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Doubly Quantum Mechanics

T0 review · 1 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Under SU_q(2), two observers cannot sharply align frames even with infinite spin exchanges.

desk verdict A serious formal framework with a clean derivation of operator-valued probabilities, but the marquee 'unavoidable fuzziness' result depends on an extra measurement postulate for probability observables, so treat the strong claim as conditional. read the letter →

arxiv 2412.05997 v3 pith:NWR6XNUQ submitted 2024-12-08 quant-ph gr-qc

Vittorio D'Esposito , Giuseppe Fabiano , Domenico Frattulillo , Flavio Mercati This is my paper · ORCID
classification quant-phgr-qc MSC 81R5081P15 PACS 03.65.Ta04.60.-m02.20.Uw
keywords doublyquantummechanicsSU_q(2)groupoperator-valuedprobabilityreferenceframesspinalignmentprotocolgravityphenomenologynoncommutativegeometryBornruledeformation
topics The Measurement ProblemQuantum Gravity
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

This paper proposes 'Doubly Quantum Mechanics,' a version of quantum theory in which the geometry of space—directions of spins, orientations of Stern–Gerlach devices, and the relative orientation between two observers' frames—is itself quantized. The authors promote the rotation group SU(2) to the quantum group SU_q(2), so that probability becomes a self-adjoint operator on the geometry Hilbert space rather than a number. Working with semi-classical geometry states, they derive that the measured spin expectation value along any axis retains an irreducible variance as the number of exchanged qubits N goes to infinity. If correct, this means two observers cannot sharply align their reference frames by exchanging arbitrarily many spin-1/2 systems—a finite, quantum-geometric limit to shared-frame communication that standard SU(2) does not impose.

What carries the argument

The key object is the quantum group SU_q(2), with generators α,γ obeying αγ=qγα, αγ*=qγ*α, γ*γ+α*α=1, and αα*−α*α=(1−q²)γ*γ. The paper promotes the complex coefficients of spin states, Pauli matrices, and observer-connecting transformations to copies of this algebra, so relative orientations are 'fuzzy points' in a noncommutative group manifold. The load-bearing computation combines the operator-valued probability (Eq. (37)) with the binomial–fuzziness joint distribution (Eq. (57)), whose variance contains the extra term N(N−1)Δ_f² that survives division by N. That identity is what converts a mild deformation of rotation symmetry into a permanent alignment floor.

What would settle it

A concrete check: construct separable semi-classical geometry states |Φ_S⟩⊗|Φ_SG⟩⊗|Φ_RO⟩ that are eigenstates of the operator-valued probability for the misaligned configurations used in the alignment protocol. If such an eigenstate exists for every rotation-matrix element R_ij, then the irreducible variance in Eq. (60) disappears and the claimed alignment limit is false.

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Extended reading notes

Core claim

The central claim is that quantizing the rotation symmetry of spin measurements deforms the Born rule. Concretely, the probability for a spin-up outcome, P(↑σ), is no longer a real number but an operator on H_SU_q(2)⊗H_SU_q(2); a measurement of that probability yields a distribution f(p) with mean p0 and generally nonzero variance Δ_f². For the alignment protocol, the measured expectation value of the spin satisfies Eq. (60), which in the N→∞ limit is [(q+$q^{{-1}}$)p0 - $q^{{-1}}$] ± (q+$q^{{-1}}$)Δ_f. Because Δ_f does not vanish for semi-classical geometry states (they are superpositions of probability eigenstates), the elements of the rotation matrix relating Alice and Bob carry an intrinsic uncertainty that cannot be removed by accumulating more data. The q=1 limit recovers the textbook $N^{{-1/2}}$ convergence.

Load-bearing premise

The load-bearing premise is that the relative orientation between two observers is a quantum geometry state, so the rotation coefficients u and v that connect their frames are operators obeying the SU_q(2) algebra (the q-deformed rotation group) rather than ordinary numbers; if they were ordinary numbers, the probability distribution would collapse to a delta and the infinite-measurement fuzziness would vanish.

Editorial extensions

If this is right

  • With q<1, the rotation-matrix elements measured by the alignment protocol converge to a finite-width distribution, not a point, as N→∞; the width is set by (q+q^{-1})Δ_f.
  • Even when the relative-orientation geometry state describes the identity rotation, the spin and Stern–Gerlach geometry states themselves introduce fuzziness, so no state of the protocol is exactly sharp.
  • The standard SU(2) result is recovered in the limit q→1, where Δ_f→0 and the uncertainty scales as N^{-1/2}.
  • Because probability is an observable, a single determination of a probability requires an apparatus and a batch of N electrons, and the geometry state collapses after the measurement; repeated identical preparations reconstruct f(p).
  • If the deformation is tied to a cosmological constant, the alignment floor provides a concrete quantum-gravitational limit on angular measurements.

