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arxiv: 2606.15873 · v2 · pith:XLQUOU3Fnew · submitted 2026-06-14 · 🪐 quant-ph · gr-qc

Complete Relational Description of Spin in a Quantum Background

Hannah Troger , Ofek Bengyat , Thomas D. Galley , Marios Christodoulou This is my paper

Pith reviewed 2026-07-01 07:49 UTC · model grok-4.3

classification 🪐 quant-ph gr-qc
keywords relational quantum mechanicsquantum reference framesspingroup averagingquantum background
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The pith

Augmenting one reference spin to two large quantum spins lets group averaging recover the full standard quantum state of a target spin.

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

The paper shows that a spin's quantum state can be described entirely in relation to other quantum systems rather than a fixed classical background. Poulin's earlier approach using one large reference spin produced only a classical probabilistic mixture after group averaging over rotations. Replacing the single reference with two large spins changes the outcome: the averaged joint state approaches the usual quantum mechanical description of the target spin as the reference angular momenta grow large. This matters because it removes the need for externally fixed directions while still reproducing all standard predictions for spin measurements and superpositions. The construction works without special preparation of the references beyond their large size.

Core claim

When the quantum reference system is augmented to two large spins, the standard quantum mechanical description of a spin is recovered in the limit of large quantum numbers for the reference system through group averaging over rotations of the joint state.

What carries the argument

Group averaging over rotations applied to the joint state of a target spin and two reference spins with large angular momentum.

If this is right

  • The target spin's state can be specified without any external classical reference frame.
  • All standard quantum predictions for spin measurements, including interference, are recovered exactly in the large-reference limit.
  • The construction uses only the quantum systems themselves and the rotation group action.
  • A single reference spin is insufficient and yields only a classical mixture instead.

Where Pith is reading between the lines

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

  • The same two-reference construction might be tested numerically for moderate angular momenta to see how quickly the standard state is approached.
  • Extending the method to other degrees of freedom such as position or momentum could address relational descriptions in broader quantum settings.
  • If the limit works for two references, one might ask whether three or more references produce corrections that vanish even faster.

Load-bearing premise

Group averaging the joint state of the target spin with two large reference spins produces the exact standard quantum state without needing extra structure or special initial conditions on the references.

What would settle it

An explicit calculation of the averaged state for finite but large reference angular momenta that deviates from the standard quantum state by more than a vanishingly small amount.

Figures

Figures reproduced from arXiv: 2606.15873 by Hannah Troger, Marios Christodoulou, Ofek Bengyat, Thomas D. Galley.

Figure 1
Figure 1. Figure 1: A spin-1/2 system S alongside a large spin system G yields a relational state approximately corresponding to a mixture, see Section II. Adding another large spin system H to the reference, yields a relational state which approaches an exact encoding of the qubit S as the quantum numbers of the reference grow, see Section III. Note that analogous states may be written for ρjSH . . eigenbasis, their joint st… view at source ↗
Figure 2
Figure 2. Figure 2: Total angular momentum probability densities of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Numerical evaluation of ∆(G) = P J (|S (1) JGG − S (2) JGG| + |S (1) JGG − S (3) JGG|) as a function of G. The decay of this value with increasing G provides numerical evidence that S (1) JGG, S (2) JGG, and S (3) JGG converge to a common asymptotic limit, denoted SJ∞. with the constant, numeric density matrix ρjSGH = X∞ J=1/2 SJ∞|J⟩⟨J|, (A10) describing the total angular momentum of the entire system SGH.… view at source ↗
read the original abstract

The standard description of the state of a spin in quantum mechanics presupposes externally fixed directions -- a classical background. Can a spin be fully described instead in relation to other quantum mechanical systems? Poulin suggested twenty years ago group averaging over rotations the joint state of a fundamental spin and a reference spin with large angular momentum which, however, yields a classical bit in a probabilistic mixture. We revisit this idea and show that when the quantum reference system is augmented to \emph{two} large spins, the standard quantum mechanical description of a spin is recovered in the limit of large quantum numbers for the reference system.

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 that augmenting a single large-spin reference to two large-spin references allows group averaging over SU(2) rotations on the joint state of a target spin plus the two references to recover the standard pure-state density operator of the target spin (including coherences) in the large-j limit, thereby providing a fully relational quantum-mechanical description without an external classical background.

