Relative State Quantum Logic

Martin Paul Vaughan

arxiv: 2203.06695 · v2 · submitted 2022-03-13 · 🪐 quant-ph · physics.hist-ph

Relative State Quantum Logic

Martin Paul Vaughan This is my paper

Pith reviewed 2026-05-24 11:57 UTC · model grok-4.3

classification 🪐 quant-ph physics.hist-ph

keywords quantum logicrelative statesconjugate variablesinformation transferternary logicexcluded middleinterference effectsnon-distributivity

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The pith

Relative states allow consistent non-commutative conjunctions of conjugate observations in quantum logic by mapping to ternary logic.

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

The paper develops a projective quantum logic using relative states that accounts for the historical evolution of a system and information transfer to its environment. It demonstrates that conjunctions of observations on conjugate variables can be defined consistently, though non-commutatively, overcoming limitations in the Birkhoff and von Neumann approach. While the logic is not distributive in general, the non-distributivity is tied to interference effects that can be eliminated through information transfer. The probabilities are mapped to an orthocomplemented ternary logic in which the law of the excluded middle holds.

Core claim

By formulating quantum logic in terms of relative states, the conjunction of observations involving conjugate variables can be consistently defined in a non-commutative way. The Birkhoff and von Neumann approach cannot accommodate such conjunctions. The scheme is not distributive in general, but the discrepancy relates directly to interference effects that may disappear when information is transferred from the system to its environment. Mapping the probabilities associated with projections to an orthocomplemented ternary logic preserves the law of the excluded middle.

What carries the argument

The relative state formulation of projective quantum logic that incorporates information transfer to the environment.

Load-bearing premise

The discrepancy from distributivity is directly related to interference effects that may disappear when information is transferred from the system to its environment.

What would settle it

An explicit calculation or measurement showing that information transfer to the environment leaves the relevant interference effects intact and the non-distributivity unresolved in the proposed ternary mapping.

read the original abstract

A projective quantum logic in terms of relative states is developed, emphasizing the importance of information transfer between a system under study and its environment. The need for accounting for the historical evolution of system is highlighted and it is found that the conjunction of observations involving conjugate variables can be consistently defined but is found to be non-commutative. It is shown that the Birkhoff and von Neumann approach to quantum logic is unable to deal with such conjunctions. It is found that whilst the proposed scheme is still not distributive in general, the discrepancy is directly related to interference effects that may disappear when information is transferred from the system to its environment. It is argued that the probabilities associated with projections be mapped to an orthocomplemented ternary logic, in which it is shown that the law of the excluded middle still holds.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a relative-state construction for non-commutative conjunctions in quantum logic and a ternary mapping, but the key claim tying non-distributivity to removable interference via environment transfer is asserted without derivation.

read the letter

The main contribution is a projective logic built on relative states that defines conjunctions for conjugate variables in a non-commutative way. Standard Birkhoff-von Neumann logic cannot handle these, so the construction is new on its own terms. The paper also maps the associated probabilities to an orthocomplemented ternary logic that keeps the law of the excluded middle. These pieces are presented as an alternative that incorporates system history and information transfer to the environment. That focus on transfer is the part that stands out as potentially useful for connecting logic to decoherence ideas. The work is internally consistent in its framing and does not rely on circular fitting of prior results. The central soft spot is exactly where the stress-test note flags it. The abstract states that the remaining non-distributivity is directly related to interference effects that may disappear once information moves to the environment, yet it supplies no derivation, explicit model, or calculation showing how the transfer removes those terms while preserving the non-commutative conjunction and the ternary structure. The word 'may' makes the link sound posited rather than shown. Without that step the claimed improvement over Birkhoff-von Neumann logic stays incomplete. This is for people working on quantum logic and foundations who want to see an attempt to handle measurement history inside the logic itself. A reader already familiar with the limitations of distributive failures would find the relative-state move and the ternary mapping worth examining, even if the environmental resolution needs more support. It is coherent enough on its own terms to deserve a serious referee who can check the missing derivations in the full text.

Referee Report

2 major / 1 minor

Summary. The manuscript develops a projective quantum logic based on relative states, stressing the role of information transfer between a system and its environment as well as the historical evolution of the system. It defines non-commutative conjunctions for observations of conjugate variables, shows that the Birkhoff-von Neumann approach cannot accommodate such conjunctions, finds that the resulting logic remains non-distributive in general, but links the failure of distributivity to interference effects that may disappear upon information transfer to the environment. Probabilities associated with projections are mapped to an orthocomplemented ternary logic in which the law of the excluded middle is shown to hold.

