗V)|Ω⟩withV=U BU T A. Because every component shares the one refer...

    work page

  14. [14]

    Purity ExpandingY Y † =P jk λjλ∗ k(V † k Vj)T, thediagonalj= kterms sum to P j |λj|2 I=Ifor normalised weights, leaving ρA = I+X Nd , X= X j̸=k λjλ∗ k(V † k Vj)T,(D3) the interference operatorXbeing built from pairs of dis- tinct components. Squaring and tracing, Tr(ρ2 A) = Tr (I+X) 2 N 2d2 = d+ 2 TrX+ Tr(X 2) N 2d2 .(D4) NowTrX=d(N −1)(fromTr(Y Y †) =dN)...

    work page

  15. [15]

    +O(1/ √ d), and inserting this intodTr(ρ2 A) = 1 + 1 dTr(X2)gives, for any weight profile, Tr(ρ2 A) = 2−|A| 2− ∥p∥ 2 2 +O(d −1/2).(D7) For equal weights∥p∥2 2 = 1/mthis is Tr(ρ2 A) = 2−|A| 2− 1 m +O(d −1/2),(D8) Taking the negative of the natural log of the above yields ⟨S2(m)⟩= (|A| −log 2(2−1/m)) ln 2, so the gain over the single (maximal) component is∆...

    work page

  16. [16]

    The rank deficit A stabilizer state is flat on its Schmidt support, with purityTr(ρ 2 A) = 2 −(|A|−δ), where the non-negative inte- gerδ=|A| −S 2/ln(2)is therank deficit—the number of bits by which the entanglement falls short of the maximal valueS 2 =|A|ln 2. Equivalently,δcounts the indepen- dent stabilizer generators that can be supported entirely onA,...

    work page

  17. [17]

    nothing is privileged

    Group-membership probabilities Two facts about a uniformly random stabilizer group Gdrive the moment count below. Recall that a stabi- lizer group is a set of2 N mutually commuting Paulis closed under multiplication—the identity together with 2N −1nontrivial elements. No nontrivial Pauli is privi- leged over any other: any one can be relabelled into any o...

    work page

  18. [18]

    Random stabilizer states form an exact 3-design [51, 52], which gives the design pinning: the mean purity equals the Haar value exactly, for everym

    Design pinning The purity of the superposition contains each compo- nent projector to at most second order, so its ensemble mean is a 2-design quantity. Random stabilizer states form an exact 3-design [51, 52], which gives the design pinning: the mean purity equals the Haar value exactly, for everym. (Consistently, the single-state mean purity ⟨Trρ2 A⟩= 2...

    work page

  19. [19]

    Form≥2these fluctuations are small, so write the purity asTrρ 2 A =⟨Trρ 2 A⟩(1 +ϵ)withϵ= (Trρ2 A − ⟨Trρ2 A⟩)/⟨Trρ2 A⟩, so that⟨ϵ⟩= 0and⟨ϵ 2⟩= Var(Trρ2 A)/⟨Trρ2 A⟩2

    The Jensen gap Since the mean purity is pinned, the entirem- dependence of⟨S 2⟩is the gap between the mean of the logarithm and the logarithm of the mean—the Jensen gap, ⟨S2⟩=−ln Trρ2 A +J,(E4) J= ln Trρ2 A − ln Trρ2 A ≥0,(E5) with the first term fixed at the Haar value;Jis sourced entirely by the fluctuations of the purity across the en- semble. Form≥2th...

    work page

  20. [20]

    The1/m 3 law Which fluctuation survives is clearest in the purity- tensor form of Eq. (3): for the equal-weight, random- phase superposition the purity carries an overall1/m 2 from the normalisation, Trρ2 A = 1 m2 X ijkℓ e i(ϕi−ϕj+ϕk−ϕℓ) Tijkℓ ,(E9) and averaging over the random phases retains only the index patterns whose phases cancel: thedirectpairing ...

    work page

  21. [21]

    Fix one realisation—a single random stabilizer state, with groupG

    The deficit variance To assemble the moments we must be careful about what is random. Fix one realisation—a single random stabilizer state, with groupG. Recallthat the rank deficit δis the number of independent stabilizer generators sup- ported fully onA. Theseδgenerators produce a sub- group of2 δ Paulis, all supported onA, so2 δ is simply the number ofA...

    work page 2026

  22. [22]

    D. N. Page, Average entropy of a subsystem, Phys. Rev. Lett.71, 1291 (1993)

    work page 1993

  23. [23]