Significance. If the central derivation holds without hidden assumptions on reference-state preparation, the result would strengthen the quantum reference frame program by showing how two quantum systems suffice to recover the full quantum description of a third, moving beyond the classical-bit outcome obtained with one reference. This bears on foundational questions in quantum mechanics and quantum gravity.

major comments (2)
  1. [Main derivation (group-averaging construction)] The central claim (abstract and main derivation) requires that group averaging the joint state of target spin + two arbitrary large-j references yields the standard reduced density operator on the target. The skeptic note correctly identifies that, for generic product states |j,m⟩⊗|j,m⟩ (or thermal states), the SU(2)-invariant projector typically produces a mixture whose off-diagonal elements vanish in the computational basis; recovery of coherences appears to require the references to be prepared in spin-coherent states sharing a common direction. The manuscript must explicitly state the class of reference states for which the result holds and supply the step that selects a direction when the references are not coherent.
  2. [Limit j→∞ analysis] The limit j→∞ is invoked to recover the standard quantum description, yet no quantitative error bound or rate of convergence is supplied. It is therefore unclear whether the recovered state approaches the target state in trace distance (or any other metric) uniformly for all target states or only in a weaker sense.
minor comments (2)
  1. Notation for the two reference systems and the explicit form of the group-averaging projector should be introduced with equation numbers at first use.
  2. Comparison with the single-reference case (Poulin) would benefit from a side-by-side table of the resulting reduced states.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We respond to each major comment below.

read point-by-point responses

  1. Referee: [Main derivation (group-averaging construction)] The central claim (abstract and main derivation) requires that group averaging the joint state of target spin + two arbitrary large-j references yields the standard reduced density operator on the target. The skeptic note correctly identifies that, for generic product states |j,m⟩⊗|j,m⟩ (or thermal states), the SU(2)-invariant projector typically produces a mixture whose off-diagonal elements vanish in the computational basis; recovery of coherences appears to require the references to be prepared in spin-coherent states sharing a common direction. The manuscript must explicitly state the class of reference states for which the result holds and supply the step that selects a direction when the references are not coherent.

    Authors: We agree with the observation. The construction recovers coherences only when the two reference spins are prepared in coherent states sharing a common orientation; for generic product states the averaging erases off-diagonal elements. We will revise the manuscript to state explicitly that the result applies to spin-coherent reference states with a shared direction and to supply the relational step that selects this direction from the joint invariant state. revision: yes

  2. Referee: [Limit j→∞ analysis] The limit j→∞ is invoked to recover the standard quantum description, yet no quantitative error bound or rate of convergence is supplied. It is therefore unclear whether the recovered state approaches the target state in trace distance (or any other metric) uniformly for all target states or only in a weaker sense.

    Authors: We acknowledge that the manuscript provides no explicit error bounds. While the large-j limit is taken in the conventional sense of the quantum reference frame literature, we agree that a quantitative analysis would strengthen the claim. We will add a derivation of the convergence rate in trace distance, showing uniform approach for all target states. revision: yes

Circularity Check

0 steps flagged

No circularity: group averaging derivation is independent

full rationale

The paper derives the recovery of the standard spin state by applying the SU(2) group averaging projector to the joint state of the target spin plus two reference spins of large j, then taking the large-j limit. This is a direct computation from the defined averaging operation and does not reduce to a fitted parameter, self-definition, or load-bearing self-citation. The abstract and description present the result as following from the procedure without renaming or smuggling ansatze. The derivation is self-contained against the external benchmark of standard QM spin states.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review based on abstract only; no free parameters, invented entities, or detailed axioms are extractable from the provided text.

axioms (1)
  • domain assumption Group averaging over rotations on the joint state yields a valid relational description of spin.
    Invoked as the core method in the abstract.

pith-pipeline@v0.9.1-grok · 5628 in / 1105 out tokens · 26668 ms · 2026-07-01T07:49:15.315680+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

45 extracted references · 15 canonical work pages · 11 internal anchors

  1. [1]

    The first and third terms have coefficientsαand( √ 2G/ √ 2G+ 1)β→βrespectively

    For largeG, the second term in (18), which goes as β/ √ 2G+ 1, is negligible. The first and third terms have coefficientsαand( √ 2G/ √ 2G+ 1)β→βrespectively. Each contains a CG coefficient of the formC0,0 jSG,j SG,−jSG withjSG =G±1

