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Frame-Dependent Traces and the Third-Particle Paradox

T0 review · 2 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read The Third-Particle Paradox is a layer mismatch, not a contradiction; the paper proves exactly which states fail in each quantum-reference-frame framework.

desk verdict A careful, self-contained paper; the counterexample to the Relational Trace is genuine and the proofs are sound, but the abstract outruns what the QI all-states theorem actually delivers. read the letter →

arxiv 2607.21703 v1 pith:RYGEFYNL submitted 2026-07-23 quant-ph gr-qc

Alessandro Palumbo , Luca Apadula This is my paper
classification quant-phgr-qc MSC 81P1681P05
keywords quantumreferenceframesthird-particleparadoxperspective-neutralapproachweakinvariancerelationaltracepartialcovariancesubsystemconsistencyedgemodes
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

The paper is trying to establish that the Third-Particle Paradox—where adding an uncorrelated particle seems to erase relational information accessible from another particle's perspective—is not a genuine contradiction but the result of comparing inequivalent layers of description. It does this by introducing a statistical consistency condition and a frame-dependent trace, the Perspective Relational Trace, that enforces agreement between discarding a particle before physicalization and discarding it after. The authors prove the condition's domain: in the Perspective-Neutral approach it fails on a characterized set that includes uncorrelated product states, so that framework can only describe a closed total system; in the Quantum-Information approach it holds for all states, so that framework supports arbitrary subsystems. In the hybrid case—PN whole, QI subsystem—the weakly invariant algebra emerges from the kinematical partial trace, but consistency still holds only on a proper subset. A sympathetic reader would care because this turns the paradox into a diagnostic for which quantum-reference-frame framework is appropriate for subsystems versus closed systems.

What carries the argument

The central object is the Perspective Relational Trace (PRT), a subsystem-discarding map defined by a statistical consistency condition: the expectation value assigned by an external frame to a relational observable of subsystem 12 must equal the expectation value computed internally after discarding particle 3. In the Perspective-Neutral approach the map has the Kraus form R_2^{(1)} Π12 K_l R_23^{(1)†}, where Π12 projects onto the trivial-charge sector of subsystem 12; its domain is Λ, the largest kinematical subspace containing the physical subspace. In the Quantum-Information approach the analogous map is just the perspectival transform of the ordinary partial trace, because the partial t

What would settle it

Take G = Z2 and the product state |ηθ⟩12 ⊗ |+⟩3 from Appendix D. The paper predicts that G(12)[Tr3(ρ)] carries the phase θ while Tr3[Φ(123)(ρ)] is θ-independent, so the state lies outside Γ; a direct computation of these two operators for any θ ≠ 0 settles whether Theorem 12 is correct. Alternatively, find any operationally motivated discarding rule that satisfies the Perspective-Neutral condition for the paper's counterexample product state |ψ⟩12 ⊗ |−⟩3, and the claim that the paradox cannot be resolved there would be undercut.

Watch

Extended reading notes

Core claim

The paper's central claim is that the Third-Particle Paradox dissolves once one compares the right objects: the externally accessible relational information about subsystem 12, obtained by tracing out particle 3 before physicalization, versus the internally accessible information obtained by tracing after physicalization. The authors define a statistical consistency condition that makes this comparison precise and prove that the unique map satisfying it—the Perspective Relational Trace—exists only on a subspace Λ in the Perspective-Neutral framework, with product states already lying outside Λ; the paradox therefore persists there. In the Quantum-Information framework the analogous condition

Load-bearing premise

The paper's conclusions rest on accepting its statistical consistency condition—comparing the external trace taken before physicalization with the internal trace taken after physicalization—as the correct formalization of what the Third-Particle Paradox is about.

Editorial extensions

If this is right

  • The Perspective-Neutral framework cannot, in general, describe a subsystem after part of a closed system is discarded; the failure already occurs for product states where the third particle is uncorrelated with subsystem 12.
  • The Quantum-Information framework resolves the paradox for every kinematical state, so adopting the weakly invariant algebra from the outset is stable under subsystem discarding.
  • Tracing out a particle from a globally physical state always produces a weakly invariant reduced state, and the kinematical partial trace maps the physical trace-class ideal onto the full weakly invariant algebra.
  • Recovering the Quantum-Information description from a Perspective-Neutral total system is not faithful state by state: the consistency condition holds only on the proper subset Γ, so some externally accessible relational information is lost.
  • The mechanism behind the paradox is classical: imposing a global constraint and then discarding a subsystem is not the same as discarding first and then imposing the constraint, so no quantum effect is needed for the obstruction.

