Observable and Unobservable in Quantum Mechanics

Marcello Poletti

REVIEW 4 major objections 4 minor 27 references

Observable and Unobservable in Quantum Mechanics

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Quantum mechanics is what probability becomes when some propositions cannot be observed.

desk verdict The paper's two-valued example is clean, but the jump from one qubit to n propositions doesn't preserve information, so the central claim isn't demonstrated. read the letter →

arxiv 2507.00098 v1 pith:ZBJEMEJH submitted 2025-06-30 quant-ph physics.hist-ph

Marcello Poletti This is my paper · ORCID

classification quant-phphysics.hist-ph MSC 81P0581P10 PACS 03.65.Ta03.65.Ca

keywords quantummechanicsonto-epistemicignorancerelativeprobabilitylogicBornrulecorrelationinequalitieslogicalindeterminacyobservability

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 sets out to show that the mathematical heart of quantum mechanics is not an assumption about nature but an epistemic necessity. When a proposition's truth cannot be observed because the chain carrying information from system to observer is objectively broken, the author argues, probabilities must be conditioned on observability and stop commuting. The only linear algebra that restores ordinary Boolean negation is the quantum one: states become rays, propositions become projectors, and probability becomes the Born rule. If the argument holds, the interpretation of the cat-in-the-box thought experiment and of the classical correlation inequalities shifts from ontology to the structure of observation itself.

What carries the argument

The central object is the map from the single-proposition truth-observability space $\mathbb{R}^3$ (coordinates T, F, U) to a ray in $\mathbb{C}^2$: $(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)\mapsto(\cos\phi,e^{i\theta}\sin\phi)$. This map linearises the relative probability $[p]=|p\land\bar{p}|/|\bar{p}|$; in the new coordinates $[p]=\langle s'|P|s'\rangle$ for $P=\mathrm{diag}(1,0)$, and negation is $I-P$. An arbitrary projector in $\mathbb{C}^2$ is read as a contextual proposition, 'p is true under measurement context U', for a unitary rotation U of the T/F/U frame; the extension to n propositions uses tomographic locality to produce a state in $\mathbb{C}^{2^n}$.

What would settle it

Run the paper's two-entity example with controlled shield transmission: vary the five configuration weights (for instance $\Pi=|WW|$ and $\epsilon=|Ww|$ over the unit simplex) and measure the visible frequencies. The map to $\mathbb{C}^2$ fixes a relation between the non-commutative gap $[p][q]_p-[q][p]_q$ and the individual relative probabilities $[p]$ and $[q]$; any observed pair falling outside that relation, or any classical T/F/U joint distribution whose relative probabilities no state/projector pair in $\mathbb{C}^{2^n}$ can reproduce, would refute the claimed equivalence.

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Extended reading notes

Core claim

The paper's central claim is that the foundational postulates of quantum mechanics—Hilbert space, state vectors, Hermitian operators, and the Born rule—are the algebra of semantic relations among propositions, including propositions whose truth cannot be observed. Starting from onto-epistemic ignorance (a broken phenomenal chain rather than subjective ignorance), it defines the empirically accessible probability $[p]$ as the probability of $p$ conditional on $p$ being observable. In the three-state space of true-observable, false-observable, and unobservable, conditional probability becomes non-commutative, and the linearisation that restores Boolean negation is exactly the passage from real directions to rays in $\mathbb{C}^2$. The paper concludes that the classical correlation inequalities are not violated as theorems of logic; what experiments test is a strengthened form with every proposition replaced by 'true and observable', and that strengthened form can fail because membership of an element in $B$ may be unobservable.

Load-bearing premise

The argument depends on treating the chosen map from (true-observable, false-observable, unobservable) probabilities to complex two-dimensional rays as the unique information-preserving linearisation; if other coordinatisations satisfy the same epistemic constraints, the claimed necessity of the quantum structure fails.

Editorial extensions

If this is right

Empirical tests of the classical correlation inequalities actually test strengthened versions in which each proposition is replaced by 'true and observable'; unobservable membership can make the strengthened inequality false without violating ordinary logic.
Non-commutative relative probabilities and the Born rule follow from decidability, so even a simple deterministic classical system becomes quantum-probabilistic for a shielded observer.
The dynamical equation of motion is not derived; it remains the Hamiltonian dynamics expressed inside the derived quantum algebra.
The projector structure of quantum logic covers arbitrary contextual propositions defined by unitary transformations of the T/F/U frame, not only the original observable ones.
The construction extends to any finite number of propositions via tomographic locality, giving a state in $\mathbb{C}^{2^n}$.

