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arxiv: 1907.03456 · v2 · pith:QGZIVG5Qnew · submitted 2019-07-08 · 🪐 quant-ph

Emergence of outcomes in quantum mechanics

Bradley A. Foreman This is my paper

Pith reviewed 2026-05-25 01:26 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum measurement problememergence of outcomesquantum statesprojection operatorsdecoherencequantum discordunitary dynamicsentropy increase
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The pith

Redefining quantum states as equivalence classes of density operators resolves the tension between unitary evolution and definite outcomes.

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

The paper distinguishes the philosophical issue of why closed systems produce definite experimental outcomes from the physical issue of showing mathematically that unitary Schrödinger dynamics can lead to reduction without contradiction under an objective criterion. It addresses the physical issue by treating a quantum state not as a ray or density operator but as the full equivalence class of all density operators within a subspace that correspond to one outcome. For subsystems integrated with their environments these classes are represented by projection operators, and the change of state upon reduction is measured in a way that becomes negligible in a suitable limit. Decoherence ideas are adapted to this definition to produce a single framework that handles the measurement problem, though the resulting theory is only weakly objective. A sympathetic reader would care because the redefinition makes the emergence of outcomes a calculable feature of the dynamics rather than an added postulate.

Core claim

The paper claims that the physical problem of outcomes is solved by redefining the quantum state as an equivalence class of all density operators in a given subspace that describe the same experimental outcome. For systems of subsystems integrated with their environments, these classes are represented by projection operators; the formalism then relates directly to von Neumann's treatment of entropy increase. Nearly isolated subsystems undergo reduction only indirectly through interaction with integrated subsystems, and their reduced states coincide with the conditional states used to define quantum discord. All central concepts of decoherence theory can be carried over to this definition, so

What carries the argument

The equivalence class consisting of all density operators in a given subspace, represented by projection operators for integrated subsystems.

If this is right

  • Nearly isolated subsystems are reduced only indirectly through their interaction with integrated subsystems.
  • The reduced states of isolated subsystems are the conditional states appearing in the definition of quantum discord.
  • Key concepts from decoherence theory can be adapted without change to the new definition of state.
  • The resulting theory addresses every aspect of the quantum measurement problem in principle.
  • The formalism is closely related to von Neumann's earlier treatment of entropy increase under the second law.

Where Pith is reading between the lines

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

  • The same equivalence-class construction could be tested by checking whether entropy production in open quantum systems matches the projection-operator predictions more closely than standard density-operator evolution.
  • This state definition might allow quantitative comparison of reduction rates across different degrees of environmental integration.
  • It suggests examining whether quantum discord measures can be reinterpreted as direct signatures of the indirect reduction process for isolated subsystems.

Load-bearing premise

That equivalence classes of density operators for subsystems integrated with their environments can be represented by projection operators.

What would settle it

An experiment on a nearly isolated subsystem that exhibits reduction without detectable interaction with any integrated environment, or a calculation showing that the projection-operator representation fails to make the unitary-reduction discrepancy negligible by the stated measure.

read the original abstract

A persistent focus on the concept of emergence as a core element of the scientific method allows a clean separation, insofar as this is possible, of the physical and philosophical aspects of the problem of outcomes in quantum mechanics. The philosophical part of the problem is to explain why a closed system has definite experimental outcomes. The physical part is to show mathematically that there exists a limit in which the contradiction between unitary Schroedinger dynamics and a reduction process leading to distinct outcomes becomes negligible according to an explicitly stated criterion, and to make this criterion as objective as possible. The physical problem is solved here by redefining the notion of a quantum state and finding a suitable measure for the change of state upon reduction. The appropriate definition of the quantum state is not as a ray or density operator in Hilbert space, but rather as an equivalence class consisting of all density operators in a given subspace, the members of which all describe the same experimental outcome. For systems containing only subsystems that are integrated with their environments, these equivalence classes can be represented mathematically by projection operators, and the resulting formalism is closely related to that used by von Neumann to study the increase of entropy predicted by the second law of thermodynamics. However, nearly isolated subsystems are reduced only indirectly, as a consequence of their interaction with integrated subsystems. The reduced states of isolated subsystems are the same conditional states used in the definition of quantum discord. The key concepts of decoherence theory can all be adapted to fit this definition of a quantum state, resulting in a unified theory capable of resolving, in principle, all aspects of the quantum measurement problem. The theory thus obtained is weakly objective but not strongly objective.

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 / 2 minor

Summary. The manuscript claims to resolve the physical aspect of the quantum measurement problem by redefining the quantum state as an equivalence class of density operators in a given subspace (represented by projection operators for integrated subsystems), introducing a suitable measure for the change of state upon reduction, adapting decoherence concepts, and relating the approach to von Neumann's entropy formalism. Nearly isolated subsystems are reduced indirectly via conditional states from quantum discord. This is asserted to exhibit a limit in which the contradiction between unitary Schrödinger evolution and reduction to distinct outcomes becomes negligible according to an objective criterion, yielding a unified, weakly objective theory.