Reading between the lines

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

  • If the paper is right, the choice between Perspective-Neutral and Quantum-Information reference frames is dictated by the physical question: reserve the PN description for a genuinely closed 'whole universe', and start from the weakly invariant algebra whenever a proper subsystem is involved.
  • The charge-superselected structure obtained after the partial trace is a minimal toy model of edge modes in gauge theories; the entropy decomposition H({p_q}) + Σ p_q (log d_q + S(τ^{(q)})) and its non-distillability may carry over to local gauge-theory subregions.
  • Because the authors give a classical analogue of the constraint-then-discard obstruction, one could test the same phase-erasure phenomenon in a purely classical reference-frame model using translation symmetry and a probability distribution over momenta.
  • A practical design principle follows: any protocol that discards the extra-particle degree of freedom inside the Quantum-Information framework will reinstate the paradox, so retaining the extra-particle is necessary and sufficient for subsystem consistency.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper analyses the Third-Particle Paradox in quantum reference frame (QRF) frameworks. It identifies two distinct origins: the failure of the partial trace to be QRF-covariant, and the failure of the physical Hilbert space to inherit the kinematical tensor-product structure. It introduces a statistical consistency condition comparing subsystem discarding before and after physicalisation, together with the associated Perspective Relational Trace (PRT) and its quantum-information variant (PRT–QI). The main theorems are: Theorem 6 characterises the subspace Λ on which the PN consistency condition holds, and gives a counterexample to the earlier Relational Trace resolution; Theorem 9 proves all-state consistency in the QI approach; Theorem 12 restricts the hybrid PN-total/QI-subsystem condition to the subspace Γ; and Proposition 10 shows that the kinematical partial trace maps the physical ideal onto the full weakly invariant trace-class ideal. The paper concludes that the PN approach is appropriate for a closed, isolated system, that the QI approach is stable under subsystem discarding when adopted from the outset, and that the paradox is a consequence of comparing inequivalent layers of description without tracking external versus internal accessibility.

Significance. If the results hold, the paper gives a precise, operational diagnosis of a known paradox, separating three levels of subsystem description and characterising exactly the states on which consistency can be achieved. The proofs in Appendix G are self-contained and use standard tools (trace-pairing, Peter–Weyl decomposition, Schur-lemma arguments); the explicit counterexample to the Relational Trace and the surjectivity result connecting to boundary-charge/edge-mode superselection are valuable contributions. The main caveat is that the all-states QI resolution (Theorem 9) is relative to the convention that the full weakly invariant algebra, including the extra-particle degree of freedom, counts as relational information. This is acknowledged in the conclusions but not sufficiently in the abstract, where the claim that the QI approach 'can accommodate arbitrary subsystems' is stated without qualification. This is a scope/presentation issue rather than a technical error.

major comments (2)
  1. [Abstract; Sec. IV.2; Conclusions] The unqualified claim that 'the QI approach can accommodate arbitrary subsystems' overstates the scope of Theorem 9. The all-states consistency result is conditional on counting the full weakly invariant algebra B(HC,2|1)^G, including the extra-particle factor, as relational information. The paper's own example in Sec. IV.2 shows that the phase θ is carried by the term B, which is non-trivial on the extra-particle factor; restricting to strict relational observables O_{2|1} removes B and reinstates the Paradox within the QI framework. The Conclusions contain the caveat, but the Abstract and similar summary statements should qualify the claim, e.g. 'relative to the full weakly invariant algebra', to avoid misleading readers about the scope of the resolution.
  2. [Sec. IV.1, Def. 4/Eq. (39)] The negative PN results (Theorem 6 and the bΛ characterisation) are characterisations of the specific unnormalised statistical condition (39), which compares subnormalised physical weights. If the operational content is instead encoded in the normalised condition (46), the solvable set changes from Λ to the larger set bΛ, and some product states outside Λ become solvable. The manuscript does discuss the normalised version, but the headline conclusion that the PN approach cannot describe subsystems is often repeated without this qualification. Please state explicitly in the Abstract and Conclusions that the in-principle obstruction is for the unnormalised condition, and that the normalised condition yields a weaker, state-dependent obstruction.
minor comments (4)
  1. [Eq. (44) and following] The hatted map bT is non-linear and not CP; calling it a 'trace' in the text may confuse readers. Suggest using 'normalised output' or 'normalised PRT' consistently.
  2. [Sec. II.4, Eq. (23)] The notation T'_{i→j} for the post-operation QRF transformation is introduced, but for the partial trace no such post-operation frame exists. The later caveat is good; the main text could state earlier that the covariance formula assumes the output frame is available.
  3. [App. G.2, proof of Thm. 6] The phrase 'the previous equality on the physical space is satisfied if and only if it is satisfied on the kinematical space' is terse. Please spell out that Φ(12) is a projection onto the physical 12 sector and that both sides of Eq. (G12) are physical operators, so equality on the physical space is the relevant condition.
  4. [Footnote 26] The continuous-group example of a state with Π3|ψ3⟩=0 but non-zero global physical projection is only sketched. A brief explicit construction or a pointer to App. C/G.5 would improve reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: PRT/PRT-QI are defined by explicit constructions, and the consistency characterizations are proven rather than assumed.

full rationale

The paper's central derivation chain is self-contained and does not reduce to its inputs by construction. Definitions 4 and 7 introduce statistical consistency conditions and name the PRT / PRT-QI as maps that may satisfy them; Theorems 6 and 9 then prove existence and uniqueness by explicit Kraus decompositions (Eqs. 42 and 53), so the maps are not assumed to be the solution of the condition. The negative PN result (Theorem 6) is a genuine characterization of the set Λ, and the paper supplies an independent counterexample to the RT (Eq. 37) showing that the RT condition trivializes. The QI all-states result (Theorem 9) relies on the nontrivial identity (G14) that the partial trace commutes with weak twirling. Proposition 10 is proven by explicit block decomposition in App. G.5, and Theorem 12 follows from trace-pairing non-degeneracy. The only self-referential text is the note that the paper is based on A.P.'s Semester Project [75]; this is not load-bearing evidence. The paper also explicitly acknowledges that discarding the extra-particle would reinstate the paradox in the QI framework, which is an openly stated scope restriction, not a hidden circular step. The status of Eq. (39) as the operational content of the Paradox is argued rather than derived, but that is an interpretive premise, not a circular derivation. No fitted parameters are renamed as predictions, and no uniqueness theorem is imported from the authors' prior work. The analysis is therefore not circular.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

