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Identifiability and minimality bounds of quantum and post-quantum models of classical stochastic processes

T0 review · 3 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read Two quantum models of the same classical process can be compared by mapping each to a canonical linear model, and the smallest possible quantum memory must have dimension at least the square root of the canonical model's size.

desk verdict A genuinely new identifiability test and dimension bound for quantum generators of classical processes; the core is a standard vectorization trick, but the paper works it through carefully, with two addressable proof gaps. read the letter →

arxiv 2509.03004 v1 pith:BOEF6RTU submitted 2025-09-03 quant-ph cond-mat.stat-mechcs.CLcs.FLcs.ITmath.IT

Paul M. Riechers , Thomas J. Elliott This is my paper
classification quant-phcond-mat.stat-mechcs.CLcs.FLcs.ITmath.IT
keywords quantumhiddenMarkovmodelsidentifiabilitygeneralizedclassicalstochasticprocessesmemoryadvantageminimaldimensionpost-quantumcanonicalGHMM
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper resolves the identifiability problem for quantum hidden Markov models (QHMMs): it shows how to decide, in finite time, whether any two generators—classical, quantum, or post-quantum—produce the same classical stochastic process. The key move is to recast every QHMM as a generalized hidden Markov model (GHMM) with at most d^2 latent dimensions, then compare both models against a canonical minimal GHMM built from minimal history and future wordlists. Along the way the paper proves that the memory dimension of the smallest possible QHMM of a process is at least the square root of the size of the minimal GHMM, and gives examples showing this square-root bound can be met or missed. A reader should care because quantum models are known to simulate classical processes with less memory than any classical model; this work supplies the missing 'same process or not' test and a hard lower bound on how far that quantum compression can go.

What carries the argument

The canonical standard GHMM: given a minimal history wordlist and a minimal future wordlist, form the HF matrix of conditional probabilities, and use its inverse to express the initial vector and the transition matrices B(x) in a one-hot basis. Any generator—HMM, QHMM, or GHMM—of the same process maps to the same standard GHMM (unique once the alphabet is ordered), so equivalence is equality of the canonical objects. The lower bound follows because a d-dimensional QHMM vectorizes to a d^2-dimensional GHMM and minimal GHMMs have exactly ell_min states.

What would settle it

Produce two d-dimensional QHMMs that agree on all words of length up to 2d^2 - 1 but differ on a longer word, or exhibit a QHMM whose minimal history or future wordlist forces a word longer than d^2 - 1; either would refute Corollary 1 and the proof of the bound.

Watch

Extended reading notes

Core claim

The central claim is Theorem 2: for any classical stochastic process, the minimal QHMM has memory Hilbert-space dimension d_min satisfying d_min >= sqrt(ell_min), where ell_min is the number of words in a minimal sufficient history/future wordlist—equivalently, the latent dimension of the minimal GHMM. The proof route is the key move: a QHMM with memory dimension d is a linear model on the d^2-dimensional space of Hermitian operators (Liouville/Bloch vectorization), so it is a GHMM of size at most d^2. Minimal GHMMs are unique up to basis, so the process's minimal GHMM size ell_min is a lower bound on d^2. Theorem 1 gives a finite list of conditional and marginal probability equalities on su

Load-bearing premise

The paper's finite equivalence test rests on Proposition 2, that a d-dimensional QHMM's minimal wordlists never require words longer than d^2 - 1, which is asserted by analogy to a lemma for GHMMs without a full proof in this paper; if that bound fails, the 2d^2 - 1 cutoff would not follow.

Editorial extensions

If this is right

  • Two QHMMs with memory dimensions d and d' generate the same process if and only if all word probabilities agree up to length 2 max(d,d')^2 - 1; this is a finite, executable check.
  • No quantum generator of the process can use fewer than sqrt(ell_min) Hilbert-space dimensions, giving a concrete floor for quantum memory-compression experiments.
  • The floor is sometimes reached: there is a process whose minimal GHMM needs four latent states yet a two-dimensional QHMM generates it, a quadratic saving.
  • The floor is not universal: some processes, such as certain unifilar co-unifilar HMMs, require a QHMM as large as the minimal GHMM.
  • Comparing a QHMM against a classical HMM or GHMM uses the same test with the classical model's state count |S| playing the role of d^2.