Reading between the lines

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

Editorial inference: if the derivation is right, a controllable information shield around a classical system should produce quantum-like non-commutative statistics in a coarse-grained observable description; this is a testable consequence the paper leaves implicit.
Editorial inference: the $\mathbb{R}^3$-to-$\mathbb{C}^2$ map is not proven unique, so the strongest next step would be to derive it from symmetry or information-preservation axioms; absent that, alternative linearisations could lead to other probabilistic theories.
Editorial inference: the paper's three-way T/F/U trichotomy is the minimal case; adding further unobservable grades or contexts may yield higher-dimensional algebras, so the restriction to qubits is a modelling choice worth probing.
Editorial inference: the earlier logical-indeterminacy programme and the present onto-epistemic condition look like two aspects of a single epistemic principle; cases combining logical independence with broken phenomenal chains might yield inequalities not covered by either treatment.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Marcello has a genuinely interesting idea: when truth is defined relative to a phenomenal chain, probabilities become conditional on observability, and that conditionalization can produce non-commutativity. The two-entity example in Section III is simple and internally consistent, and the definition of relative probability [p] captures something real. The paper is also honest about its scope: no dynamics, no continuous observables, no claim to solve the full measurement problem. That's fine.

The problem is the complexification step in Section IV.C. For a single proposition, the map from R^3 to C^2 is a bijection between projective spaces, and [p] = cos²φ falls out. But the extension to n propositions is not information-preserving, and the stress-test note is correct: an independent joint T/F/U assignment lives in R^{3^n}, a projective space of real dimension 3^n - 1, while a pure state in C^{2^n} modulo phase has real dimension 2^{n+1} - 2. For n = 2, that's 8 versus 6. You cannot pack all the semantic content into fewer degrees of freedom without imposing extra constraints, and none are given. Instead, the paper borrows tomographic locality from Hardy and from Chiribella et al., which effectively assumes the quantum structure it is trying to derive. The claim that 'an arbitrary projector corresponds to a contextual proposition' is asserted, not proven.

So the central claim—that onto-epistemic ignorance necessarily yields the projection-valued algebra of QM—doesn't hold as written. The single-qubit case is a nice toy model, but the necessity argument collapses at the multi-proposition step. The four self-citations are background rather than load-bearing, so I don't count them against the paper.

Would I send it to review? Yes, actually. The question is worth asking, the prose is clear, and the flaw is precisely the kind of subtle dimensional gap a good referee could help the author see. It shouldn't be published in current form, but it deserves more than a desk reject.

Referee Report

4 major / 4 minor

Summary. The paper argues that a specific form of ignorance, called onto-epistemic ignorance—the inability to evaluate a proposition because the phenomenal chain transmitting information is broken—forces probabilities to be conditional on observability and leads to a non-commutative algebra formally equivalent to quantum mechanics. After defining a three-valued (T, F, U) model in Section III and showing an example of non-commuting conditional products, Section IV attempts to represent these relative probabilities first in a real-vector-space form and then, via a complexification map, in C^2 and C^{2^n}. The paper concludes that the foundational principles of quantum mechanics define the algebra of semantic relations among propositions, including unobservable ones.

Significance. If the derivation were sound, the paper would offer a striking result: the projective structure and Born rule of quantum mechanics would follow from a purely epistemic condition, and Bell-type inequalities would be reinterpreted as strengthened inequalities involving observability. The paper is clearly written and its single-proposition geometric construction is instructive. However, the central necessity claim is not established: the complexification step is an assumption rather than a derivation, the many-proposition extension has a dimensional gap, and the Section III non-commutativity is an artifact of conditioning on different events rather than a violation of classical probability. The paper therefore does not provide a working derivation of quantum mechanics from onto-epistemic ignorance.

major comments (4)