Significance. If the explicit construction of the measure, the limit parameter, and the demonstration of negligibility were supplied and verified, the redefinition of states as equivalence classes could offer a framework for making outcomes emergent while preserving a connection to established entropy considerations. The separation of physical and philosophical aspects and the adaptation of decoherence are potentially useful, but the significance hinges on whether the redefinition produces independent dynamical consistency rather than relabeling.

major comments (2)
  1. [Abstract] Abstract: The central claim that the physical problem is solved by 'finding a suitable measure for the change of state upon reduction' and exhibiting a mathematical limit where the unitary-reduction contradiction becomes negligible is not supported by any explicit definition of the measure, limit parameter, or calculation demonstrating negligibility. This is load-bearing for the asserted mathematical solution.
  2. [Definition of quantum state] The section defining the equivalence class of density operators: No derivation is given showing that this redefinition yields dynamical consistency or quantitative suppression of the contradiction in a stated limit; the equivalence classes appear chosen to match outcomes, raising a correctness risk that must be tested against an independent criterion.
minor comments (2)
  1. [Abstract] The terms 'weakly objective' and 'strongly objective' appear in the final sentence but receive no definition or reference, impairing evaluation of the claimed objectivity level.
  2. [Abstract] Spelling: 'Schroedinger' should be 'Schrödinger'.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful review and constructive comments on our manuscript. We address each major comment below and will revise the manuscript to strengthen the explicit constructions and derivations as requested.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim that the physical problem is solved by 'finding a suitable measure for the change of state upon reduction' and exhibiting a mathematical limit where the unitary-reduction contradiction becomes negligible is not supported by any explicit definition of the measure, limit parameter, or calculation demonstrating negligibility. This is load-bearing for the asserted mathematical solution.

    Authors: The manuscript defines the measure for the change upon reduction as the distance (in trace norm) between the distinct projection operators that label the equivalence classes for integrated subsystems. The limit parameter is the size of the environment (or number of degrees of freedom) in the decoherence regime, where the overlap between distinct classes vanishes exponentially. The negligibility follows because the unitary evolution maps the full state into a mixture whose reduction cost, measured by the adapted von Neumann entropy increase, becomes arbitrarily small. We agree, however, that a self-contained calculation with an explicit model is not sufficiently highlighted and will add a dedicated subsection with an analytic or numerical demonstration in the revised version. revision: yes

  2. Referee: [Definition of quantum state] The section defining the equivalence class of density operators: No derivation is given showing that this redefinition yields dynamical consistency or quantitative suppression of the contradiction in a stated limit; the equivalence classes appear chosen to match outcomes, raising a correctness risk that must be tested against an independent criterion.

    Authors: The equivalence classes are not chosen ad hoc but are operationally defined: two density operators belong to the same class precisely when they yield identical statistics for every observable of the integrated subsystem (i.e., they share the same support on the relevant projector). Dynamical consistency follows because unitary Schrödinger evolution on the composite system preserves membership in these classes once decoherence has occurred. Quantitative suppression is obtained by adapting the decoherence measures to the conditional states of nearly isolated subsystems (via quantum discord) and showing that the reduction cost vanishes in the same limit that makes the pointer states stable. We will insert an explicit derivation in the revised manuscript that verifies this consistency against the independent criterion of monotonic increase of von Neumann entropy for the integrated subsystem, as already indicated in the text. revision: yes

Circularity Check

1 steps flagged

Redefinition of quantum state as equivalence class incorporates outcomes by construction

specific steps
  1. self definitional [Abstract]
    "The physical problem is solved here by redefining the notion of a quantum state and finding a suitable measure for the change of state upon reduction. The appropriate definition of the quantum state is not as a ray or density operator in Hilbert space, but rather as an equivalence class consisting of all density operators in a given subspace, the members of which all describe the same experimental outcome."

    The definition is constructed so that members of the equivalence class 'all describe the same experimental outcome,' which is the phenomenon the paper claims to derive mathematically via a limit where the unitary-reduction contradiction becomes negligible. The outcomes are therefore presupposed in the state definition itself rather than obtained from the dynamics or an independent criterion.

full rationale

The paper states that the physical problem of outcomes is solved by redefining the quantum state as an equivalence class whose members 'all describe the same experimental outcome.' This directly builds the target phenomenon into the foundational definition rather than deriving a limit in which unitary dynamics and reduction become consistent according to an independent criterion. The abstract presents this redefinition as the solution without exhibiting an explicit mathematical limit or objective measure showing negligibility of the contradiction. The connection to von Neumann's entropy formalism is described only as 'closely related,' providing no independent derivation. This matches the self-definitional pattern at the core of the claimed resolution.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the existence of a suitable measure for state change upon reduction and on the assumption that the equivalence-class definition makes the unitary-reduction contradiction negligible according to an objective criterion; these are introduced without independent derivation in the abstract.

axioms (2)
  • domain assumption Unitary Schrödinger dynamics governs closed quantum systems.
    Invoked as the dynamics that must be reconciled with reduction.
  • domain assumption Experimental outcomes are definite and distinct.
    Core premise of the problem of outcomes stated in the abstract.
invented entities (1)
  • Equivalence class of density operators as the quantum state no independent evidence
    purpose: To represent experimental outcomes directly so that reduction does not contradict unitary evolution.
    Newly introduced definition that is the load-bearing change in the formalism.

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Reference graph

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