The ledger is clean: no free parameters and no invented physical entities. The PRT is a definition; the extra-particle degree of freedom is inherited from Ref. [9]. The axioms are standard QRF domain assumptions (compact G, ideal complete frames) plus standard mathematical facts (Peter-Weyl, trace-pairing non-degeneracy).

assumptions (5)
  • domain assumption Compact Lie group G; each particle is an ideal complete QRF with Hilbert space L^2(G)
    Sec. II, Eq. (1). The main theorems hold for compact G; non-compact cases are heuristic only.
  • standard math The coherent group averaging Π of Eq. (3) is a bounded, orthogonal projector onto the physical (strongly invariant) subspace
    Requires compactness and Haar invariance; footnotes 4 and App. A note the non-compact failure.
  • standard math Peter-Weyl decomposition and Schur's lemma: the G-invariant subspace of H^(q)_L ⊗ H^(q̄)_L is at most one-dimensional
    Used in App. C to derive the projector decomposition (C5) and Proposition 10.
  • domain assumption For ideal complete frames the Schrödinger reduction map R^(i) is a unitary isomorphism H_phys → H_{S|i}
    Sec. II.1, Eq. (10), citing Ref. [10]; needed for Lemma 5 and Theorem 6.
  • standard math Non-degeneracy of the trace pairing between B(H) and B_1(H), and between a von Neumann algebra and its predual
    Used in App. G.1-G.6 to promote state equalities from observable statistics.
invented entities (1)
  • Perspective Relational Trace (PRT) and PRT-QI
    purpose: Frame-dependent map discarding a subsystem while preserving the statistical consistency condition comparing external and internal QRFs.
    A new mathematical construction (Defs. 4, 7); it is justified by the consistency condition it satisfies, not by an external falsifiable prediction. The paper itself notes (Eq. 48) that the map equals the Relational Trace dressed by isometries.

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Pith. "Pith review of Frame-Dependent Traces and the Third-Particle Paradox." pith.science (2026). https://pith.science/paper/RYGEFYNL

@misc{pith2026260721703,
  author       = {Pith},
  title        = {Pith review of: Frame-Dependent Traces and the Third-Particle Paradox},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RYGEFYNL}},
  note         = {Machine review of arXiv:2607.21703}
}
read the original abstract

The Paradox of the Third Particle arises when comparing subsystem descriptions across Quantum Reference Frame (QRF) perspectives. We isolate two distinct origins of the Paradox: the QRF covariance of the partial trace and the failure of the physical Hilbert space to inherit the kinematical tensor-product structure. We give an explicit counterexample to the Relational Trace (RT) resolution: an uncorrelated product state for which the RT statistical condition trivialises. We then introduce a new statistical consistency condition comparing subsystem discarding between external and internal QRFs, together with an associated frame-dependent map, the Perspective Relational Trace (PRT). We argue that our condition captures the operational content of the Paradox: rather than imposing consistency on the whole state space, we characterise exactly the states on which it holds in the Perspective-Neutral (PN) and Quantum-Information (QI) approaches. This separates three levels of description: a PN subsystem of a PN whole, where consistency fails on a characterised set that includes product states; a QI subsystem of a QI whole, where it holds for all states; and a QI subsystem obtained from a PN whole by kinematical partial trace, where the full weakly invariant algebra is recovered, yet consistency holds only on a proper subset. These results show that the PN approach can consistently describe only a closed, isolated system, while the QI approach can accommodate arbitrary subsystems. Tracing out a subsystem from a globally PN state yields a charge-superselected algebra, reproducing in a minimal QRF model the boundary-charge structure of edge modes. We understand the Paradox not as a genuine contradiction, but as the consequence of comparing inequivalent physical layers without tracking which information is externally and which internally accessible.

Figures

Figures reproduced from arXiv: 2607.21703 by the authors.

Figure 1
Figure 1. Schematic representation of the strong twirling (left) and weak twirling (right), in the charge basis associated with [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Heuristic explanation of the clash between tracing before and after projection onto the physical Hilbert space. The [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. We now characterise the Perspective Relational Trace and its uniqueness via the following Theorem, whose proof is given in App. G.2. Theorem 6 (Characterisation of the PRT). Let Λ = n ρ123 ∈ B1(H123) [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

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Works this paper leans on

82 extracted references · 24 canonical work pages

  1. [65]

    Why gauge?

    C. Rovelli, “Why gauge?”Foundations of Physics, vol. 44, no. 1, pp. 91–104, 2014. https://doi.org/10.1007/s10701-013-9768-7

    work page doi:10.1007/s10701-013-9768-7 2014

  2. [80]

    Superselection rules and quantum protocols,

    A. Kitaev, D. Mayers, and J. Preskill, “Superselection rules and quantum protocols,”Physical Review A, vol. 69, no. 5, p. 052326, may 2004. https://doi.org/10.1103/PhysRevA.69.052326

    doi:10.1103/physreva.69.052326 2004

  3. [1]

    Quantum mechanics and the covariance of physical laws in quantum reference frames,

    F. Giacomini, E. Castro-Ruiz, and Č. Brukner, “Quantum mechanics and the covariance of physical laws in quantum reference frames,”Nature Communications, vol. 10, p. 494, jan 2019. https://doi.org/10.1038/s41467-018-08155-0 22

    doi:10.1038/s41467-018-08155-0 2019

  4. [2]