Reading between the lines

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

  • The argument is essentially linear-algebraic: any probabilistic theory whose memory states live in a finite-dimensional real vector space and whose transitions are linear would inherit the same square-root bound, so the result likely extends beyond quantum theory to other generalized probabilistic theories.
  • The canonical fingerprint could serve as a learning objective: if a process is sampled rather than known, constructing the standard GHMM from estimated probabilities of short words would yield a bound on the quantum memory needed, provided the word-length bound holds.
  • The square-root bound being only a bound suggests a finer invariant—perhaps tied to positivity or complete positivity—controls when quantum compression actually achieves it; finding that invariant would sharpen Theorem 2 to an equality.
  • The same canonicalization should transfer to input–output transducers, as the paper notes; one testable extension is whether the 2d_max^2 - 1 word bound survives when transitions depend on inputs.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper addresses identifiability and minimality of quantum and post-quantum generators of classical stochastic processes. It proposes to map any QHMM (and any GHMM) to a canonical standard GHMM via vectorization of the quantum memory space, so that two models are identifiable as generating the same process iff they share this canonical representation. The main results are: (i) Theorem 1, an equivalence criterion for two QHMMs based on agreement on sufficient history/future wordlists; (ii) Corollary 1, a finite check on words of length at most 2d_max^2−1; (iii) Theorem 2, the lower bound d_min ≥ sqrt(ℓ_min) on the memory Hilbert-space dimension of any QHMM generating a process with minimal GHMM size ℓ_min; and (iv) Theorem 3, with examples showing the bound can be tight (a 2-dimensional QHMM for a 4-state process) and sometimes not (a 4-state process needing dimension 4). The proof strategy is to reduce QHMMs to GHMMs in d^2 real dimensions and then invoke known results on GHMM minimality from D. Upper's 1997 thesis.

Significance. If the main claims hold, the paper provides a unified identifiability framework across classical, quantum, and post-quantum stochastic generators, and a simple, physically meaningful lower bound on quantum memory dimension in terms of the minimal GHMM size. The bound is sometimes tight, demonstrating a genuine quadratic quantum advantage in memory dimension. The paper also gives explicit algorithms and worked examples. The core reduction of QHMMs to GHMMs via Liouville/Bloch vectorization is standard and appears sound. However, the manuscript relies heavily on an unpublished thesis (Ref. [7]) for key structural results, and at least one proposition that underpins the finite identification check is not proved self-containedly. These gaps are local and repairable, but they need to be addressed before the results can be considered fully established.