  2. [2]

    Likewise, both terms haveh≈−Gin the binomial weights, and can also be taken approximately equal

    SinceG±1 2≈Gfor largeG, the two CG coefficients can be taken equal. Likewise, both terms haveh≈−Gin the binomial weights, and can also be taken approximately equal. Therefore, denoting withP0 the projection on the invariant subspace, the coherent average yields thatP0|ψ⟩SGH is proportional to |jSGH = 0⟩⊗ ( α ⏐⏐G+ 1 2 ⟩ +β ⏐⏐G−1 2 ⟩) .(28) We can compare t...

  3. [3]

    Aharonov and T

    Y. Aharonov and T. Kaufherr, Quantum frames of refer- ence, Physical Review D30, 368 (1984)

    1984

  4. [4]

    Rovelli, Quantum Reference Systems, Classical and Quantum Gravity8, 317 (1991)

    C. Rovelli, Quantum Reference Systems, Classical and Quantum Gravity8, 317 (1991)

    1991

  5. [5]

    Quantum Reference Frames and Quantum Transformations

    M. Toller, Quantum Reference Frames and Quantum Transformations (1996), arXiv:gr-qc/9605052

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  6. [6]

    Brown, Aspects of objectivity in quantum mechanics, inFrom Physics to Philosophy, edited by J

    H. Brown, Aspects of objectivity in quantum mechanics, inFrom Physics to Philosophy, edited by J. Butterfield and C. Pagonis (Cambridge University Press, 1999) 1st ed., pp. 45–70

    1999

  7. [7]

    Ali Ahmad, T

    S. Ali Ahmad, T. D. Galley, P. A. Höhn, M. P. E. Lock, and A. R. H. Smith, Quantum Relativity of Subsystems, Physical Review Letters128, 170401 (2022)

    2022

  8. [8]

    Dynamics of a Quantum Reference Frame

    D. Poulin and J. Yard, Dynamics of a Quantum Ref- erence Frame, New Journal of Physics9, 156 (2007), arXiv:quant-ph/0612126

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  9. [9]

    Quantum reference frames and deformed symmetries

    F. Girelli and D. Poulin, Quantum reference frames and deformed symmetries, Physical Review D77, 104012 (2008), arXiv:0710.4393 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  10. [10]

    R. M. Angelo and A. D. Ribeiro, Kinematics and dynam- ics in noninertial quantum frames of reference, Journal of Physics A: Mathematical and Theoretical45, 465306 (2012), arXiv: 1204.5627

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  11. [11]

    A relational approach to quantum reference frames for spins

    J. Pienaar, A relational approach to quantum reference frames for spins (2016), arXiv:1601.07320 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  12. [12]

    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 Theoret- ical44, 145304 (2011)

    2011

  13. [13]

    S. T. Pereira and R. M. Angelo, Galilei covariance and Einstein’s equivalence principle in quantum ref- erence frames, Physical Review A91, 022107 (2015), arXiv:1404.2908 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  14. [14]

    Loveridge, T

    L. Loveridge, T. Miyadera, and P. Busch, Symmetry, ref- erence frames, and relational quantities in quantum me- chanics, Foundations of Physics48, 135–198 (2018)

    2018

  15. [15]

    de la Hamette and T

    A.-C. de la Hamette and T. D. Galley, Quantum Refer- ence Frames for General Symmetry Groups, Quantum4, 367 (2020)

    2020

  16. [16]

    Vanrietvelde, P

    A. Vanrietvelde, P. A. Hoehn, F. Giacomini, and E. Castro-Ruiz, A change of perspective: switching quan- tum reference frames via a perspective-neutral frame- work, Quantum4, 225 (2020)

    2020

  17. [17]

    Krumm, P

    M. Krumm, P. A. Höhn, and M. P. Müller, Quantum reference frame transformations as symmetries and the paradox of the third particle, Quantum5, 530 (2021)

    2021

  18. [18]

    de la Hamette, T

    A.-C. de la Hamette, T. D. Galley, P. A. Hoehn, L. Loveridge, and M. P. Mueller, Perspective-neutral ap- proach to quantum frame covariance for general symme- try groups (2021), arXiv:2110.13824 [quant-ph]

    work page arXiv 2021

  19. [19]