    Physics within a quantum reference frame,

    R. M. Angelo, N. Brunner, S. Popescu, A. J. Short, and P. Skrzypczyk, “Physics within a quantum reference frame,”Journal of Physics A: Mathematical and Theoretical, vol. 44, no. 14, p. 145304, Mar 2011. https://doi.org/10.1088/1751-8113/44/14/145304

    work page doi:10.1088/1751-8113/44/14/145304 2011

  5. [3]

    Quantum reference frame transformations as symmetries and the paradox of the third particle,

    M. Krumm, P. A. Höhn, and M. P. Müller, “Quantum reference frame transformations as symmetries and the paradox of the third particle,”Quantum, vol. 5, p. 530, Aug 2021. https://doi.org/10.22331/q-2021-08-27-530

    work page doi:10.22331/q-2021-08-27-530 2021

  6. [4]

    A change of perspective: switching quantum reference frames via a perspective-neutral framework,

    A. Vanrietvelde, P. A. Höhn, F. Giacomini, and E. Castro-Ruiz, “A change of perspective: switching quantum reference frames via a perspective-neutral framework,”Quantum, vol. 4, p. 225, Jan 2020. https://doi.org/10.22331/q-2020-01-27-225

    doi:10.22331/q-2020-01-27-225 2020

  7. [5]

    Switching quantum reference frames in the N-body problem and the absence of global relational perspectives,

    A. Vanrietvelde, P. A. Höhn, and F. Giacomini, “Switching quantum reference frames in the N-body problem and the absence of global relational perspectives,”Quantum, vol. 7, p. 1088, aug 2023. https://doi.org/10.22331/q-2023-08-22-1088

    work page doi:10.22331/q-2023-08-22-1088 2023

  8. [6]

    Relativistic quantum reference frames: The operational meaning of spin,

    F. Giacomini, E. Castro-Ruiz, and Č. Brukner, “Relativistic quantum reference frames: The operational meaning of spin,” Physical Review Letters, vol. 123, p. 090404, Aug 2019. https://link.aps.org/doi/10.1103/PhysRevLett.123.090404

    work page doi:10.1103/physrevlett.123.090404 2019

Show all 82 references
  1. [7]

    Quantum clocks and the temporal localisability of events in the presence of gravitating quantum systems,

    E. Castro-Ruiz, F. Giacomini, A. Belenchia, and Č. Brukner, “Quantum clocks and the temporal localisability of events in the presence of gravitating quantum systems,”Nature Communications, vol. 11, p. 2672, may 2020. https://doi.org/10.1038/s41467-020-16013-1

    2020 doi

  2. [8]

    Quantum reference frames for general symmetry groups,

    A.-C. de la Hamette and T. D. Galley, “Quantum reference frames for general symmetry groups,”Quantum, vol. 4, p. 367, nov 2020. https://doi.org/10.22331/q-2020-11-30-367

    2020 doi

  3. [9]

    Relative subsystems and quantum reference frame transformations,

    E. Castro-Ruiz and O. Oreshkov, “Relative subsystems and quantum reference frame transformations,”Communications Physics, vol. 8, no. 1, p. 187, 2025. https://doi.org/10.1038/s42005-025-02036-x

    2025 doi

  4. [10]

    Perspective-neutral approach to quantum frame covariance for general symmetry groups,

    A.-C. de la Hamette, T. D. Galley, P. A. Höhn, L. Loveridge, and M. P. Müller, “Perspective-neutral approach to quantum frame covariance for general symmetry groups,”arXiv preprint arXiv:2110.13824, 2021. https://arxiv.org/abs/2110.13824

    2021 arXiv

  5. [11]

    Entanglement-asymmetry correspondence for internal quantum reference frames,

    A.-C. de la Hamette, S. L. Ludescher, and M. P. Müller, “Entanglement-asymmetry correspondence for internal quantum reference frames,”Phys. Rev. Lett., vol. 129, p. 260404, Dec 2022. https://link.aps.org/doi/10.1103/PhysRevLett.129.260404

    2022 doi

  6. [12]

    whole universe

    It is therefore weakly invariant under translations of12. Therefore, tracing out part of a global physical state leads to a weakly invariant subsystem state. This discussion suggests that, once one starts from the PN approach but wishes to describe a proper subsystem, a weakly...

  7. [13]

    Indefinite causal order and quantum coordinates,

    A.-C. de la Hamette, V. Kabel, M. Christodoulou, and Č. Brukner, “Indefinite causal order and quantum coordinates,” Phys. Rev. Lett., vol. 135, no. 14, p. 141402, Oct. 2025. https://doi.org/10.1103/bnkn-4p3f

    2025 doi

  8. [14]

    Quantum reference frames for Lorentz symmetry,

    L. Apadula, E. Castro-Ruiz, and Č. Brukner, “Quantum reference frames for Lorentz symmetry,”Quantum, vol. 8, p. 1440, aug 2024. https://doi.org/10.22331/q-2024-08-14-1440

    2024 doi

  9. [15]

    What can we do in a symmetry-constrained perspective? The importance of the total charge’s status in quantum reference frame frameworks,

    G. Doat and A. Vanrietvelde, “What can we do in a symmetry-constrained perspective? The importance of the total charge’s status in quantum reference frame frameworks,”Quantum, vol. 10, p. 2126, Jun 2026. https://doi.org/10.22331/q-2026-06-08-2126