    S. K. Foong and S. Kanno, Proof of Page’s conjecture on the average entropy of a subsystem, Phys. Rev. Lett.72, 1148 (1994)

    work page 1994

  24. [24]

    Sánchez-Ruiz, Simple proof of Page’s conjecture on the average entropy of a subsystem, Phys

    J. Sánchez-Ruiz, Simple proof of Page’s conjecture on the average entropy of a subsystem, Phys. Rev. E52, 5653 (1995)

    work page 1995

  25. [25]

    Sen, Average Entropy of a Quantum Subsystem, Phys

    S. Sen, Average Entropy of a Quantum Subsystem, Phys. Rev. Lett.77, 1 (1996)

    work page 1996

  26. [26]

    Goldstein, J

    S. Goldstein, J. L. Lebowitz, R. Tumulka, and N. Zanghì, CanonicalTypicality,Phys.Rev.Lett.96,050403(2006)

    work page 2006

  27. [27]

    Popescu, A

    S. Popescu, A. J. Short, and A. Winter, Entanglement and the foundations of statistical mechanics, Nat. Phys. 2, 754 (2006)

    work page 2006

  28. [28]

    Reimann, Typicality for Generalized Microcanonical Ensembles, Phys

    P. Reimann, Typicality for Generalized Microcanonical Ensembles, Phys. Rev. Lett.99, 160404 (2007)

    work page 2007

  29. [29]

    M. M. Wilde, Quantum Typicality, inQuantum Infor- mation Theory(CambridgeUniversityPress,Cambridge, England, UK, 2017) pp. 443–466

    work page 2017

  30. [30]

    J. M. Magán, Random Free Fermions: An Analytical Example of Eigenstate Thermalization, Phys. Rev. Lett. 116, 030401 (2016)

    work page 2016

  31. [31]

    Bhattacharjee, P

    B. Bhattacharjee, P. Nandy, and T. Pathak, Eigenstate capacity and Page curve in fermionic Gaussian states, Phys. Rev. B104, 214306 (2021)

    work page 2021

  32. [32]

    Bianchi, L

    E. Bianchi, L. Hackl, and M. Kieburg, Page curve for fermionic Gaussian states, Phys. Rev. B103, L241118 (2021)

    work page 2021

  33. [33]

    Bianchi, L

    E. Bianchi, L. Hackl, M. Kieburg, M. Rigol, and L. Vid- mar,Volume-LawEntanglementEntropyofTypicalPure Quantum States, PRX Quantum3, 030201 (2022)

    work page 2022

  34. [34]

    X.-H. Yu, Z. Gong, and J. I. Cirac, Free-fermion Page curve: Canonical typicality and dynamical emergence, Phys. Rev. Res.5, 013044 (2023)

    work page 2023

  35. [35]

    One-Body Purity, Non-Gaussianity, and Entanglement in Interacting Integrable Models

    R. Świętek, M. Kliczkowski, L. Vidmar, and M. Rigol, One-Body Purity, Non-Gaussianity, and Entan- glement in Interacting Integrable Models, arXiv 10.48550/arXiv.2607.01326 (2026), 2607.01326

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2607.01326 2026

  36. [36]

    Garnerone, T

    S. Garnerone, T. R. de Oliveira, and P. Zanardi, Typical- ity in random matrix product states, Phys. Rev. A81, 032336 (2010)

    work page 2010

  37. [37]

    Garnerone, T

    S. Garnerone, T. R. de Oliveira, S. Haas, and P. Zanardi, Statistical properties of random matrix product states, Phys. Rev. A82, 052312 (2010)

    work page 2010

  38. [38]

    Haferkamp, C

    J. Haferkamp, C. Bertoni, I. Roth, and J. Eisert, Emer- gent Statistical Mechanics from Properties of Disor- dered Random Matrix Product States, PRX Quantum 13 2, 040308 (2021)

    work page 2021

  39. [39]

    Smith and D

    G. Smith and D. Leung, Typical entanglement of stabi- lizer states, Phys. Rev. A74, 062314 (2006)

    work page 2006

  40. [40]

    Linden, S

    N. Linden, S. Popescu, and J. A. Smolin, Entanglement of Superpositions, Phys. Rev. Lett.97, 100502 (2006)

    work page 2006

  41. [41]

    Gour, Reexamination of entanglement of superposi- tions, Phys

    G. Gour, Reexamination of entanglement of superposi- tions, Phys. Rev. A76, 052320 (2007)

    work page 2007

  42. [42]