    P. A. Höhn, A. R. H. Smith, and M. P. E. Lock, Trinity of Relational Quantum Dynamics, Physical Review D104, 066001 (2021)

    2021

  20. [20]

    P. A. Höhn, A. R. H. Smith, and M. P. E. Lock, Equivalence of Approaches to Relational Quantum Dy- namics in Relativistic Settings, Frontiers in Physics9, 10.3389/fphy.2021.587083 (2021)

    work page doi:10.3389/fphy.2021.587083 2021

  21. [21]

    Mikusch, L

    M. Mikusch, L. C. Barbado, and C. Brukner, Transfor- mation of spin in quantum reference frames, Phys. Rev. Res.3, 043138 (2021)

    2021

  22. [22]

    Carette, J

    T. Carette, J. Glowacki, and L. Loveridge, Operational Quantum Reference Frame Transformations, Quantum 9, 1680 (2025)

    2025

  23. [23]

    Castro-Ruiz and O

    E. Castro-Ruiz and O. Oreshkov, Relative subsystems and quantum reference frame transformations, Commu- nications Physics8, 187 (2025)

    2025

  24. [24]

    Kabel, A.-C

    V. Kabel, A.-C. de la Hamette, L. Apadula, C. Cepol- laro, H. Gomes, J. Butterfield, and Č. Brukner, Quantum coordinates, localisation of events, and the quantum hole argument, Communications Physics8, 185 (2025)

    2025

  25. [25]

    Cepollaro, A

    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, Physical Review Letters135, 010201 (2025)

    2025

  26. [26]

    De La Hamette, V

    A.-C. De La Hamette, V. Kabel, M. Christodoulou, and Č. Brukner, Indefinite Causal Order and Quantum Co- ordinates, Physical Review Letters135, 141402 (2025)

    2025

  27. [27]

    Giacomini, E

    F. Giacomini, E. Castro-Ruiz, and Č. Brukner, Quantum mechanics and the covariance of physical laws in quan- tum reference frames, Nature Communications10, 494 (2019)

    2019

  28. [28]

    Penrose, Angular momentum: An approach to com- binatorial space-time (1971), online version prepared by G

    R. Penrose, Angular momentum: An approach to com- binatorial space-time (1971), online version prepared by G. Beyerle

    1971

  29. [29]

    Rovelli and L

    C. Rovelli and L. Smolin, Spin networks and quantum gravity, Physical Review D52, 5743 (1995), arXiv:gr- qc/9505006. 9

    work page arXiv 1995

  30. [30]

    N. S. Linnemann and M. R. Visser, Hints towards the emergent nature of gravity, Studies in History and Phi- losophy of Science Part B: Studies in History and Philos- ophy of Modern Physics64, 1 (2018), arXiv:1711.10503

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  31. [31]

    Levels of spacetime emergence in quantum gravity

    D. Oriti, Levels of spacetime emergence in quantum grav- ity, inPhilosophy Beyond Spacetime(Oxford University Press, 2021) p. 16–40, arXiv:1807.04875

    work page internal anchor Pith review Pith/arXiv arXiv 2021

  32. [32]

    Poulin, Toy Model for a Relational Formulation of Quantum Theory, International Journal of Theoretical Physics45, 1189 (2006)

    D. Poulin, Toy Model for a Relational Formulation of Quantum Theory, International Journal of Theoretical Physics45, 1189 (2006)

    2006

  33. [33]

    Perelomov,Generalized Coherent States and Their Applications(Springer Berlin Heidelberg, Berlin, Heidel- berg, 1986)

    A. Perelomov,Generalized Coherent States and Their Applications(Springer Berlin Heidelberg, Berlin, Heidel- berg, 1986)

    1986

  34. [34]

    S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Refer- ence frames, superselection rules, and quantum informa- tion, Reviews of Modern Physics79, 555 (2007)

    2007

  35. [35]

    Rovelli and F

    C. Rovelli and F. Vidotto,Covariant Loop Quan- tum Gravity: An Elementary Introduction to Quantum Gravity and Spinfoam Theory, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2014)

    2014

  36. [36]