    2026 doi

  10. [16]

    The perspectives of non-ideal quantum reference frames,

    S. C. Garmier, L. Hausmann, and E. Castro-Ruiz, “The perspectives of non-ideal quantum reference frames,” 2025. https://arxiv.org/abs/2512.19343

    2025 arXiv

  11. [17]

    Frame perspectives for process matrices: from coordinate parametrization to spacetime representation,

    L. Apadula, A. Grinbaum, and Č. Brukner, “Frame perspectives for process matrices: from coordinate parametrization to spacetime representation,”arXiv preprint arXiv:2604.02873, 2026. https://arxiv.org/abs/2604.02873

    2026 arXiv

  12. [18]

    Accessibility of global properties from internal quantum reference frame perspectives,

    A.-C. de la Hamette, V. Kabel, and Č. Brukner, “Accessibility of global properties from internal quantum reference frame perspectives,” 2025. https://arxiv.org/abs/2510.09100

    2025

  13. [19]

    The group structure of dynamical transformations between quantum reference frames,

    A. Ballesteros, F. Giacomini, and G. Gubitosi, “The group structure of dynamical transformations between quantum reference frames,”Quantum, vol. 5, p. 470, Jun 2021. https://doi.org/10.22331/q-2021-06-08-470

    2021 doi

  14. [20]

    Sum of entanglement and subsystem coherence is invariant under quantum reference frame transformations,

    C. Cepollaro, A. Akil, P. Cieśliński, A.-C. de la Hamette, and Č. Brukner, “Sum of entanglement and subsystem coherence is invariant under quantum reference frame transformations,”Phys. Rev. Lett., vol. 135, p. 010201, Jun 2025. https://link.aps.org/doi/10.1103/h6b3-y4vt

    2025 doi

  15. [21]

    Reference frames, superselection rules, and quantum information,

    S. D. Bartlett, T. Rudolph, and R. W. Spekkens, “Reference frames, superselection rules, and quantum information,” Reviews of Modern Physics, vol. 79, no. 2, pp. 555–609, Apr 2007. https://doi.org/10.1103/RevModPhys.79.555

    2007 doi

  16. [22]

    Symmetry, Reference Frames, and Relational Quantities in Quantum Mechanics,

    L. Loveridge, T. Miyadera, and P. Busch, “Symmetry, Reference Frames, and Relational Quantities in Quantum Mechanics,” Foundations of Physics, vol. 48, no. 2, pp. 135–198, Feb. 2018. http://link.springer.com/10.1007/s10701-018-0138-3

    2018 doi

  17. [23]

    Relativity of quantum states and observables,

    L. Loveridge, P. Busch, and T. Miyadera, “Relativity of quantum states and observables,”EPL (Europhysics Letters), vol. 117, no. 4, p. 40004, feb 2017. https://doi.org/10.1209%2F0295-5075%2F117%2F40004

    2017

  18. [24]

    Operational Quantum Reference Frame Transformations,

    T. Carette, J. Glowacki, and L. Loveridge, “Operational Quantum Reference Frame Transformations,”Quantum, vol. 9, p. 1680, mar 2025. https://doi.org/10.22331/q-2025-03-27-1680

    2025 doi

  19. [25]

    Observer-dependent entropy and diagonal Rényi invariants in quantum reference frames,

    A.-C. de la Hamette, “Observer-dependent entropy and diagonal Rényi invariants in quantum reference frames,”arXiv preprint arXiv:2603.23598, 2026. https://arxiv.org/abs/2603.23598

    2026

  20. [26]

    The resource theory of quantum reference frames: manipulations and monotones,

    G. Gour and R. W. Spekkens, “The resource theory of quantum reference frames: manipulations and monotones,”New Journal of Physics, vol. 10, no. 3, p. 033023, mar 2008. http://stacks.iop.org/1367-2630/10/i=3/a=033023?key=crossref. 82bca9378b047fb39f6b06f2f8078c1f

    2008

  21. [27]

    Changing quantum reference frames,

    M. C. Palmer, F. Girelli, and S. D. Bartlett, “Changing quantum reference frames,”Physical Review A, vol. 89, no. 5, p. 052121, may 2014. https://doi.org/10.1103/PhysRevA.89.052121

    2014 doi

  22. [28]

    Quantum reference frames associated with noncompact groups: The case of translations and boosts and the role of mass,

    A. R. H. Smith, M. Piani, and R. B. Mann, “Quantum reference frames associated with noncompact groups: The case of translations and boosts and the role of mass,”Physical Review A, vol. 94, no. 1, p. 012333, jul 2016. https://doi.org/10.1103/PhysRevA.94.012333

    2016 doi

  23. [29]

    Dynamics of a quantum reference frame,

    D. Poulin and J. Yard, “Dynamics of a quantum reference frame,”New J. Phys., vol. 9, no. 5, p. 156, May 2007. https://doi.org/10.1088/1367-2630/9/5/156

    2007 doi

  24. [30]

    Approximating relational observables by absolute quantities: a quantum accuracy-size trade-off,

    T. Miyadera, L. Loveridge, and P. Busch, “Approximating relational observables by absolute quantities: a quantum accuracy-size trade-off,”Journal of Physics A: Mathematical and Theoretical, vol. 49, no. 18, p. 185301, may 2016. https://doi.org/10.1088/1751-8113/49/18/185301

    2016 doi

  25. [31]

