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arxiv: 2605.09014 · v2 · pith:I4UX2DYHnew · submitted 2026-05-09 · 🪐 quant-ph · math-ph· math.MP

Resource theory of coherence in continuous position basis from measurement-induced dephasing

Karol Sajnok , Fabio Costa This is my paper

Pith reviewed 2026-05-19 14:56 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MP
keywords quantum coherencecontinuous variablesresource theorydephasing channelposition basismeasurement back-actioninterference visibility
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The pith

Quantum coherence in the continuous position basis is disturbance under a momentum-kick dephasing channel rather than distance from diagonal states.

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

The paper develops a resource theory for quantum coherence directly in the continuous position basis. Standard dephasing maps from finite dimensions cannot apply because position eigenstates are non-normalizable. It therefore defines incoherence through the fixed points of a physically motivated dephasing channel built from random momentum kicks, which is equivalent to the unconditional back-action of a finite-resolution position measurement. For realistic kernels this fixed-point set contains no normal states, so coherence is characterized by the amount of disturbance the channel produces. The work introduces two loss-based quantifiers, threshold witnesses linked to interference visibility, and an example with a Gaussian wave packet in gravity.

Core claim

A resource-theoretic framework for coherence in the continuous position basis is constructed by introducing a dephasing channel based on random momentum kicks. This channel yields a fixed-point notion of incoherence together with dephasing-covariant free operations. For physically relevant kernels the fixed-point set contains no normal states, showing that continuous-basis coherence is tied to dephasing disturbance rather than distance from a nonempty set of diagonal states.

What carries the argument

The dephasing channel generated by random momentum kicks, equivalently the unconditional back-action of a finite-resolution position measurement, which supplies both the fixed-point definition of incoherence and the class of free operations.

If this is right

  • A natural class of dephasing-covariant free operations is available for the resource theory.
  • The relative-entropy dephasing loss satisfies the main resource-theoretic properties under these operations.
  • Threshold witnesses certify coherence above a finite value and connect directly to interference visibility in two-path settings.
  • The framework applies to concrete dynamics such as a Gaussian wavepacket evolving in a gravitational potential.

Where Pith is reading between the lines

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

  • The same disturbance-based approach could be applied to other continuous observables whose eigenstates are non-normalizable.
  • Experiments could test the quantifiers by varying the resolution of position measurements and recording the resulting dephasing loss.
  • The gravitational-potential example suggests possible links between this coherence notion and gravitational decoherence models.

Load-bearing premise

The dephasing channel from random momentum kicks supplies a suitable and physically motivated definition of incoherence for normal states in the continuous position basis.

What would settle it

Finding any normal state that remains invariant under the random-momentum-kick dephasing channel for a physically relevant kernel would falsify the claim that the fixed-point set contains no normal states.

Figures

Figures reproduced from arXiv: 2605.09014 by Fabio Costa, Karol Sajnok.

Figure 1
Figure 1. Figure 1: Time dependence of the lower-bound expression for the [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
read the original abstract

We develop a resource-theoretic framework for quantum coherence directly in continuous basis, with emphasis on the position representation. Since position eigenstates are non-normalizable generalized eigenstates, the standard finite-dimensional dephasing map cannot be transferred directly to normal states. We therefore introduce a physically motivated dephasing channel based on random momentum kicks, equivalently described as the unconditional back-action of a finite-resolution position measurement. This yields a fixed-point notion of incoherence and a natural class of dephasing-covariant free operations. For physically relevant kernels, however, the fixed-point set contains no normal states, showing that continuous-basis coherence is tied to dephasing disturbance rather than to distance from a nonempty set of diagonal states. We study two quantifiers built from the channel action: a relative-entropy dephasing loss and a Hilbert-Schmidt dephasing loss. The former satisfies the main resource-theoretic properties under the free operations considered, while the latter is convex and experimentally transparent but fails monotonicity and strong monotonicity. We also formulate threshold witnesses for certifying coherence above a finite value and connect them, in a two-path setting, with interference visibility. Finally, we illustrate the framework with a Gaussian wavepacket evolving in a gravitational potential. The resulting theory provides a mathematically consistent and physically motivated treatment of coherence in continuous-variable systems.

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

1 major / 2 minor

Summary. The paper develops a resource theory of quantum coherence in the continuous position basis. Standard finite-dimensional dephasing maps cannot be used directly due to non-normalizable position eigenstates. Instead, a dephasing channel is defined via random momentum kicks (equivalently, unconditional back-action from finite-resolution position measurements). Incoherence is identified with fixed points of this channel. The central result is that for physically relevant kernels this fixed-point set contains no normal states, so coherence is tied to dephasing disturbance rather than distance to a nonempty set of diagonal states. Two quantifiers are introduced (relative-entropy dephasing loss, which satisfies the main resource-theoretic properties, and Hilbert-Schmidt dephasing loss, which is convex but fails monotonicity). Threshold witnesses are formulated and linked to interference visibility; the framework is illustrated with a Gaussian wave packet in a gravitational potential.

