Quantum state change in light of changes in valuational entropies

Arkady Bolotin

REVIEW 5 major objections 3 minor 21 references

Quantum state change in light of changes in valuational entropies

T0 review · 5 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper argues that in any Hilbert space of dimension two or more, some proposition about a quantum state is always informationally indefinite, so both smooth reversible evolution and abrupt measurement collapse are inescapable features…

desk verdict A novel framing for an old debate, but the core argument relies on an unsupported stipulation and an ill-defined entropy; the conclusion that two processes are inevitable does not follow. read the letter →

arxiv 1908.02887 v1 pith:VYZ4YZ2N submitted 2019-08-08 quant-ph

Arkady Bolotin This is my paper · ORCID

classification quant-ph MSC 81P1081P1503B50 PACS 03.65.Ta03.65.Ca

keywords valuationalentropypredicatevaguenesssetmembershipHilbertspaceKochen-Speckertheoremquantummeasurementproblemstatecollapsetruthvaluegaps

topics The Measurement Problem

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 tries to establish that two distinct ways of changing a pure quantum state are not optional but forced by the mathematics of Hilbert space. It defines an entropy for the predicate '... is an element of ...' on a vector and a closed linear subspace: zero when membership is definitely true or false, and positive when membership is borderline. The author argues that in every Hilbert space of dimension at least two there is always some vector and some family of subspaces for which this entropy cannot be zero. If that is right, deterministic and reversible evolution and nondeterministic and irreversible measurement collapse both necessarily exist. Consequently, interpretations that try to keep only a single process for quantum state change cannot remove the measurement problem.

What carries the argument

The central object is the entropy of the predicate $P_\in$ over a vector $|\Psi\rangle$ and a closed linear subspace $P$, defined by $$H(P_\in(|\Psi\rangle, P)) = \log_\$\beta$ N - \max(|M|, |M^\perp|) \frac{\log_\$\beta$ \max(|M|, |M^\perp|)}{N},$$ where $M$ and $M^\perp$ count component matches with vectors in $P$ and $P^\perp$. This quantity does the argument's work by separating determined membership (entropy zero) from undetermined membership (entropy positive), and the paper links that distinction to whether the truth-value transformation map is bijective and reversible or not. The contextuality theorem supplies the existence of subspaces with positive entropy, making the two-process conclusion follow.

What would settle it

Evaluate the entropy formula on a known uncolorable set of rays in a three-dimensional Hilbert space for a fixed state: if every subspace in the set returns zero entropy, then the claim that positive entropy is unavoidable is wrong. More concretely, the paper would be refuted by producing any finite family of closed subspaces and a vector for which the contextuality theorem still applies but formula (16) gives zero for every member.

Watch

Extended reading notes

Core claim

The central claim is that, in any Hilbert space $H$ with dimension $N \geq 2$, there exists a nonempty set $S$ of closed linear subspaces of $H$ such that, for a given vector $|\Psi\rangle$ in $H$, the entropy of the predicate $P_\in$ on $|\Psi\rangle$ and every subspace in $S$ cannot be zero. The entropy $H(P_\in(|\Psi\rangle, P))$ is constructed from how many components of $|\Psi\rangle$ match vectors in $P$ or in the orthogonal complement $P^\perp$; it vanishes exactly when set membership is determined, true or false, and is positive when membership is neither. Since every closed subspace represents an atomic proposition, this entropy is a valuational entropy measuring uncertainty about assigning truth values. Zero change in these entropies over time corresponds to deterministic reversible evolution, while nonzero change corresponds to nondeterministic irreversible processes, namely, quantum state collapse or information loss. The paper concludes that two separate processes of quantum state change are proper to the Hilbert space formalism rather than optional, citing the known contextuality theorem that rules out two-valued assignments to all rays in dimension three and higher.

Load-bearing premise

The claim depends on stipulating that the statement '$|\Psi\rangle \in P$' can be neither true nor false, a truth-value gap or many-valued truth value introduced without argument, and if that stipulation fails, the nonzero entropy and the two-process conclusion collapse.