    Quantum reference frames on finite homogeneous spaces,

    J. Głowacki, L. Loveridge, and J. Waldron, “Quantum reference frames on finite homogeneous spaces,”Int. J. Theor. Phys., 23 vol. 63, no. 5, p. 137, 2024. https://doi.org/10.1007/s10773-024-05650-7

    2024 doi

  26. [32]

    Spacetime quantum reference frames and superpositions of proper times,

    F. Giacomini, “Spacetime quantum reference frames and superpositions of proper times,”Quantum, vol. 5, p. 508, Jul 2021. https://doi.org/10.22331/q-2021-07-22-508

    2021 doi

  27. [33]

    Reduced quantum-reference-frame channels for open quantum systems,

    P. Luppi, V. Kabel, F. Giacomini, and A. Smirne, “Reduced quantum-reference-frame channels for open quantum systems,” arXiv preprint arXiv:2607.05578, 2026. https://arxiv.org/abs/2607.05578

    2026 arXiv

  28. [34]

    Trinity of relational quantum dynamics,

    P. A. Höhn, A. R. H. Smith, and M. P. E. Lock, “Trinity of relational quantum dynamics,”Physical Review D, vol. 104, no. 6, p. 066001, Sep 2021. https://doi.org/10.1103/PhysRevD.104.066001

    2021 doi

  29. [35]

    Quantum relativity of subsystems,

    A. S. Ahmad, T. D. Galley, P. A. Höhn, M. P. E. Lock, and A. R. H. Smith, “Quantum relativity of subsystems,”Phys. Rev. Lett., vol. 128, p. 170401, Apr 2022. https://link.aps.org/doi/10.1103/PhysRevLett.128.170401

    2022 doi

  30. [36]

    Transformation of spin in quantum reference frames,

    M. Mikusch, L. C. Barbado, and Č. Brukner, “Transformation of spin in quantum reference frames,”Phys. Rev. Research, vol. 3, p. 043138, Nov 2021. https://link.aps.org/doi/10.1103/PhysRevResearch.3.043138

    2021 doi

  31. [37]

    Internal quantum reference frames for finite Abelian groups,

    P. A. Höhn, M. Krumm, and M. P. Müller, “Internal quantum reference frames for finite Abelian groups,”J. Math. Phys., vol. 63, no. 11, p. 112207, 2022. https://doi.org/10.1063/5.0088485

    2022 doi

  32. [38]

    Quantum generalisation of Einstein’s equivalence principle can be verified with entangled clocks as quantum reference frames,

    C. Cepollaro and F. Giacomini, “Quantum generalisation of Einstein’s equivalence principle can be verified with entangled clocks as quantum reference frames,”Classical and Quantum Gravity, vol. 41, no. 18, p. 185009, aug 2024. https://doi.org/10.1088/1361-6382/ad6d26

    2024 doi

  33. [39]

    Quantum reference frames for an indefinite metric,

    A.-C. de la Hamette, V. Kabel, E. Castro-Ruiz, and Č. Brukner, “Quantum reference frames for an indefinite metric,” Communications Physics, vol. 6, no. 1, p. 231, 2023. https://doi.org/10.1038/s42005-023-01344-4

    2023 doi

  34. [40]

    Quantum conformal symmetries for spacetimes in superposition,

    V. Kabel, A.-C. de la Hamette, E. Castro-Ruiz, and Č. Brukner, “Quantum conformal symmetries for spacetimes in superposition,”Quantum, vol. 8, p. 1547, dec 2024. https://doi.org/10.22331/q-2024-12-04-1547

    2024 doi

  35. [41]

    Quantum reference frames at the boundary of spacetime,

    V. Kabel, Č. Brukner, and W. Wieland, “Quantum reference frames at the boundary of spacetime,”Phys. Rev. D, vol. 108, p. 106022, Nov 2023. https://link.aps.org/doi/10.1103/PhysRevD.108.106022

    2023 doi

  36. [42]

    Quantum Frame Relativity of Subsystems, Correlations and Thermodynamics,

    P. A. Höhn, I. Kotecha, and F. M. Mele, “Quantum Frame Relativity of Subsystems, Correlations and Thermodynamics,” arXiv preprint arXiv:2308.09131, Aug. 2023. https://arxiv.org/abs/2308.09131

    2023 arXiv

  37. [43]

    Quantum coordinates, localisation of events, and the quantum hole argument,

    V. Kabel, A.-C. de la Hamette, L. Apadula, C. Cepollaro, H. Gomes, J. Butterfield, and Č. Brukner, “Quantum coordinates, localisation of events, and the quantum hole argument,”Communications Physics, vol. 8, no. 1, p. 185, apr 2025. https://doi.org/10.1038/s42005-025-02084-3

    2025 doi

  38. [44]

    Quantum reference fields transformations in linearized quantum gravity,

    L.-Q. Chen and F. Giacomini, “Quantum reference fields transformations in linearized quantum gravity,”arXiv preprint arXiv:2606.09344, 2026. https://arxiv.org/abs/2606.09344

    2026 arXiv

  39. [45]

    Switching Internal Times and a New Perspective on the ‘Wave Function of the Universe’,

    P. A. Höhn, “Switching Internal Times and a New Perspective on the ‘Wave Function of the Universe’,”Universe, vol. 5, no. 5, p. 116, May 2019. https://doi.org/10.3390/universe5050116

    2019 doi

  40. [46]

    Equivalence of approaches to relational quantum dynamics in relativistic settings,