Significance. If the emptiness result for normal fixed points holds rigorously, the work supplies a physically motivated and mathematically consistent extension of coherence resource theories to continuous-variable systems. It supplies experimentally accessible quantifiers, connects them to observable interference, and provides a concrete example in a gravitational setting. These elements address a genuine gap between finite-dimensional resource theories and continuous-basis quantum optics or quantum information.

major comments (1)
  1. [Fixed-point analysis following channel definition] The load-bearing claim that 'for physically relevant kernels, however, the fixed-point set contains no normal states' (abstract and the section defining the dephasing channel) must be supported by an explicit proof that the integral operator Φ(ρ) = ρ admits no trace-class, positive, unit-trace solutions for kernels such as Gaussians or L1 functions with compact momentum support. The current statement leaves open the possibility that suitable position-space wave functions satisfy the fixed-point integral equation; without ruling this out under only the stated kernel assumptions, the reinterpretation that coherence is 'tied to dephasing disturbance rather than to distance from a nonempty set of diagonal states' remains unverified.
minor comments (2)
  1. [Channel construction] Clarify the precise action of the dephasing channel on density operators in the position representation, including the explicit integral kernel and any regularity conditions imposed on it.
  2. [Quantifier properties] Supply a short counter-example or explicit calculation showing why the Hilbert-Schmidt dephasing loss fails strong monotonicity under the dephasing-covariant operations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need to strengthen the fixed-point analysis. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: The load-bearing claim that 'for physically relevant kernels, however, the fixed-point set contains no normal states' (abstract and the section defining the dephasing channel) must be supported by an explicit proof that the integral operator Φ(ρ) = ρ admits no trace-class, positive, unit-trace solutions for kernels such as Gaussians or L1 functions with compact momentum support. The current statement leaves open the possibility that suitable position-space wave functions satisfy the fixed-point integral equation; without ruling this out under only the stated kernel assumptions, the reinterpretation that coherence is 'tied to dephasing disturbance rather than to distance from a nonempty set of diagonal states' remains unverified.

    Authors: We agree that the claim requires an explicit proof under the stated kernel assumptions to be fully rigorous. In the revised manuscript we will insert a dedicated subsection containing a detailed proof by contradiction: assume a trace-class positive unit-trace ρ satisfies Φ(ρ)=ρ; the integral kernel then forces the position-space matrix elements to obey an equation whose only solutions are either non-normalizable or have vanishing trace, for both Gaussian kernels and L1 kernels with compact momentum support. This establishes that the fixed-point set contains no normal states and thereby justifies the reinterpretation of coherence as tied to dephasing disturbance. revision: yes

Circularity Check

0 steps flagged

No significant circularity: framework derived from external physical dephasing model with independent mathematical result on fixed points.

full rationale

The paper motivates the dephasing channel via random momentum kicks (equivalently, finite-resolution position measurement back-action) as a physically motivated replacement for standard dephasing maps, which cannot be directly transferred due to non-normalizable position eigenstates. It then derives the fixed-point notion of incoherence and proves that for physically relevant kernels this set contains no normal states. This emptiness result is presented as a mathematical consequence of the integral operator defined by the kernel, not as a redefinition or statistical fit. No self-citations, ansatzes smuggled via prior work, or fitted inputs renamed as predictions appear in the derivation chain. The resource quantifiers (relative-entropy and Hilbert-Schmidt dephasing loss) and witnesses are constructed from the channel action after the fixed-point analysis, preserving independence from the target claims. The overall structure is self-contained against external physical benchmarks rather than internally circular.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central addition is a new conceptual channel and fixed-point definition rather than numerical fitting; the main premise is a domain assumption about position eigenstates.

axioms (1)
  • domain assumption Position eigenstates are non-normalizable generalized eigenstates, so the standard finite-dimensional dephasing map cannot be transferred directly to normal states.
    Explicitly stated in the abstract as the reason a new channel is required.
invented entities (1)
  • Dephasing channel based on random momentum kicks (or finite-resolution position measurement back-action) no independent evidence
    purpose: To define a fixed-point notion of incoherence and dephasing-covariant free operations in the continuous position basis
    Introduced as the physically motivated replacement for standard dephasing; independent evidence is not provided in the abstract.

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