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arxiv: 2605.19706 · v3 · pith:HBZELC25new · submitted 2026-05-19 · 🪐 quant-ph · math-ph· math.MP

Finite-Precision Quantum Mechanics

Abbas Edalat This is my paper

Pith reviewed 2026-05-20 06:17 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MP
keywords finite precisionquantum parcelsinterval quantum mechanicsdensity matrix setsmeasurement updategeometric informationfoundational puzzles
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The pith

Quantum states are finite-precision parcels of density matrices rather than exact points, resolving foundational puzzles while recovering standard predictions only in the unattainable infinite-precision limit.

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

The paper argues that standard quantum mechanics idealizes states as exact points and measurements as sharp, but real experiments always involve finite resolution and access only to macroscopic observables. It elevates the macroscopic state to a fundamental concept by defining the quantum state as a parcel: an open set of density matrices consistent with finitely many measured expectation intervals. Unitary evolution lifts to a deterministic flow on these parcels, while measurement becomes a volume-contracting update that refines the set and increases geometric information measured by Hilbert-Schmidt volume. A double-parcel construction resolves the entropy paradox by ensuring monotonic information growth. If this holds, wave-particle duality reduces to a smooth precision trade-off, Schrödinger's cat avoids literal superposition, and entanglement becomes a purely epistemic geometric update rather than spooky action at a distance, with all laboratory predictions of standard quantum mechanics preserved exactly only in the unphysical infinite-precision limit.

Core claim

The state of a quantum system is a quantum parcel, a basic weak-star open set of density matrices defined by finitely many open expectation intervals. This parcel is the exact mathematical representation of the set of all microscopic states compatible with the measured values of a finite set of macroscopic observables. Unitary evolution lifts to a deterministic flow on parcels, and a finite-precision measurement process is represented by a volume-contracting update that refines the initial parcel into a more constrained open set, strictly increasing the geometric information defined as the Hilbert-Schmidt volume of the parcel. By introducing a second impossible set, we obtain a double-parcel

What carries the argument

The quantum parcel, defined as a basic weak-star open set of density matrices by finitely many open expectation intervals, which represents the compatible microscopic states and carries the deterministic evolution flow plus the volume-contracting measurement update.

If this is right

  • Wave-particle duality becomes a continuous trade-off in the precision of position versus momentum rather than a fundamental paradox.
  • Schrödinger's cat is never in a literal superposition of alive and dead states because finite precision prevents the required point-like macroscopic state.
  • Entanglement loses its spooky action at a distance and is replaced by a purely epistemic geometric update of the joint parcel.
  • The von Neumann entropy paradox is resolved because information, measured by parcel volume, increases monotonically under the double-parcel construction.
  • All empirical predictions of standard quantum mechanics are recovered exactly only in the infinite-precision limit, which is never physically attained.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Simulation algorithms for quantum systems could be redesigned to track parcel volumes from the outset rather than adding precision as an afterthought.
  • Tests of the framework could measure whether repeated finite-precision observations on the same system show the predicted strict increase in geometric information.
  • Macroscopic quantum phenomena in solid-state devices or biological systems might exhibit different scaling once parcel refinement rather than point-state collapse is modeled.
  • The approach opens a route to derive classical limits directly from the volume-contraction dynamics without separate decoherence postulates.

Load-bearing premise

That a quantum parcel defined by open expectation intervals exactly represents the set of all microscopic states compatible with finite macroscopic observable measurements, and that the volume-contracting update precisely captures the effect of finite-precision measurement.