Editorial extensions

If this is right

In any quantum system with a Hilbert space of dimension at least two, there will always be some atomic proposition whose truth value is not simply yes or no, so measurement-induced information gain or loss cannot be eliminated.
Deterministic reversible evolution and nondeterministic irreversible collapse are both mathematical consequences of the Hilbert space formalism, not artifacts of a particular interpretation.
Single-process interpretations of quantum mechanics cannot, by themselves, dissolve the measurement problem; the two-process structure is built into the underlying geometry.
The change in valuational entropy gives quantitative meaning to the gain of information in collapse and the loss of information in measurement.
The contextuality theorem acts as the existence proof: no global two-valued assignment of truth values to all propositions is possible, so positive predicate entropy is unavoidable.

Reading between the lines

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

A direct testable extension would be to compute the entropy formula on explicit finite sets of rays, such as those used in known contextuality proofs, and check exactly which rays produce positive entropy for a given state; the paper predicts at least one positive value in every such uncolorable set.
If the two-process conclusion is correct, the paper implicitly offers a quantitative account of the quantum-to-classical transition: information gain about one observable forces information loss about an incompatible one, with the entropy changes equal and opposite.
The entropy definition appears to depend on a choice of basis through the component matching, so a natural question the paper leaves open is whether the distinction between zero and nonzero entropy is basis-invariant; testing that dependence would clarify the robustness of the claim.
The argument suggests a bridge between contextuality and thermodynamics: the nonzero predicate entropy may be interpreted as an information-theoretic resource that cannot be removed without a non-reversible state change.

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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

Colleague,

This paper tries to ground the distinction between unitary evolution and measurement collapse in a purely mathematical property of Hilbert space. The idea is to define a 'valuational entropy' for the predicate '...is an element of...' that is zero when membership is true or false and positive when membership is supposedly undetermined, then argue that such undetermined cases always exist, forcing two kinds of state change. That's a tidy conceptual move, and the paper is clearly written and honest about what it wants to prove. I also think the framing—'is the two-process account optional?'—is a genuinely interesting question.

But the argument has a hole at its center. The entropy H(P∈(|Ψ⟩,P)) in Eq. (16) is defined via sets M and M⊥ that depend on particular representative vectors p and p⊥ chosen from P and P⊥. Change the representatives and the entropy changes, so it isn't a property of the subspaces and state alone. The claim that membership can be 'neither true nor false' is stipulated in Eqs. (3)–(5) with no supporting semantics; the example with P3 and |Ψ1⟩ actually shows ordinary non-membership: no multiple of (1,0,0,0) equals (1,1,1,1), so the statement is simply false. The paper's argument that |Ψ1⟩ can't be regarded as not in P3 because it isn't in P⊥3 is a non-sequitur—non-membership doesn't require membership in the orthogonal complement. So the positivity case in Eq. (17) is unsupported. The appeal to Kochen-Specker in Section 4 doesn't repair this: that theorem forbids global two-valued assignments over a set of projection operators, but it says nothing about any individual membership statement being vague. Finally, the two processes are defined in Eqs. (24) and (28) as exactly the cases of zero and nonzero entropy change, so the conclusion is built into the definitions.

Where does that leave us? The paper doesn't deliver a valid derivation that Hilbert space alone forces two state-change processes. The defect is load-bearing, not a minor gap. Still, I'd point a philosophical reader to the paper as a provocative attempt to connect quantum logic's 'undefined' propositions with information-theoretic entropy; there's a kernel of a research program there, but it needs a carefully defined semantics for vague membership and an entropy that's independent of arbitrary choices.

My recommendation: reject, but with an encouraging note that the question is worth pursuing.