major comments (3)
  1. [Section III, Proposition 2 (and Corollary 1)] Proposition 2 is the linchpin of the finite identification check: Corollary 1's statement that agreement on words of length 2d_max^2−1 is sufficient depends entirely on the existence of minimal wordlists containing no word longer than d^2−1. The proof given is a single sentence, 'analogous to Lem. 4.2.14 of Ref. [7]', with no self-contained argument. The claim is plausible — the nested subspaces V_k = span{A_w(σ0) : |w|≤k} in the d^2-dimensional Hermitian-operator space must stabilize by k=d^2−1, after which appending symbols no longer enlarges the span — but this argument is not supplied. Please provide a full proof or state explicitly which lemma from Ref. [7] is being used and prove that it transfers verbatim to QHMMs. Without this, Corollary 1 and the wordlist construction in Section IV do not follow.
  2. [Section III, Theorem 1] The proof of the main identifiability theorem is very terse: it says a QHMM can be recast as a GHMM, then 'See Thm. 4.3.3 of Ref. [7], from which the remainder of the proof follows.' Since this theorem is the paper's central contribution for quantum models, the reduction should be made explicit. In particular, the reader needs to see the correspondence between the history/future spaces of the original QHMM and those of the vectorized GHMM (Appendix A provides two vectorization methods), and how the three conditions in Theorem 1 are exactly Upper's equivalence conditions under that correspondence. As written, the proof is a citation plus assertion.
  3. [Section IV, Eqs. (16)–(18)] The construction of the standard GHMM uses the inverse (HF)^{-1} without any justification of invertibility. For minimal wordlists, the pairing between the history and future quotient spaces should be nondegenerate and hence HF nonsingular, but this is neither proved nor cited. Since the canonicity and backwards-compatibility claims depend on this construction, a sentence or lemma justifying invertibility is needed.
minor comments (4)
  1. [Corollary 1] The phrase 'agree on the probability all words of length 2d_max^2−1' is ambiguous; it could be read as only words of exactly that length. Since agreement on exactly that length implies agreement on all shorter words by marginalization, the statement is correct, but it should be clarified (e.g., 'all words of length at most 2d_max^2−1' or an explicit note that shorter words follow by summing over continuations).
  2. [Theorem 3, first paragraph] The sentence 'That the bound is not always tight is already evidenced by prior results showing that certain processes can be generated by a finite GHMM but no finite QHMM' is not evidence for non-tightness of a lower bound for processes that do have finite QHMMs. The example in App. C2 already demonstrates non-tightness; the sentence should be removed or reframed.
  3. [Proposition 1] The proof says 'there cannot be more than d^2 linearly-independent density matrices', but the objects Aw(σ0) are not necessarily density matrices (they are subnormalized positive operators). The statement should say 'linearly independent Hermitian operators' (or 'elements of the d^2-dimensional real operator space').
  4. [Appendix A.2, Eq. (A6)] The construction of G(x) using d^2 independent inputs would be clearer if the inputs {ρ^(n)} are explicitly chosen to be physically allowed density operators that are linearly independent as operators (e.g., a basis of Hermitian operators). The current text says 'd^2 independent inputs' without specifying this.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central bound follows from an explicit QHMM-to-GHMM vectorization combined with an external theorem on GHMM minimality.

full rationale

The paper's main result, d_min ≥ sqrt(ℓ_min), is derived by an explicit construction: a QHMM of memory dimension d is mapped to a GHMM acting on the d^2-dimensional (Liouville/Bloch) space of operators (Appendix A), and then Upper's theorem (Ref. [7], Thm. 4.3.10) is invoked to say the minimal GHMM for the process has ℓ_min latent states. Neither input contains the conclusion: ℓ_min is a process quantity (minimal wordlist length) independent of any QHMM, and the vectorization is a standard linear-algebra isomorphism, not a fit. The identifiability criterion (Theorem 1) similarly transfers Upper's GHMM equivalence theorem to QHMMs via the same explicit mapping; it does not assume the equivalence it proves. The proof of Proposition 2 (bound on word length d^2−1) is delegated to an analogy with Lemma 4.2.14 of Upper's thesis and is not self-contained; that is a proof gap affecting Corollary 1's finite-window check, but it is a citation to an external, parameter-free theorem, not a self-citation and not a circular definition. The paper contains self-citations (e.g., Refs. [4,21,24,37]) but none are load-bearing for the central derivation; they provide examples or pointers to parallel work. No fitted parameter is relabeled as a prediction, and no quantity is defined in terms of the bound it is used to prove. Therefore the derivation chain is not circular.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claims rest on prior GHMM theory from Upper's thesis and on standard quantum mechanics. No new postulates, free parameters, or invented physical entities appear. The main unproved ingredient is the word-length bound for QHMM wordlists (Proposition 2).

assumptions (4)
  • domain assumption Upper's Theorem 4.3.3: two GHMMs generate the same process iff they agree on sufficient wordlists.
    Invoked as the basis for Theorem 1 after converting QHMMs to GHMMs.
  • domain assumption Upper's Theorem 4.3.10: the minimal GHMM of a process has ℓ_min states, equal to the minimal wordlist size.
    Used in the proof of Theorem 2 to conclude ℓ_min ≤ d^2.
  • domain assumption Upper's Lemma 4.2.14: minimal wordlists for GHMMs need no words longer than the latent dimension.
    Extended by analogy to QHMMs in Proposition 2; not proven in this paper.
  • domain assumption Quantum operations are linear completely positive maps on density operators, with Kraus decompositions.
    Enables the QHMM to GHMM mapping via Liouville space or Bloch vectorization; standard quantum mechanics.