[Section III, Eqs. (6)–(10)] The claimed non-commutativity is not a violation of classical probability. In Eq. (9), [q]_p is defined using only the joint states TT and TF, which means it conditions not only on p being true and observable but also on q being observable; it excludes the TU state. Similarly, [p]_q conditions on p being observable. Equation (6) is the classical identity for a fixed joint distribution, and the two products in Eq. (16) are not both equal to the same joint probability. The inequality [p][q]_p ≠ [q][p]_q therefore follows from changing the conditioning event, not from any breakdown of the classical joint-probability theorem. This undermines the paper's motivation for abandoning classical probability.
[Section IV.C, complexification map] The map F: (sinθ cosφ, sinθ sinφ, cosθ) → (cosφ, e^{iθ} sinφ) is a bijection of projective spaces, but it is introduced deliberately to linearize the conditional probability [p]; it is not forced by the definitions of onto-epistemic ignorance. The subsequent claim that an arbitrary projector in C^2 corresponds to a contextual proposition 'truth of p under measurement context U' defines the semantics of rotated projectors into existence. Without an independent argument that every unitary image of the canonical TFU frame corresponds to a meaningful proposition in the original phenomenal-chain model, the full projector algebra is assumed rather than derived.
[Section IV.C, n-proposition extension] The assertion that the extension to n propositions in C^{2^n} is 'entirely natural' is not supported. A joint T/F/U assignment for n propositions is a normalized vector in R^{3^n}, i.e., a point in RP^{3^n−1} of real dimension 3^n−1, while a pure state in C^{2^n} modulo global phase is a point in CP^{2^n−1} of real dimension 2^{n+1}−2. For n=2 these dimensions are 8 and 6, and the gap grows with n. The proposed complexification therefore cannot preserve all information unless additional constraints are imposed on the R^{3^n} space. The paper imposes none; the appeal to tomographic locality in Refs. [23,24] imports an axiom from quantum-information derivations and does not supply the missing constraints. The footnote 7 disclaimer that independence details are 'not pursued here' confirms that the many-proposition step is left unproved.
[Section IV.C, concluding claim] The central claim that 'The foundational principles of quantum mechanics define the algebra of semantic relations among propositions—even unobservable ones' is an overclaim. The paper shows that one particular representation of relative probabilities exists in C^2, but it does not prove uniqueness: it does not exclude other state spaces (e.g., real or quaternionic Hilbert spaces, or generalized probabilistic theories with more degrees of freedom) that would also accommodate the T/F/U model. The necessity of the quantum projection structure is therefore not established by the arguments given.

minor comments (4)

[Section IV.A heading and text] The acronym 'CTP' appears where 'CPT' (classical probability theory) is intended; please correct this consistently.
[Section V, Eq. (37)] The barred sets such as \bar{A} in Eq. (37) are not defined; the notation should be introduced explicitly, for example as 'A is true and observable'.
[Section IV.C, Fig. 1] The text references 'FIG. 1' but no figure appears in the manuscript; either include the figure or remove the reference.
[References] Reference [5] is a personal account rather than a standard citation for Bell's theorem; consider replacing it with a primary or standard secondary source.

Circularity Check

1 steps flagged · score 7.0 of 10

The C^2 complexification is chosen specifically to linearize [p] into a Born-rule expression, so the claimed derivation of QM's projection structure is built into the representation.

self definitional [Sec. IV.B–IV.C, Eqs. (26)–(27) and (35)–(36)]
""Equation 26 is problematic in that it is non-linear; that is, there exists no operator P such that: [p] = ⟨s|P |s⟩ (27). ... To resolve this algebraically in a satisfactory manner, it is necessary to linearise the conditional probabilities; ... A space of this type exists: it is C2. ... |s′⟩ preserves the entire information content of |s⟩ in a form that allows direct extraction of [p] through a trivial projector: P = (1 0; 0 0).""

The Born-rule expression [p] = ⟨s′|P|s′⟩ is not derived from the onto-epistemic model; it is the condition used to select the C^2 representation. The nonlinear conditional probability of Eq. (26) is a ratio of two expectations in R^3; the paper declares that this must be linearised and asserts that a suitable space exists, namely C^2. The map (sinθ cosφ, sinθ sinφ, cosθ) → (cosφ, e^{iθ} sinφ) is chosen precisely so that [p] = cos²φ equals the expectation of the projector diag(1,0). The additional phase θ is inserted as an extra degree of freedom and later promoted to a physical phase by allowing arbitrary projectors.

full rationale

The early sections of the paper (II–III) develop a conditional-probability model with T/F/U states and exhibit a non-commutativity of conditional probabilities; those parts are not circular, since Eqs. (7)–(10) follow from the stated definitions. The circularity enters at the complexification step in Sec. IV.C. Having found that the R^3 representation yields the nonlinear expression [p] = ⟨s|P̂|s⟩/⟨s|P̄|s⟩ (Eq. 26), the paper announces that linearisation is necessary and asserts that 'a space of this type exists: it is C2'. The map from R^3 to C^2 is then chosen so that the squared modulus of the first component of the qubit state reproduces the previously defined [p]. The Born rule is therefore an input selected to match the earlier conditional probability, not an output forced by onto-epistemic ignorance. Arbitrary projectors on C^2 are then identified with 'contextual propositions' without a uniqueness or semantic-completeness proof. The extension to n propositions is also asserted as 'entirely natural' via tomographic locality, but a joint T/F/U distribution lives in R^{3^n} (real dimension 3^n − 1), whereas a C^{2^n} pure state modulo phase has real dimension 2^{n+1} − 2 (e.g., 8 vs 6 for n = 2), so the complexification cannot be information-preserving without additional quantum assumptions. The author's prior works are cited only as background and are not load-bearing. Because the central claim that the projection structure of QM follows necessarily reduces to the chosen representation, the analysis finds substantial circularity.

