bibkey: raveh2024dicke
authors: David Raveh and Rafael I. Nepomechie
year: 2024
title: Dicke states as matrix product states
doi: 10.1103/PhysRevA.110.052438
claim: Fixed-occupation uniform qudit words have orthonormal occupation sectors across each cut, positive multinomial-ratio Schmidt weights, rank equal to the feasible-sector count, and exact sequential occupation-isometry preparation.
strata_touched:
  - D5/S3/Quantum/Entanglement/OccupancyWordSectors
  - D5/S3/Quantum/Entanglement/CoherentHistorySchmidt
  - D5/S3/Quantum/Entanglement/SequentialOccupationHistory
license: citation-only
triage: anchor

Dicke states as matrix product states

Verified locator

DOI 10.1103/PhysRevA.110.052438 resolves to the Physical Review A article, volume 110, article 052438, published 26 November 2024. Both the DOI CSL metadata and https://arxiv.org/html/2408.04729v2 were retrieved successfully on 7 September 2026. Section 4 treats qudit Dicke states, defined by uniform superposition of permutations of a multiset. It gives normalized sector vectors, the product of prefix and suffix multinomial counts divided by the full multinomial count, and the minimal MPS bond dimension.

Delivered W foundation

The formal subset of this delivery is OccupancyWordSectors (W). It uses actual words Fin n -> alphabet and multiset occupations. Multiplicity is the cardinality of the actual word fiber; its multinomial and factorial formulas are proved from actual word counting. The sector Gram identity proves normalization and disjoint-support orthogonality. Boundary contains all feasible submultisets of the specified cut length. A restriction of Mathlib’s existing Sym.equivNatSumOfFintype identifies these actual Boundary carriers with the existing bounded TimeSlice coordinates and transports their counts. This W layer supplies a reusable foundation, with no whole-atom coverage claim.

C/S decomposition and sequential-chain mapping

CoherentHistorySchmidt (C) and SequentialOccupationHistory (S) extend the frozen OccupancyWordSectors foundation. C proves the arbitrary-cut normalized coefficient decomposition, positive binomial weights, and rank of the actual word coefficient matrix. S constructs occupation transitions and contractions, derives cut factorization for arbitrary finite algebraic chains, and proves that the least achievable maximum bond is attained. The complete coherent-history atom remains open because a pure-state circuit on one fixed physical register has not been constructed.

C’s cut coefficient matrix has the constant inverse-square-root amplitude on legal concatenations and zero elsewhere. The actual sector cardinality is identified with the factorial multinomial, and this bridge gives the product of coordinate binomial coefficients divided by the binomial cut count as the squared Schmidt coefficient. The actual 5040 rank sequence and unique maximum 12 at cut 4 then reuse BoundedTimeSlice, without a new count table.

schmidt_coefficient_sq_binomial consumes the actual-cardinality theorem and Mathlib’s Nat.choose_mul_factorial_mul_factorial on every coordinate and cut. schmidt_coefficient_eq_sqrt_binomial uses the positive coefficient to select the nonnegative root. These algebraic consequences introduce no new model premise.

On the frozen W base, C’s counting and two Gram formulas are companions of the coefficient construction; they do not supply a new W-count/W-gram first-freeze witness. Its support factorization remains live, and representative-word rows and columns produce the lower-bound diagonal minor. S’s Gram, word contraction and arbitrary-chain cut induction are separate live constructions. The channel consumes Gram but is not a premise of the exact occupation preparation or of the attained-minimum theorem.

Sequential construction mapping

Section 4 equations 4.7-4.10 were inspected in the retained HTML on 8 September 2026. Set paper l=t+1, total occupation k=a, prefix occupation a=b, and emitted label m=i. Equation 4.9 supports exactly c=b+{i}; equation 4.10 gives gamma squared as M(a-b-{i})/M(a-b), equal to the remaining count divided by the remaining length. The positive square root gives nextStep on the actual Boundary carriers. Its identity Gram proof uses unique extension support, singleton cancellation, and the sum of remaining counts. A singleton Kraus family reuses the repository’s canonical quantum channel API.