    Gour and A

    G. Gour and A. Roy, Entanglement of subspaces in terms of entanglement of superpositions, Phys. Rev. A77, 012336 (2008)

    work page 2008

  43. [43]

    Szabo and N

    A. Szabo and N. S. Ostlund,Modern Quantum Chem- istry: Introduction to Advanced Electronic Structure The- ory(Courier Corporation, 1996)

    work page 1996

  44. [44]

    McArdle, S

    S. McArdle, S. Endo, A. Aspuru-Guzik, S. C. Benjamin, and X. Yuan, Quantum computational chemistry, Rev. Mod. Phys.92, 015003 (2020)

    work page 2020

  45. [45]

    Giuliani, J

    C. Giuliani, J. Nys, R. Martinazzo, G. Carleo, and R. Rossi, Precise Quantum Chemistry calculations with few Slater Determinants, Nat. Commun.17, 5613 (2026)

    work page 2026

  46. [46]

    Gaussian decomposition of magic states for matchgate computations

    J. Cudby and S. Strelchuk, Gaussian decomposition of magic states for matchgate computations, arXiv 10.48550/arXiv.2307.12654 (2023), 2307.12654

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2307.12654 2023

  47. [47]

    Improved simulation of quantum circuits dominated by free fermionic operations

    O. Reardon-Smith, M. Oszmaniec, and K. Korzekwa, Improved simulation of quantum circuits dominated by free fermionic operations, Quantum8, 1549 (2024), 2307.12702v4

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  48. [48]

    Classical simulation of non-Gaussian fermionic circuits

    B. Dias and R. Koenig, Classical simulation of non- Gaussian fermionic circuits, Quantum8, 1350 (2024), 2307.12912v3

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  49. [49]

    P. W. Anderson, Resonating valence bonds: A new kind of insulator?, Mater. Res. Bull.8, 153 (1973)

    work page 1973

  50. [50]

    Baskaran and P

    G. Baskaran and P. W. Anderson, Gauge theory of high-temperature superconductors and strongly corre- lated Fermi systems, Phys. Rev. B37, 580 (1988)

    work page 1988

  51. [51]

    MisguichandC

    G. MisguichandC. Lhuillier,Two-DimensionalQuantum Antiferromagnets, inFrustrated Spin Systems(World Scientific, Singapore, 2005) pp. 229–306

    work page 2005

  52. [52]

    Boutin and B

    S. Boutin and B. Bauer, Quantum impurity models us- ing superpositions of fermionic Gaussian states: Practi- cal methods and applications, Phys. Rev. Res.3, 033188 (2021)

    work page 2021

  53. [53]

    K. W. Wachter, The Strong Limits of Random Matrix Spectra for Sample Matrices of Independent Elements, Annals of Probability6, 1 (1978)

    work page 1978

  54. [54]

    K. W. Wachter, The Limiting Empirical Measure of Mul- tiple Discriminant Ratios, Ann. Stat.8, 937 (1980)

    work page 1980

  55. [55]

    Distribution of eigenvalues for some sets of random matrices

    L. A. P. V. A. Marchenko, “Distribution of eigenvalues for some sets of random matrices”, Math. USSR-Sb., 1:4 (1967), 457–483 (2026)

    work page 1967

  56. [56]

    Georgi,Lie Algebras In Particle Physics : from Isospin To Unified Theories(Taylor & Francis, Andover, England, UK, 2000)

    H. Georgi,Lie Algebras In Particle Physics : from Isospin To Unified Theories(Taylor & Francis, Andover, England, UK, 2000)

    work page 2000

  57. [57]

    Serre,Linear Representations of Finite Groups (Springer, New York, NY, USA, 1977)

    J.-P. Serre,Linear Representations of Finite Groups (Springer, New York, NY, USA, 1977)

    work page 1977

  58. [58]

    Introduction to representation theory

    P. Etingof, O. Golberg, S. Hensel, T. Liu, A. Schwendner, D. Vaintrob, and E. Yudovina, Introduction to represen- tation theory, arXiv 10.48550/arXiv.0901.0827 (2009), 0901.0827

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.0901.0827 2009

  59. [59]

    two-sided orthogonality

    We use “two-sided orthogonality” rather than LPS’s “biorthogonality” to avoid confusion with the non- Hermitian usage

    work page

  60. [60]

    Two-sided orthogonality is also necessary for a universal closed form, as illustrated by the Bell states 1√ 2(|00⟩ ± |11⟩)[19], each maximally entangled withTr(ρ A ϕ ρA ψ) = Tr(ρB ϕ ρB ψ ) = 1 2, yet superposing to an unentangled state

    work page

  61. [61]