    P. A. Höhn, A. R. H. Smith, and M. P. E. Lock, “Equivalence of approaches to relational quantum dynamics in relativistic settings,”Frontiers in Physics, vol. 9, p. 181, jun 2021. https://doi.org/10.3389/fphy.2021.587083

    2021

  41. [47]

    Relativistic Bell test within quantum reference frames,

    L. F. Streiter, F. Giacomini, and Č. Brukner, “Relativistic Bell test within quantum reference frames,”Phys. Rev. Lett., vol. 126, p. 230403, Jun 2021. https://link.aps.org/doi/10.1103/PhysRevLett.126.230403

    2021 doi

  42. [48]

    Gravitational entropy is observer-dependent,

    J. De Vuyst, S. Eccles, P. A. Höhn, and J. Kirklin, “Gravitational entropy is observer-dependent,”Journal of High Energy Physics, vol. 2025, no. 7, p. 146, jul 2025. https://doi.org/10.1007/JHEP07(2025)146

    2025 doi

  43. [49]

    Quantum reference frames, measurement schemes and the type of local algebras in quantum field theory,

    C. J. Fewster, D. W. Janssen, L. D. Loveridge, K. Rejzner, and J. Waldron, “Quantum reference frames, measurement schemes and the type of local algebras in quantum field theory,”Communications in Mathematical Physics, vol. 406, no. 1, p. 19, 2025. https://doi.org/10.1007/s0022...

    2025 doi

  44. [50]

    Crossed products and quantum reference frames: on the observer-dependence of gravitational entropy,

    J. De Vuyst, S. Eccles, P. A. Höhn, and J. Kirklin, “Crossed products and quantum reference frames: on the observer-dependence of gravitational entropy,”JHEP, vol. 07, p. 063, 2025. https://doi.org/10.1007/JHEP07(2025)063

    2025 doi

  45. [51]

    Error correction in lattice quantum electrodynamics with quantum reference frames,

    E. Rothlin, C. Ferradini, and L.-Q. Chen, “Error correction in lattice quantum electrodynamics with quantum reference frames,”arXiv preprint arXiv:2604.06149, 2026. https://arxiv.org/abs/2604.06149

    2026 arXiv

  46. [52]

    Gauss law codes and vacuum codes from lattice gauge theories,

    J. P. Lacambra, A. Chatwin-Davies, M. Honda, and P. A. Höhn, “Gauss law codes and vacuum codes from lattice gauge theories,”arXiv preprint arXiv:2604.06087, 2026. https://arxiv.org/abs/2604.06087

    2026 arXiv

  47. [53]

    Relational entanglement entropies and quantum reference frames in gauge theories,

    G. Araujo-Regado, P. A. Höhn, and F. Sartini, “Relational entanglement entropies and quantum reference frames in gauge theories,”arXiv preprint arXiv:2506.23459, 2025. https://arxiv.org/abs/2506.23459

    2025 arXiv

  48. [54]

    Quantum reference frames on homogeneous spaces,

    J. Głowacki, “Quantum reference frames on homogeneous spaces,” 2024. https://arxiv.org/abs/2409.07231

    2024 arXiv

  49. [55]

    Quantization of dynamical symplectic reduction,

    M. Bojowald and A. Tsobanjan, “Quantization of dynamical symplectic reduction,”Communications in Mathematical Physics, vol. 382, pp. 547–583, 2021. https://arxiv.org/abs/1906.04792

    2021 arXiv

  50. [56]

    Algebraic properties of quantum reference frames: Does time fluctuate?

    M. Bojowald and A. Tsobanjan, “Algebraic properties of quantum reference frames: Does time fluctuate?”Quantum Reports, vol. 5, no. 1, pp. 22–37, 2023. https://arxiv.org/abs/2211.04520

    2023 arXiv

  51. [57]

    An algebraic approach to the “frozen formalism

    M. Bojowald and A. Tsobanjan, “An algebraic approach to the “frozen formalism” problem of time,”Physical Review D, vol. 107, p. 024003, 2023. https://arxiv.org/abs/2212.13961

    2023 arXiv

  52. [58]

    Effective constraints for quantum systems,

    M. Bojowald, B. Sandhoefer, A. Skirzewski, and A. Tsobanjan, “Effective constraints for quantum systems,”Reviews in Mathematical Physics, vol. 21, pp. 111–154, 2009. https://arxiv.org/abs/0804.3365

    2009 arXiv

  53. [59]

    Effective constraints for relativistic quantum systems,

    M. Bojowald and A. Tsobanjan, “Effective constraints for relativistic quantum systems,”Physical Review D, vol. 80, p. 125008, 2009. https://arxiv.org/abs/0906.1772

    2009 arXiv

  54. [60]

    An effective approach to the problem of time,

    M. Bojowald, P. A. Höhn, and A. Tsobanjan, “An effective approach to the problem of time,”Classical and Quantum Gravity, vol. 28, p. 035006, 2011. https://arxiv.org/abs/1009.5953

    2011 arXiv

  55. [61]

    Effective approach to the problem of time: general features and examples,

    M. Bojowald, P. A. Höhn, and A. Tsobanjan, “Effective approach to the problem of time: general features and examples,” Physical Review D, vol. 83, p. 125023, 2011. https://arxiv.org/abs/1011.3040

    2011 arXiv

  56. [62]

    Effective relational dynamics of a nonintegrable cosmological model,

    P. A. Höhn, E. Kubalova, and A. Tsobanjan, “Effective relational dynamics of a nonintegrable cosmological model,”Physical 24 Review D, vol. 86, p. 065014, 2012. https://arxiv.org/abs/1111.5193