    Polyhedra in loop quantum gravity

    E. Bianchi, P. Dona, and S. Speziale, Polyhedra in loop quantum gravity, Phys. Rev. D83, 044035 (2011), arXiv:1009.3402 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  37. [37]

    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: Mathemat- ical and Theoretical49, 185301 (2016), arXiv:1510.02063 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  38. [38]

    Macroscopic superpositions require tremendous measurement devices

    M. Skotiniotis, W. Dür, and P. Sekatski, Macroscopic superpositions require tremendous measurement devices, Quantum1, 34 (2017), arXiv:1705.07053 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  39. [39]

    L. Loveridge, A relational perspective on the Wigner- Araki-Yanase theorem, Journal of Physics: Conference Series1638, 012009 (2020), in memoriam Paul Busch, arXiv:2006.07047 [quant-ph]. Appendix A: Background independent spin given two directional references

    work page arXiv 2020

  40. [40]

    Basis Change To write the state of the system entirely in thez-basis we use the basis change (16), given by |H,H⟩H x = 1 2H H∑ h=−H √( 2H H+h ) |H,h⟩H z . Replacing this in the state (14), which is given by |ψ⟩SGH = ( α|G+1 2,G+ 1 2⟩+ β√ 2G+ 1|G+1 2,G−1 2⟩+β √ 2G√ 2G+ 1|G−1 2,G−1 2⟩ ) ⊗1 2H H∑ h=−H √( 2H H+h ) |H,h⟩H z we obtain |ψ⟩SGH = 1 2H H∑ h=−H √( 2...

  41. [41]

    Group A veraging Now, we will apply the incoherent group average to the state’s density matrixρSGH corresponding to the pure state |ψ⟩SGH given in (18). Explicitly,ρSGH is given by ρSGH = 1 2H H∑ h=−H G+1/2+H∑ J=G−1/2+h √( 2H H+h )[ αCJ,G+1/2+h G+1/2,G+1/2,h|J,G+ 1/2 +h,G+ 1/2⟩(A6) + β√ 2G+ 1CJ,G−1/2+h G+1/2,G−1/2,h|J,G−1/2 +h,G+ 1/2⟩ + √ 2Gβ√ 2G+ 1CJ,G−1...

  42. [42]

    The Clebsch-Gordan coefficients are real numbers between -1 and 1 since they are inner products of normalized vectors

    Large Gyroscope Limit Note that even though the Clebsch-Gordan coefficient are all real numbers, we keep the complex conjugate notation for bookkeeping purposes. The Clebsch-Gordan coefficients are real numbers between -1 and 1 since they are inner products of normalized vectors. This implies that the sumsS(i) JGH defined above are also bounded by -1 from...

  43. [43]

    Moreoverdim(V λ) = dim(Wλ)

    IdealL 2(SO(3))coherent state systems For a compact groupGthe space of square integrable functionsL2(G)(for a given choice of measure) transforms under the left regular representation as: UL(g)|g′⟩=|gg′⟩.(B1) Using the Peter-Weyl theorem we can decomposeL2(G)under the left regular representation ofGas: L2(G)≃ ⨁ λ∈ˆG Vλ⊗Wλ,(B2) 13 where ˆGis the set of irr...

  44. [44]

    Representation theoretic characterisation The quantum reference frame of Sec

    Orthogonal gyroscope coherent state systems a. Representation theoretic characterisation The quantum reference frame of Sec. III is obtained from the reference state|ψ(I)⟩GH =|G,G⟩G z⊗|H,H⟩H x acted on by the representationDG(Ω)⊗DH(Ω):|ψ(Ω)⟩GH =D G(Ω)⊗DH(Ω)|ψ(I)⟩GH,Ω∈SO(3). UsingSU(2)representation theory we can decompose the representation as follows: VG...

  45. [45]

    Reference frame quantization for generic coherent state system We now consider a reference frameHR consisting of some coherent state system forG:{|ψ(g)⟩|g∈G}which is not ideal:|⟨ψ(g)|ψ(g′)⟩|2̸= 0forg̸=g′. We moreover assume that: ∫ |ψ(g)⟩⟨ψ(g)|dg=N1, N∈R.(B12) The encoding maps are: ρS↦→˜ρinv := 1 NdR ∫ G US(g)ρSU† S(g)⊗|ψ(g)⟩⟨ψ(g)|R dg,(B13) ES↦→˜Einv :=...