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Pith. "Pith review of Identifiability and minimality bounds of quantum and post-quantum models of classical stochastic processes." pith.science (2026). https://pith.science/paper/BOEF6RTU

@misc{pith2026250903004,
  author       = {Pith},
  title        = {Pith review of: Identifiability and minimality bounds of quantum and post-quantum models of classical stochastic processes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BOEF6RTU}},
  note         = {Machine review of arXiv:2509.03004}
}
read the original abstract

To make sense of the world around us, we develop models, constructed to enable us to replicate, describe, and explain the behaviours we see. Focusing on the broad case of sequences of correlated random variables, i.e., classical stochastic processes, we tackle the question of determining whether or not two different models produce the same observable behavior. This is the problem of identifiability. Curiously, the physics of the model need not correspond to the physics of the observations; recent work has shown that it is even advantageous -- in terms of memory and thermal efficiency -- to employ quantum models to generate classical stochastic processes. We resolve the identifiability problem in this regime, providing a means to compare any two models of a classical process, be the models classical, quantum, or `post-quantum', by mapping them to a canonical `generalized' hidden Markov model. Further, this enables us to place (sometimes tight) bounds on the minimal dimension required of a quantum model to generate a given classical stochastic process.

Figures

Figures reproduced from arXiv: 2509.03004 by the authors.

Figure 1
Figure 1. FIG. 1. A classical stochastic process can be generated by [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. We address identifiability and minimality of quan [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4. HMM representation of the example process for [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

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

  1. Quantum Dimension Reduction of Hidden Markov Models

    quant-ph 2026-01 conditional novelty 7.0 of 10

    Any finite ergodic HMM can be made deterministic by labelling transitions, yielding a normal iMPS that can be variationally compressed into a smaller quantum model.

Reference graph

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    Example: Tightness of bound For the first example, consider the process depicted by the HMM in Fig. 3. This is inspired by the communi- cation task described in Ref. [37]. We will show for this example that the minimal wordlist contains four words, whilst also prescribing an explicit QHMM construction of dimension two. This thus substantiates the tightnes...

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    PMq (wf |wh) = PMq ′(wf |wh) for all wh ∈ Ωh ∪ Ω′ h and all wf ∈ Ωf ∪ Ω′ f,

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    PMq (xwf |wh) = PMq ′(xwf |wh) for all x ∈ X, all wh ∈ Ωh ∪ Ω′ h, and all wf ∈ Ωf ∪ Ω′ f, and

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    For a quantum system of arbitrary finite dimension—i.e., a qudit ρ acting on a d-dimensional Hilbert space Hd—we can make a gener- alized Bloch decomposition [34, 35]

    Generalized Bloch representation of density matrices The state of a qubit ρ can be expressed via its Bloch vector ⃗ a: ρ = I/2 + ⃗ a· ⃗ σ/2, where ⃗ σ= ( σx, σy, σz) is the vector of Pauli matrices. For a quantum system of arbitrary finite dimension—i.e., a qudit ρ acting on a d-dimensional Hilbert space Hd—we can make a gener- alized Bloch decomposition ...

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    Marginalizing over the output events, the overall channel is CP, and also trace preserving (CPTP)

    Generalized Bloch representation of QHMMs The superoperators {Ax} of a QHMM correspond to completely positive (CP) maps on the quantum memory system. Marginalizing over the output events, the overall channel is CP, and also trace preserving (CPTP). Each of the |X |subchannels described by the superoperators can be fully determined, using linearity, from m...

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    That is, sup- pose our QHMM has superoperators {Ax}x described by a set of Kraus operators {Kxy}x,y

    Liouville space representation An alternative, more direct approach to vectoriza- tion is simply to ‘reshape’ density operators into vec- tors and superoperators into matrices. That is, sup- pose our QHMM has superoperators {Ax}x described by a set of Kraus operators {Kxy}x,y. A GHMM can be constructed with transition operators {G(x)}x∈X = {P y∈Y Kxy ⊗ K ...

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