What would settle it

An experiment that produces a literal macroscopic superposition, such as a cat in a coherent alive-and-dead state that cannot be explained away by finite resolution, or a sequence of measurements whose information gain fails to increase monotonically with the predicted volume contraction.

read the original abstract

Standard quantum mechanics is an idealisation based on infinite-precision objects: point states, exact probabilities, and sharp measurements. Yet every real experiment has finite resolution, and for macroscopic systems we never have access to the microscopic state. Following Heisenberg's call for a theory built only on observable quantities, and von Neumann's insight that a complete description of a macroscopic system is neither possible nor necessary, we elevate the macroscopic state to a fundamental concept. We introduce Interval Quantum Mechanics (IQM), in which the state of a quantum system is never a point but a quantum parcel -- a convex weak open set of density matrices defined by finitely many open expectation intervals, representing exactly the set of microscopic states compatible with a finite set of macroscopic measurements. We show that unitary evolution lifts to a deterministic flow on parcels, and that a fuzzy measurement is represented by a volume-contracting update, strictly increasing the geometric information defined as the inverse of the Hilbert-Schmidt volume. By introducing a second impossible set we obtain a double-parcel whose geometric information increases monotonically -- dissolving the entropy stagnation problem, since von Neumann entropy is defined on point states that no finite-precision experiment can certify. The framework reformulates foundational puzzles without additional interpretational assumptions: wave-particle duality becomes a smooth trade-off; Schroedinger's cat is described by a parcel containing many microscopically distinct states; and spooky action at a distance disappears, replaced by a purely epistemic geometric update. All empirical predictions of standard quantum mechanics are recovered in the infinite-precision limit, which is never physically attained.

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.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes Interval Quantum Mechanics (IQM) in which the fundamental state is a quantum parcel: a basic weak-star open set of density matrices defined by finitely many open expectation intervals. This parcel is asserted to be exactly the set of all microscopic states compatible with finite-precision measurements of macroscopic observables. Unitary evolution is lifted to a deterministic flow on parcels, while finite-precision measurement is represented by a volume-contracting update that refines the parcel and strictly increases its Hilbert-Schmidt volume (geometric information). A double-parcel construction yields monotonic information growth, resolving the von Neumann entropy paradox. The framework claims to eliminate wave-particle duality, literal superpositions (e.g., Schrödinger's cat), and entanglement non-locality (replaced by epistemic geometric updates) without extra interpretational assumptions, recovering all standard quantum predictions exactly in the infinite-precision limit.

Significance. If the central identifications and derivations hold, the work would constitute a significant foundational reformulation by elevating finite-precision macroscopic descriptions to the fundamental level, following Heisenberg and von Neumann. It offers a geometric, epistemic account that could dissolve several interpretational puzzles into artifacts of the infinite-precision idealization while preserving empirical content in the appropriate limit. The volume-contraction and monotonic-information mechanisms provide a concrete handle on measurement and entropy that might prove useful beyond foundations.

major comments (2)
  1. [Abstract / parcel definition] Abstract and the section defining the quantum parcel: the assertion that a basic weak-star open set delimited by finitely many open expectation intervals is exactly the set of all density matrices compatible with finite macroscopic data is presented without a derivation from the operational meaning of finite resolution. This identification is load-bearing for the claims that superposition and non-locality are eliminated and that the volume-contracting update captures measurement without additional assumptions.
  2. [Measurement update and double-parcel construction] The paragraph introducing the volume-contracting update and double-parcel: the claim that finite-precision measurement is represented by the unique volume-contracting refinement that strictly increases Hilbert-Schmidt volume and reproduces standard QM in the infinite-precision limit requires an explicit proof that this update follows necessarily from the finite-resolution axioms rather than being chosen to produce the desired monotonic information increase and paradox resolutions.
minor comments (1)
  1. [Notation and definitions] The notation for the double-parcel and the precise definition of geometric information (Hilbert-Schmidt volume of an open set) would benefit from an explicit formula or example computation to clarify how the volume is evaluated on the space of density matrices.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our proposal for Interval Quantum Mechanics. Below we respond to each major comment, providing clarifications and indicating planned revisions to the manuscript.

read point-by-point responses

  1. Referee: [Abstract / parcel definition] Abstract and the section defining the quantum parcel: the assertion that a basic weak-star open set delimited by finitely many open expectation intervals is exactly the set of all density matrices compatible with finite macroscopic data is presented without a derivation from the operational meaning of finite resolution. This identification is load-bearing for the claims that superposition and non-locality are eliminated and that the volume-contracting update captures measurement without additional assumptions.