    2012 arXiv

  57. [63]

    Interpreting quantum reference frame transformations through a simple example,

    E. Castro-Ruiz, T. D. Galley, and L. Loveridge, “Interpreting quantum reference frame transformations through a simple example,” 2025. https://arxiv.org/abs/2508.09540

    2025

  58. [64]

    On the relation between perspective-neutral, algebraic, and effective quantum reference frames,

    J. De Vuyst, P. A. Höhn, and A. Tsobanjan, “On the relation between perspective-neutral, algebraic, and effective quantum reference frames,”arXiv preprint arXiv:2507.14131, 2025. https://arxiv.org/abs/2507.14131

    2025 arXiv

  59. [66]

    The paradox of the third particle is classical,

    L. Hausmann and R. Renner, “The paradox of the third particle is classical,”arXiv preprint arXiv:2607.15351, 2026. https://arxiv.org/abs/2607.15351

    2026 arXiv

  60. [67]

    Decomposition of entanglement entropy in lattice gauge theory,

    W. Donnelly, “Decomposition of entanglement entropy in lattice gauge theory,”Physical Review D, vol. 85, p. 085004, 2012. https://doi.org/10.1103/PhysRevD.85.085004

    2012 doi

  61. [68]

    Entanglement entropy and nonabelian gauge symmetry,

    W. Donnelly, “Entanglement entropy and nonabelian gauge symmetry,”Class. Quant. Grav., vol. 31, no. 21, p. 214003,

  62. [69]

    Remarks on entanglement entropy for gauge fields,

    H. Casini, M. Huerta, and J. A. Rosabal, “Remarks on entanglement entropy for gauge fields,”Phys. Rev. D, vol. 89, no. 8, p. 085012, 2014. https://doi.org/10.1103/PhysRevD.89.085012

    2014 doi

  63. [70]

    Local subsystems in gauge theory and gravity,

    W. Donnelly and L. Freidel, “Local subsystems in gauge theory and gravity,”Journal of High Energy Physics, vol. 2016, no. 9, p. 102, sep 2016. https://doi.org/10.1007/JHEP09(2016)102

    2016 doi

  64. [71]

    The entanglement of distillation for gauge theories,

    K. Van Acoleyen, N. Bultinck, J. Haegeman, M. Marien, V. B. Scholz, and F. Verstraete, “The entanglement of distillation for gauge theories,”Phys. Rev. Lett., vol. 117, no. 13, p. 131602, 2016. https://doi.org/10.1103/PhysRevLett.117.131602

    2016 doi

  65. [72]

    Aspects of entanglement entropy for gauge theories,

    R. M. Soni and S. P. Trivedi, “Aspects of entanglement entropy for gauge theories,”JHEP, vol. 01, p. 136, 2016. https://doi.org/10.1007/JHEP01(2016)136

    2016 doi

  66. [73]

    Edge modes as dynamical frames: charges from post-selection in generally covariant theories,

    S. Carrozza, S. Eccles, and P. A. Höhn, “Edge modes as dynamical frames: charges from post-selection in generally covariant theories,”SciPost Physics, vol. 17, no. 2, p. 048, 2024. https://doi.org/10.21468/SciPostPhys.17.2.048

    2024 doi

  67. [74]

    On entanglement entropy in non-abelian lattice gauge theory and 3d quantum gravity,

    C. Delcamp, B. Dittrich, and A. Riello, “On entanglement entropy in non-abelian lattice gauge theory and 3d quantum gravity,”JHEP, vol. 11, p. 102, 2016. https://doi.org/10.1007/JHEP11(2016)102

    2016 doi

  68. [75]

    Frame-dependent traces and the paradox of the third particle,

    A. Palumbo, “Frame-dependent traces and the paradox of the third particle,” Semester Project, Institute for Theoretical Physics, ETH Zürich, Oct. 2025, supervisors: Luca Apadula and Renato Renner

    2025

  69. [76]

    Soft edges: the many links between soft and edge modes,

    G. Araujo-Regado, P. A. Höhn, F. Sartini, and B. Tomova, “Soft edges: the many links between soft and edge modes,” Journal of High Energy Physics, vol. 2025, no. 7, p. 180, 2025. https://doi.org/10.1007/JHEP07(2025)180

    2025 doi

  70. [77]

    I. M. Gel’fand and N. Y. Vilenkin,Generalized Functions, Volume 4: Applications of Harmonic Analysis, ser. AMS Chelsea Publishing. Providence, RI: American Mathematical Society, 2016, vol. 380

    2016

  71. [78]

    Schwartz,Théorie des distributions, Tome 1

    L. Schwartz,Théorie des distributions, Tome 1. Paris: Hermann, 1950

    1950

  72. [79]

    J. M. Renes,Quantum Information Theory: Concepts and Methods. Berlin; Boston: De Gruyter, 2022

    2022

  73. [82]

    Compositionality in quantum reference frame perspectives,

    B. Sahdo and E. Castro-Ruiz, “Compositionality in quantum reference frame perspectives,” upcoming. Appendix A: Perspective Neutral Approach: mathematical details We use the trace pairing between trace-class and bounded operators, ⟨ρ,A⟩ tr := Tr[ρA], ρ∈ B 1(H),A ∈ B(H).(A1) For...

  74. [2014]

    https://doi.org/10.1088/0264-9381/31/21/214003

    doi

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