Referee Report

5 major / 3 minor

Summary. This paper aims to show that the standard two-process description of quantum state change is not optional but is forced by the Hilbert-space formalism. It introduces a 'valuational entropy' H(P∈(|Ψ⟩,P)) for the membership predicate applied to a state vector and a closed subspace, proposes that this entropy is zero when the membership statement is bivalent and positive when it is 'neither true nor false,' and argues on the basis of the Kochen–Specker theorem that such nonzero cases always exist in dimension ≥2. From the dichotomy ΔH=0 vs. ΔH≠0 it then derives deterministic reversible evolution versus nondeterministic nonreversible measurement processes.

Significance. The question addressed is significant: if the argument worked, it would give a mathematical derivation of the measurement/evolution dichotomy from Hilbert-space structure and would undercut interpretations with a single dynamical law. The paper's strengths are that it formulates a concrete entropy expression and works through explicit two-qubit examples, and it engages a real literature on quantum logic and many-valued semantics. However, the central construction is not sound: the entropy is not well defined, the illustrative 'borderline' case is an ordinary false membership statement, and the Kochen–Specker inference is unsupported. The conclusion is therefore not established.

major comments (5)

[Section 2, Eqs. (3)–(5)] The paper stipulates that the predicate P∈ may be partial, so that a statement like |Ψ⟩∈P can be neither true nor false, but it gives no independent semantic argument or formal theory for this possibility. The phrase 'Suppose that . . .' at the start of the borderline-case discussion is not a justification. Since the nonzero-entropy branch of Eq. (17) and every later conclusion depend on this premise, the main claim presently rests on an assumption rather than on a result.
[Section 2, Eqs. (14)–(16)] The sets M and M⊥ are not well defined. Equation (14) defines M as {au_i : au_i = p_i}, but p is an arbitrary vector in P and a is an arbitrary nonzero scalar; choosing a different representative p or a different scalar changes the equality conditions and therefore changes |M| and |M⊥|. Since the valuational entropy (16) is a function of these cardinalities, the distinction between zero and positive entropy in Eq. (17) is not an invariant property of the pair (|Ψ⟩,P) but an artifact of the chosen representatives.
[Section 2, Eqs. (8)–(12)] The P3 example does not exhibit vagueness. For |Ψ1⟩=(1,0,0,0)^T and P3={a(1,1,1,1)^T}, there is no scalar a with (a,a,a,a)=(1,0,0,0), so |Ψ1⟩∉P3 is simply true. The additional fact that |Ψ1⟩∉P3⊥ is irrelevant to that determination; a vector can fail to be in both a subspace and its orthogonal complement. Thus the example supports the zero-entropy, bivalent-false case, not the claimed 'neither true nor false' case, and Eq. (17)'s positive-entropy row remains unsupported.
[Section 4] The inference from the Kochen–Specker theorem to the existence of a set S with nonzero valuational entropy is not drawn anywhere in the text. The theorem rules out certain global assignments of sharp truth values to projection operators; it does not imply that a given membership statement such as |Ψz+⟩∈X+ is indeterminate. Indeed, in the paper's own example, |Ψz+⟩=(1,0)^T is simply not an element of X+={a(1,1)^T}, so H(P∈(|Ψz+⟩,X+))=logβ2 is the entropy of an ordinary false statement under the paper's interpretation, not a proof of vagueness.
[Section 3, Eqs. (24) and (28)] The conclusion that two distinct processes exist is substantially circular. The deterministic reversible process is defined as one with ΔH=0 and the nondeterministic nonreversible process as one with ΔH≠0; after these definitions, finding that both cases are represented in the examples is a restatement of the definitions. The paper would need an independent proof that both cases necessarily occur, but that proof is exactly what the flawed P3 example and the unsupported Kochen–Specker assertion fail to supply.

minor comments (3)

[Section 2, Eq. (16)] The formula for H is undefined when max(|M|,|M⊥|)=0 because logβ0 is not defined; the paper should either exclude this case or define the entropy by a limit.
[Section 2, Eq. (17)] The symbol Hmax is introduced as the maximum of H but is never computed or characterized; the statement that H takes values in (0,Hmax] is not substantiated.
[Section 2, Eq. (22)] The alignment of truth-value gaps with probabilities strictly between 0 and 1 is presented as an inference, but the correspondence is asserted rather than derived from a probability model, and the notation [ [P] ]v /∈ B2 / Pr(X=x) ∈ (0,1) is not a well-formed formula of the paper's formalism.