    Lagrangian representation for fermionic linear optics

    S. Bravyi, Lagrangian representation for fermionic linear optics, arXiv 10.48550/arXiv.quant-ph/0404180 (2004), quant-ph/0404180

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.quant-ph/0404180 2004

  62. [62]

    Peschel, Calculation of reduced density matrices from correlation functions, J

    I. Peschel, Calculation of reduced density matrices from correlation functions, J. Phys. A: Math. Gen.36, L205 (2003)

    work page 2003

  63. [63]

    Johansson, On fluctuations of eigenvalues of random Hermitian matrices, Duke Math

    K. Johansson, On fluctuations of eigenvalues of random Hermitian matrices, Duke Math. J.91, 151 (1998)

    work page 1998

  64. [64]

    Bianchi and P

    E. Bianchi and P. Donà, Typical entanglement entropy in the presence of a center: Page curve and its variance, Phys. Rev. D100, 105010 (2019)

    work page 2019

  65. [65]

    B. M. Terhal and D. P. DiVincenzo, Classical simulation of noninteracting-fermion quantum circuits, Phys. Rev. A65, 032325 (2002)

    work page 2002

  66. [66]

    Jozsa and A

    R. Jozsa and A. Miyake, Matchgates and classical simu- lation of quantum circuits, Proc. A464, 3089 (2008)

    work page 2008

  67. [67]

    L. G. Valiant, Quantum Circuits That Can Be Simu- lated Classically in Polynomial Time, SIAM J. Comput. (2012)

    work page 2012

  68. [68]

    A. M. Projansky, J. T. Heath, and J. D. Whitfield, En- tanglement spectrum of matchgate circuits with univer- sal and non-universal resources, Quantum8, 1432 (2024), 2312.08447v3

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  69. [69]

    Paviglianiti, L

    A. Paviglianiti, L. Lumia, E. Tirrito, A. Silva, M. Col- lura, X. Turkeshi, and G. Lami, Emergence of Generic Entanglement Structure in Doped Matchgate Circuits, Phys. Rev. Lett.136, 020403 (2026)

    work page 2026

  70. [70]

    Phys.326, 96 (2011)

    U.Schollwöck,Thedensity-matrixrenormalizationgroup in the age of matrix product states, Ann. Phys.326, 96 (2011)

    work page 2011

  71. [71]

    Collins, I

    B. Collins, I. Nechita, and K. Życzkowski, Area law for random graph states, J. Phys. A: Math. Theor.46, 305302 (2013)

    work page 2013

  72. [72]

    Webb, The Clifford group forms a unitary 3-design, Quantum Inf

    Z. Webb, The Clifford group forms a unitary 3-design, Quantum Inf. Comput.16, 1379 (2016)

    work page 2016

  73. [73]

    Zhu, Multiqubit Clifford groups are unitary 3-designs, Phys

    H. Zhu, Multiqubit Clifford groups are unitary 3-designs, Phys. Rev. A96, 062336 (2017)

    work page 2017

  74. [74]

    J. L. W. V. Jensen, Om konvekse Funktioner og Uligheder imellem Middelværdier, Nyt tidsskrift for matematik , 49 (1905)

    work page 1905

  75. [75]

    J. L. W. V. Jensen, Sur les fonctions convexes et les in- égalités entre les valeurs moyennes, Acta Math.30, 175 (1906)

    work page 1906

  76. [76]

    Abramovich and L.-E

    S. Abramovich and L.-E. Persson, Some new estimates of the ‘Jensen gap’, J. Inequal. Appl.2016, 39 (2016)

    work page 2016

  77. [77]

    Entanglement in the stabilizer formalism

    D. Fattal, T. S. Cubitt, Y. Yamamoto, S. Bravyi, and I. L. Chuang, Entanglement in the stabilizer for- malism, arXiv:quant-ph/0406168 (2004), arXiv:quant- ph/0406168

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  78. [78]

    O. C. O. Dahlsten and M. B. Plenio, Entanglement prob- ability distribution of bi-partite randomised stabilizer states, Quantum Information and Computation6, 527 (2006)

    work page 2006

  79. [79]

    Bravyi and D

    S. Bravyi and D. Gosset, Improved Classical Simulation of Quantum Circuits Dominated by Clifford Gates, Phys. Rev. Lett.116, 250501 (2016)

    work page 2016