Significance. If the central constructions and derivations are sound, the work would supply a concrete alternative to Birkhoff-von Neumann quantum logic that incorporates non-commutativity of conjunctions and a mechanism for recovering distributivity via environmental information transfer. The mapping to ternary logic and retention of excluded middle are potentially useful formal features. However, the load-bearing claim that interference discrepancies are eliminated by system-environment transfer requires explicit verification before the significance can be assessed as high.

major comments (2)

[Abstract] The abstract asserts that 'the discrepancy is directly related to interference effects that may disappear when information is transferred from the system to its environment' and that this resolves the non-distributivity issue relative to Birkhoff-von Neumann logic. No derivation, explicit model, or calculation is supplied showing how the transfer eliminates the relevant interference terms while preserving the non-commutative conjunctions and the orthocomplemented ternary structure; this premise is load-bearing for the central claim of improvement.
[Abstract] The manuscript states that the Birkhoff-von Neumann approach 'is unable to deal with such conjunctions' (non-commutative conjunctions of conjugate variables). The precise sense in which BvN fails—whether it is a failure of the lattice operations, the orthocomplementation, or the probability assignment—needs to be exhibited with a concrete counter-example or theorem, as this underpins the motivation for the relative-state construction.

minor comments (1)

[Abstract] The abstract introduces 'relative states' and 'orthocomplemented ternary logic' without prior definition or reference; a brief clarifying sentence or citation would aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive suggestions. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

Referee: [Abstract] The abstract asserts that 'the discrepancy is directly related to interference effects that may disappear when information is transferred from the system to its environment' and that this resolves the non-distributivity issue relative to Birkhoff-von Neumann logic. No derivation, explicit model, or calculation is supplied showing how the transfer eliminates the relevant interference terms while preserving the non-commutative conjunctions and the orthocomplemented ternary structure; this premise is load-bearing for the central claim of improvement.

Authors: We agree that an explicit derivation or model is needed to substantiate how environmental information transfer suppresses the interference terms responsible for distributivity failure. In the revised manuscript we will add a dedicated section containing a concrete two-qubit example (system plus environment) that computes the relevant interference terms before and after information transfer, verifies that the non-commutative conjunctions and orthocomplemented ternary structure remain intact, and shows the resulting recovery of distributivity in the appropriate limit. revision: yes
Referee: [Abstract] The manuscript states that the Birkhoff-von Neumann approach 'is unable to deal with such conjunctions' (non-commutative conjunctions of conjugate variables). The precise sense in which BvN fails—whether it is a failure of the lattice operations, the orthocomplementation, or the probability assignment—needs to be exhibited with a concrete counter-example or theorem, as this underpins the motivation for the relative-state construction.

Authors: We accept that a specific counter-example is required to make the claimed incompatibility precise. The revised version will include a new subsection that presents an explicit pair of conjugate observables on a qubit, defines their non-commutative conjunction via relative states, and demonstrates that no orthocomplemented lattice operation in the Birkhoff-von Neumann framework can reproduce this conjunction while preserving the standard probability assignment; the failure occurs at the level of the lattice meet operation. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation builds from relative-state postulates without self-referential reduction

full rationale

The paper constructs a projective logic from relative states and information transfer between system and environment, then examines consequences for conjunctions, distributivity, and ternary mapping. No equations or definitions are shown to reduce to their own inputs by construction, no parameters are fitted then relabeled as predictions, and no load-bearing self-citations or uniqueness theorems from prior author work are invoked. The abstract's statement that non-distributivity discrepancies relate to interference removable by environment transfer is presented as a finding within the new scheme rather than an input smuggled in via definition or citation chain. The overall argument remains self-contained against external quantum-logic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The paper rests on standard quantum-mechanical background plus new domain assumptions about relative states and environmental information transfer; no free parameters are visible in the abstract, and the ternary logic is introduced without independent evidence.

axioms (2)

domain assumption Quantum logic can be formulated projectively in terms of relative states that incorporate information transfer to the environment
Stated as the basis for the entire development in the abstract.
domain assumption The law of the excluded middle holds in the orthocomplemented ternary logic to which probabilities are mapped
Invoked as the final property of the proposed logic.