Circularity Check

2 steps flagged · score 8.0 of 10

The two processes are stipulated into existence: deterministic/reversible is defined as ΔH=0 and nondeterministic/nonreversible as ΔH≠0, then their coexistence is presented as a Hilbert-space result.

self definitional [Section 3, Eqs. (24) and (28); Section 4, opening paragraph]
"To sum up, if processes of the quantum state change yield no differences in the valuational entropies of the propositions (i.e., cause neither gain nor loss of information about the propositions), then such processes are deterministic and reversible. By contrast, if processes of the quantum state change bring forth changes in the valuational entropies of the propositions (i.e., cause the gain or loss of information about the propositions), then these processes are nondeterministic and nonreversible."

Equations (24) and (28) define the two classes of state change as, respectively, ΔH=0 and ΔH≠0. Section 4 then argues that because Hilbert space admits both zero and nonzero predicate entropies (via Kochen–Specker), two distinct processes must exist. That inference is the definitional dichotomy itself: the 'deterministic and reversible' process just is the no-entropy-change case, and the 'nondeterministic and nonreversible' process just is the entropy-change case. No independent physical criterion is supplied to show that these coincide with Schrödinger evolution and measurement collapse; the coincidence is stipulated in the summary text.
renaming known result [Section 3, final paragraph]
"The aforesaid deterministic and reversible processes can be regarded as normal or Schrödinger’s evolutions. As to the nondeterministic and nonreversible processes, those, which are associated with the minimizing of the uncertainty about the truth value assignment, can be regarded as the 'collapse' of the quantum state, while those, which are associated with the maximizing of the uncertainty about the truth value assignment, can be regarded as the loss of information in a quantum measurement."

The standard two processes (Schrödinger evolution and collapse) are identified with the paper's entropy categories by an explicit 'can be regarded as.' This is a renaming, not a derivation: the existence of two processes was already built into the zero/nonzero entropy partition, and the match to standard quantum mechanics is asserted rather than shown.

full rationale

The paper contains no fitted parameters and no load-bearing self-citations; the Kochen–Specker appeal is external and non-circular, though arguably a non-sequitur. The circularity lies in the central conclusion: the two processes are not derived from independent properties of Hilbert space but are stipulated as the two cases of zero versus nonzero change in valuational entropy. Section 3 explicitly labels ΔH=0 as deterministic-reversible and ΔH≠0 as nondeterministic-nonreversible; Section 4's conclusion then restates that partition. The additional 'can be regarded as Schrödinger evolution/collapse' is renaming. Accordingly the main claim reduces by construction to the definition of the process types, while the nontrivial existence of nonzero entropy values is a separate mathematical claim. Score 8 rather than 10 because the paper does provide a substantive (if questionable) Kochen–Specker-based argument that nonzero entropy occurs, so not literally every step is a tautology.

Assumptions & free parameters 2 free parameters · 3 assumptions · 1 invented entities

The central claim relies on a new entropy measure that is not well-defined due to arbitrary representative vectors, and on the ad hoc assumption that set membership can be vague. The Kochen-Specker theorem is invoked in a way that does not follow from the theorem itself. No free parameters are fitted to data, but the definitions introduce arbitrary choices.

free parameters (2)

Representative vectors p and p⊥
The sets M and M⊥ in Eqs. (14)-(15) are defined using a particular vector p in P and p⊥ in P⊥, but these vectors are arbitrary and unspecified. The entropy value changes with this choice, so the measure is not well-defined.
Logarithm base β
The base of the logarithm in Eq. (16) is arbitrary; it affects numerical values but not the zero/nonzero distinction. It is a free choice that does not correspond to any physical input.