    Authors: The definition of the quantum parcel is intended to capture precisely the operational content of finite-precision measurements. Each finite-resolution measurement of a macroscopic observable corresponds to constraining its expectation value to an open interval, which defines a basic open set in the weak-star topology on the convex set of density matrices. The parcel is the intersection of finitely many such sets. We will expand the manuscript with an explicit derivation in a new subsection, starting from the operational axioms and showing how this eliminates literal superpositions and non-local effects as artifacts of the point-state idealization. revision: yes

  2. Referee: [Measurement update and double-parcel construction] The paragraph introducing the volume-contracting update and double-parcel: the claim that finite-precision measurement is represented by the unique volume-contracting refinement that strictly increases Hilbert-Schmidt volume and reproduces standard QM in the infinite-precision limit requires an explicit proof that this update follows necessarily from the finite-resolution axioms rather than being chosen to produce the desired monotonic information increase and paradox resolutions.

    Authors: We acknowledge the need for a formal proof of uniqueness and necessity. The update rule is derived by requiring that the post-measurement parcel remains the set of states compatible with the observed finite-precision outcome, while contracting the volume to reflect increased information. In the revision, we will add a theorem proving that this is the canonical choice satisfying the axioms, with the double-parcel ensuring monotonicity of geometric information. This will be shown to recover standard quantum predictions in the infinite-precision limit without ad hoc choices. revision: yes

Circularity Check

2 steps flagged

Parcel as exact compatible-state set and volume-contracting update are asserted by definition rather than derived from finite-precision axioms

specific steps
  1. self definitional [Abstract]
    "Such a parcel is the exact mathematical representation of the set of all microscopic states that are compatible with the measured values of a finite set of macroscopic observables."

    The paper first defines the parcel as a basic weak-star open set given by open expectation intervals; it then states without separate derivation that this set is precisely the collection of all compatible microscopic states. The subsequent claims that the framework eliminates foundational puzzles rest on this asserted equivalence, making the resolutions follow by construction from the chosen representation rather than from axioms of finite precision.

  2. self definitional [Abstract]
    "a finite-precision (fuzzy) measurement process is represented by a volume-contracting update that refines the initial parcel into a more constrained open set, strictly increasing the geometric information defined as the Hilbert-Schmidt volume of the parcel."

    The measurement update is introduced as the operation that contracts volume and increases the newly defined geometric information; the monotonic increase and resolution of the von Neumann entropy paradox via the double-parcel are then direct consequences of this choice, without an external benchmark showing the update is the unique or necessary refinement dictated by finite-resolution measurement.

full rationale

The paper's central chain begins by defining a quantum parcel as a weak-star open set of density matrices via finitely many open expectation intervals, then immediately asserts this construction is exactly the set of all microscopic states compatible with finite macroscopic data. The fuzzy measurement is then represented by a volume-contracting refinement that increases Hilbert-Schmidt volume by construction. These identifications allow the claimed resolutions of superposition, entanglement non-locality, and the entropy paradox to follow directly from the framework's own definitions and the infinite-precision limit recovery, without an independent derivation showing the parcel geometry is forced by operational finite-resolution axioms. This produces partial circularity (score 6) while still leaving room for independent content in the unitary flow and limit behavior.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The central claims rest on redefining states via parcels and postulating specific liftings of evolution and measurement; these are introduced without derivation in the abstract and lack independent external grounding.

axioms (2)
  • ad hoc to paper Unitary evolution lifts to a deterministic flow on parcels
    Stated as shown in the abstract without further justification provided here.
  • ad hoc to paper Finite-precision measurement is a volume-contracting update that refines the parcel and strictly increases geometric information
    Presented as a core property of the framework in the abstract.
invented entities (2)
  • quantum parcel no independent evidence
    purpose: Exact representation of all microscopic states compatible with finite macroscopic expectation intervals
    New basic object elevating the macroscopic state to fundamental status
  • double-parcel no independent evidence
    purpose: Construction yielding monotonic information increase to resolve von Neumann entropy paradox
    Introduced specifically to address the entropy issue

pith-pipeline@v0.9.0 · 5812 in / 1537 out tokens · 86083 ms · 2026-05-20T06:17:02.403030+00:00 · methodology

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