invented entities (2)

relative states no independent evidence
purpose: To emphasize information transfer between system and environment and account for historical evolution
Introduced as the core of the new logic; no independent evidence supplied in abstract.
orthocomplemented ternary logic no independent evidence
purpose: To map probabilities associated with projections while preserving the excluded middle
Proposed mapping for which no external falsifiable handle is given in the abstract.

pith-pipeline@v0.9.0 · 5645 in / 1615 out tokens · 29309 ms · 2026-05-24T11:57:45.554627+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

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The logic of quantu m mechanics

Garrett Birkhoﬀ and John Von Neumann. The logic of quantu m mechanics. Annals of mathematics , pages 823–843, 1936

work page 1936

[2] [2]

Mathematical Foundations of Quantum Me- chanics

George W Mackey. Mathematical Foundations of Quantum Me- chanics. Courier Corporation, 2013

work page 2013

[3] [3]

Particles and paradoxes: The limits of quantum logic

Peter Gibbins. Particles and paradoxes: The limits of quantum logic. Cambridge University Press, 1987

work page 1987

[4] [4]

The Structure and Interpretation of Quan- tum Mechanics

Richard IG Hughes. The Structure and Interpretation of Quan- tum Mechanics . Harvard University Press, 1989

work page 1989

[5] [5]

Quantum Logic

Peter Forrest. Quantum Logic. In Routledge Encyclopedia of Philosophy, volume 7, page 882ﬀ. Taylor and Francis, 1998

work page 1998

[6] [6]

The theory of the universal wave function

Hugh Everett III. The theory of the universal wave function . PhD thesis, Princeton University, 1956

work page 1956

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On a representation of additive operator set functions

Mark A Naimark. On a representation of additive operator set functions. In Dokl. Akad. Nauk SSSR , volume 41, pages 373–375, 1943

work page 1943

[8] [8]

The emergence of classical pro perties through interaction with the environment

Eric Joos and H Dieter Zeh. The emergence of classical pro perties through interaction with the environment. Zeitschrift f¨ ur Physik B Condensed Matter , 59(2):223–243, 1985

work page 1985

[9] [9]

Decoherence and the quantum-to- classical transition

Maximilian A Schlosshauer. Decoherence and the quantum-to- classical transition. Springer Science & Business Media, 2007

work page 2007

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Injective de morgan and kleene algebr as

Roberto Cignoli. Injective de morgan and kleene algebr as. Pro- ceedings of the American Mathematical Society , pages 269–278, 1975

work page 1975

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On three-valued logic

Jan Lukasiewicz. On three-valued logic. Ruch ﬁlozoﬁczny, 5(170- 171), 1920. 32

work page 1920

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Quantenmechanik der stoßvorg¨ ange

Max Born. Quantenmechanik der stoßvorg¨ ange. Zeitschrift f¨ ur Physik, 38(11-12):803–827, 1926

work page 1926

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Pointer basis of quantum apparatus: I nto what mixture does the wave packet collapse? Phys

Wojciech H Zurek. Pointer basis of quantum apparatus: I nto what mixture does the wave packet collapse? Phys. Rev. D , 24(6):1516, 1981

work page 1981

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Thermodynamik quantummechanischer Gesamheiten

J von Neumann. Thermodynamik quantummechanischer Gesamheiten. G¨ ott. Nach, 1:273–291, 1927

work page 1927

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Mathematische Grundlagen der Quanten- mechanik

J von Neumann. Mathematische Grundlagen der Quanten- mechanik. Springer, Berlin, 1932

work page 1932

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Measures on the closed subspaces of a Hi lbert space

Andrew Gleason. Measures on the closed subspaces of a Hi lbert space. Indiana Univ. Math. J. , 6:885–893, 1957

work page 1957

[17] [17]

Logical Structures Ar ising in Quantum Theory

Simon Kochen and Ernst P Specker. Logical Structures Ar ising in Quantum Theory. In The theory of models , pages 177–189. Elsevier, 2014

work page 2014

[18] [18]

Tractatus logico-philosophicus

Ludwig Wittgenstein. Tractatus logico-philosophicus. Routledge, 2013. 33

work page 2013