Assumptions & free parameters 1 free parameters · 5 assumptions · 1 invented entities

The argument rests on several unproved premises: von Neumann's psycho-physical parallelism, the three-state T/F/U model of observability, cited results on logical indeterminacy, and tomographic locality. The only free parameter is the illustrative epsilon in the example. The central complexification step is a modeling choice, not a consequence forced by the stated axioms.

free parameters (1)

epsilon (|Ww|) = >0, hand-set in example
Introduced ad hoc in Section III to create a non-vanishing invisible configuration; all noncommutativity disappears when epsilon = 0. It is a parameter in the illustrative example, not a measured constant.

assumptions (5)

standard math Classical joint probability theorem for absolute probabilities
Equation (6) is used as the baseline that relative probability will deviate from.
domain assumption Psycho-physical parallelism and the existence of a phenomenal chain
Section II takes von Neumann's principle as given; it is necessary for defining onto-epistemic ignorance as a break in the phenomenal chain.
ad hoc to paper T/F/U three-state model captures all relevant observability conditions
Section III introduces the triple true/false/unobservable without deriving it; the very definition of relative probability depends on this modeling choice.
domain assumption Brukner et al.'s logical indeterminacy results
The paper builds on refs [18-20] to connect logical independence to quantum randomness; these are cited, not rederived.
domain assumption Tomographic locality or local tomography
The n-proposition extension in Section IV.C invokes refs [23,24]; no proof is given that the relative-probability model satisfies this axiom.

invented entities (1)

Phenomenal chain
purpose: An information-transmitting medium whose absence defines onto-epistemic ignorance
No independent operational definition or test is provided; it is used as a conceptual device in the shield thought experiments.

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Cite this review

Pith. "Pith review of Observable and Unobservable in Quantum Mechanics." pith.science (2026). https://pith.science/paper/ZBJEMEJH

@misc{pith2026250700098,
  author       = {Pith},
  title        = {Pith review of: Observable and Unobservable in Quantum Mechanics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZBJEMEJH}},
  note         = {Machine review of arXiv:2507.00098}
}

read the original abstract

This work explores the connection between logical independence and the algebraic structure of quantum mechanics. Building on results by Brukner et al., it introduces the notion of \textit{onto-epistemic ignorance}: situations in which the truth of a proposition is not deducible due to an objective breakdown in the phenomenal chain that transmits information from a system A to a system B, rather than to any subjective lack of knowledge. It is shown that, under such conditions, the probabilities accessible to a real observer are necessarily conditioned by decidability and obey a non-commutative algebra, formally equivalent to the fundamental postulates of quantum mechanics.

Figures

Figures reproduced from arXiv: 2507.00098 by the authors.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

First, CTP and QM will be brought closer by re- formulating CTP within a pseudo-quantum alge- bra, composed of vector spaces, state vectors, and linear operators

work page

[2] [2]

The resulting algebra will then be used as a step- ping stone to provide an analogous operator-based representation of relative probability

work page

[3] [3]

p is true and observable,

Finally, the identity between this algebra and QM will be established. A. Geometric Representation of Classical Probability Theory Equation 6, by introducing a new algebraic symbol, highlights a typical semantic issue in classical probabil- ity theory. The probability of “ p ∧ q” is not an algebraic function of the probabilities of p and q, but depends, i...

work page

[4] [4]

J. S. Bell, On the einstein podolsky rosen paradox, Physics Physique Fizika 1, 195 (1964)

1964

[5] [5]

J. S. Bell, Bertlmann’s socks and the nature of reality, Le Journal De Physique Colloques 42, 41 (1981)

work page 1981

[6] [6]

J. S. Bell, Speakable and unspeakable in quantum mechan- ics: Collected papers on quantum philosophy (Cambridge university press, 2004)

work page 2004

[7] [7]

Aspect, P

A. Aspect, P. Grangier, and G. Roger, Experimental tests of realistic local theories via bell’s theorem, Phys. Rev. Lett. 47, 460 (1981)

1981

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Cirillo, Il teorema di bell raccontato a modo mio (

M. Cirillo, Il teorema di bell raccontato a modo mio (... pi` u o meno ...) (2015)

work page 2015