Equation 4.8 is implemented by contraction, which recursively multiplies step entries and sums over each intervening Boundary, with a terminal occupation basis-vector cap. contraction_eq_sector proves the stronger suffix equality by induction using multiplicity_erase_mul, the public actual-word consequence of the already ported erasure recurrence. history_sequential_preparation identifies the full contraction with the uniform word state in equation 4.1. The actual FiniteChain contraction supplies the factorization for every exact finite algebraic chain, with arbitrary complex caps. Taking the finite maximum of actual cut-bond cardinalities gives universal necessity. The occupationChain with occupationInitial attains the target boundary maximum and prepares the state exactly; IsLeast of actual achievable maximum-bond values records both claims together. Its concrete 5040 minimum is 12, by the general proof and the transported landed maximum.

The attaining construction has variable memory carriers and actual isometric steps. This proves finite MPS and variable-bond sequential-isometry optimality. Compatible embeddings and unitary extensions on a single fixed 12-dimensional physical register remain unconstructed. No time-homogeneous memory claim or identification with the separate 16-dimensional prediction model is made.

Lean prerequisite provenance

The normalized finite-class indicator argument is adapted from QuAIR/Lean-QIT, QIT/Symmetry/SymmetricSubspace.lean at immutable commit c1d59b133b56e3d79efb11ee46a728d290f761f5, specifically tensorPowerProfileUnitVector_trace_rankOne_eq_one and tensorPowerProfileUnitVector_inner. Copyright (c) 2026 QuAIR; authors QuAIR Team. The full Apache-2.0 license appears below. The upstream repository has no NOTICE file anywhere in its complete recursive tree. The adaptation changes recursive tensor words and realized-profile subtypes to Fin-indexed words, multisets, and explicit length proofs.

The upstream pins Lean v4.30.0 and Mathlib c5ea00351c28e24afc9f0f84379aa41082b1188f, incompatible with this repository’s v4.33.0 and db584cd6d46c92f209a44c0f1c829460d327499d. Retire the adapted proofs when this repository’s pinned Mathlib contains equivalent normalized finite-fiber indicator declarations, replacing them with direct imports. These are known results used as prerequisites, with no novelty claim.

The counting proof is a narrow A17.2 adaptation of the same immutable source: tensorPowerProfileClass_succ_card becomes multiplicity_succ; tensorPowerProfile_tail_factorial_prod_mul becomes erase_factorial_prod_mul; tensorPowerProfile_multinomial_tail_mul_length becomes multinomial_erase_mul; tensorPowerProfile_multinomial_succ_recurrence becomes multinomial_succ; tensorPowerProfileClass_card_eq_multinomial becomes sector_words_card_multinomial. These source declarations occupy lines 501-744. The proof keeps their induction on word length, head/tail recurrence, and positive factorial cancellation. The carrier adaptation replaces recursive tensor words with Fin-indexed words (upstream QIT/Util/TensorPower.lean supplies precisely this Fin.cons equivalence), and replaces realized profiles and tailAfterHead with multisets and erase. Pinned Multiset count/card/erase lemmas discharge the profile infrastructure. multiplicity_eq_factorial unfolds the resulting Mathlib multinomial.

Reverified immutable source SHA256: 72c5267dbd2cfebd6f68adec837ed8dc2ee1985926c053d34d1e06597cb47108. LICENSE SHA256: f3af7f9a7b5239d74bdc8bb43047820c1370caad7bd361777eda6ecca6bd874b. The GitHub recursive tree query returned the same commit, truncated=false, and no NOTICE paths. Copyright and license are retained below. Retire the counting adaptation when this repository’s pinned Mathlib provides equivalent fixed-occupation actual-word cardinality declarations, replacing it by imports.

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