assumptions (3)

domain assumption Each closed linear subspace uniquely represents an atomic proposition about the quantum system.
Standard quantum logic assumption, stated in Section 2 after Eq. (17).
ad hoc to paper The predicate P∈ can be partial, admitting truth value gaps or many-valued truth values.
Introduced in Section 2 (Eqs. (3)-(5)) without proof; it is the core stipulation that makes nonzero entropy possible.
ad hoc to paper Kochen-Specker theorem implies the existence of a set S of subspaces for which the entropy is nonzero.
Asserted in Section 4 without derivation. The theorem concerns noncontextual hidden variable assignments, not vagueness of individual membership.

invented entities (1)

Valuational entropy H(P∈(...))
purpose: Quantifies uncertainty in the truth value of the statement that a vector belongs to a subspace.
A new quantity with no operational definition or connection to measurement. It is constructed from component matchings and depends on arbitrary representatives.

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Cite this review

Pith. "Pith review of Quantum state change in light of changes in valuational entropies." pith.science (2026). https://pith.science/paper/VYZ4YZ2N

@misc{pith2026190802887,
  author       = {Pith},
  title        = {Pith review of: Quantum state change in light of changes in valuational entropies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VYZ4YZ2N}},
  note         = {Machine review of arXiv:1908.02887}
}

read the original abstract

In the statement "The vector is an element of the closed linear subspace of the Hilbert space H", the predicate "... is an element of ..." might be not only determined, that is, either true or false (depending on whether set membership is applicable or inapplicable to the specified vector and subspace) but also undetermined, that is, neither true nor false. To evaluate the vagueness of set membership among arbitrary vectors and closed linear subspaces of H, the notion of the entropy of the predicate "... is an element of ..." is introduced in the present paper. Since each closed linear subspace in H uniquely represents the atomic proposition P about a quantum system, the entropy of this predicate can also be considered as the valuational entropy that measures the uncertainty about the assignment of truth values to the proposition P. As it is demonstrated in the paper, in the Hilbert space H of the dimension greater than or equal to 2, there always exists a nonempty set S of the closed linear subspaces in H, such that the entropy of the predicate "... is an element of ..." on the given vector of H and all the subspaces of S cannot be zero. This implies the existence of two different processes of the pure quantum state change: the process which yields no changes in the valuational entropies of the propositions (corresponding to the deterministic and reversible evolution) and the process which brings forth changes in the valuational entropies (corresponding to the gain or loss of information in a quantum measurement).

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Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

A.M. Dirac. Quantum Mechanics. Clarendon Press, Oxford, 1958

work page 1958

[2] [2]

von Neumann

J. von Neumann. Mathematical Foundations of Quantum Mechanics . Princeton University Press, Princeton, 1955

work page 1955

[3] [3]

A phenomenological solution to the measu rement problem? Husserl and the foundations of quantum mechanics

Steven French. A phenomenological solution to the measu rement problem? Husserl and the foundations of quantum mechanics. Studies in History and Philosophy of Modern Physics , 33:467–491, 2002

work page 2002

[4] [4]

Everett III

H. Everett III. Relative State Formulation of Quantum Me chanics. Rev. Modern Physics , 29:454–462, 1957

work page 1957

[5] [5]

J. S. Bell. Speakable and Unspeakable in Quantum Mechanics . Oxford University Press, 1987

work page 1987

[6] [6]

Saunders, J

S. Saunders, J. Barrett, A. Kent, and D. Wallace. Many Worlds? Everett, Quantum Theory and Reality. Oxford University Press, 2010

work page 2010

[7] [7]

Many-Worlds interpretation of Quantum Mechanics

Lev Vaidman. Many-Worlds interpretation of Quantum Mechanics. In Edward N. Zalta, editor, The Stanford Encyclopedia of Philosophy . Metaphysics Research Lab, Stanford University, 2018

work page 2018

[8] [8]

A Logical Introduction to Proof

Daniel Cunningham. A Logical Introduction to Proof. Chapter 2: Predicate Logic . Springer- Verlag, New York, 2013. 11

work page 2013