Reading between the lines

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

  • If the paper is right, shared-reference-frame protocols face a per-physical-qubit capacity bound that standard shot-noise analysis misses: even with unlimited qubits, the classical bits extracted per physical qubit would not reach unity because the frame itself cannot be learned perfectly.
  • The same operator-valued probability formalism could be applied to other quantum groups than SU_q(2); for groups with a classical limit one would expect similar N→∞ fuzziness whenever the geometry states are not probability eigenstates.
  • A natural testbed is a doubly-quantum CHSH game: if the deformed probabilities violate Tsirelson's bound, the model would lie outside standard quantum theory and violate information causality—this is only suggested in the paper, not derived.
  • The discretization θ(n)=2 arcsin(q^n) implies that only a countable set of relative rotation angles is physical; as q→1 the angles become dense, but for any q<1 the closest allowed angle differs from the classical one, so even the mean rotation matrix is slightly deformed—an effect that might appear as a systematic, not statistical, residual.
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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

1 major / 3 minor

Summary. The paper proposes a framework of 'Doubly Quantum Mechanics' for spin-1/2 systems in which the SU(2) group of spatial rotations is promoted to the quantum group SU_q(2). The coefficients x,y of spin states, the matrix elements a,c of Pauli observables, and the relative rotation parameters u,v between reference frames are all promoted to operators acting on a 'geometry' Hilbert space H_SUq(2). Axioms 0-4 define geometry states, pre-measurement states, observables, expectation values, and measurement collapse for spin measurements. A key consequence is that probabilities become self-adjoint operators P(↑σ). The paper studies semi-classical geometry states and an alignment protocol in which Alice sends N spins and Bob measures them; it derives that the measured spin expectation value approaches [(q+q^{-1})p0-q^{-1}] ± (q+q^{-1})Δ_f as N→∞, so that the rotation matrix elements cannot be determined with arbitrary precision. Appendices provide spectral analysis of the probability eigenstates and numerical examples.

Significance. The paper is clearly written and offers a self-contained axiomatic construction, with explicit algebraic derivations, a detailed spectral analysis of the probability operator in Appendix A, and reproducible numerical examples in Section 6.2 and Appendix B. The operator-valued probability idea is a potentially novel contribution to quantum-foundational and quantum-gravity-motivated research. However, the central physical claim of unavoidable fuzziness in reference-frame alignment relies on a measurement rule for operator-valued probabilities that is not derived from the axioms; as argued in Major Comment 1, an alternative plausible rule would eliminate the residual uncertainty. The significance is therefore conditional: the framework is interesting, but the headline prediction is not yet forced by the stated postulates.

major comments (1)
  1. [Section 6.1, Eq. (57) and Appendix C.3] The joint distribution P(k,N,p) = C(N,k) p^k(1-p)^{N-k} f(p) is introduced via a projective-eigenvalue rule for the probability operator P_A(↑σ^B): a probability measurement collapses the geometry state to an eigenstate |p,r⟩ with probability f(p). This rule is not a consequence of Axioms 1-4. Axiom 4 specifies the collapse of the spin state onto |↑σ⟩ or |↓σ⟩ in a single spin measurement; Axiom 3 defines P(↑σ) as the operator-valued expectation value of the projector, but neither axiom prescribes how the outcome of a batch of N spin measurements relates to the spectrum of P. Under an alternative measurement prescription in which the N spin outcomes are used to estimate the fixed expectation value p0 = ⟨Φ|P_A(↑σ^B)|Φ⟩ without collapsing the geometry state, the variance would be Var(k) = Np0(1-p0), so Eq. (60) would tend to (q+q^{-1})p0 - q^{-1} with no residual ±(q+q^{-1})Δ_f and sharp alignment would be restored. The authors must either derive (57) from the stated axioms or explicitly add the projective-eigenvalue rule as an additional axiom. Until this is done, the central claim of an unavoidable infinite-N fuzziness is not a forced consequence of the framework.
minor comments (3)
  1. [Eq. (24), Axiom 1] The printed normalization ⟨ψ′|ψ′⟩ = q²xx* + yy* = 1 is inconsistent with the definition |ψ′⟩ = -q|↑⟩y* + |↓⟩x*; expanding the bra-ket gives q²yy* + xx*. The correct expression (using (15)) still equals 1, so this is likely a typographical transposition, but it should be corrected.
  2. [Section 4.2, Eq. (46)] The statement that x′,y′ 'realiz[e] an isomorphism C(SUq(2))⊗C(SUq(2))≃C(SUq(2))' is mathematically imprecise: the displayed map is the coproduct, which is an algebra homomorphism from C(SUq(2)) into the tensor product, not an isomorphism of the tensor-product algebra with C(SUq(2)).
  3. [Section 6.1, Eq. (60)] The notation ⟨σ^q_B⟩^{meas}_A would benefit from an explicit definition; as written it is used both for the measured value and for the random variable describing it.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the central fuzziness result follows from the stated axioms plus an explicit (non-derived) probability-measurement postulate, and self-citations are not load-bearing.

full rationale

I walked the derivation chain from Axioms 0–4 through Section 6. The probability operator P(↑σ) is defined in Axiom 3 and Eq. (37) as an operator-valued expectation value; its eigenstates and spectrum are computed in Appendix A, not imported from prior work. The semi-classical conditions (51)–(52) select a class of states, and Appendix A/B show that misaligned factorizable semi-classical states are not eigenstates of P, so the variance Δ_f is computed rather than fitted. Eq. (60)'s residual uncertainty (q+q^{-1})Δ_f follows mathematically from the joint distribution (57), and no parameter is adjusted to produce the predicted values. The main caveat is that Section 4.3 and Appendix C.3 add an explicit operational rule — a probability measurement collapses the geometry state to a P eigenstate and returns p with density f(p) — which is not derived from Axioms 1–4. This is a postulate-dependence or underdetermination issue, not a circular reduction: an alternative collective measurement rule could in principle remove Δ_f, but the paper openly states its rule as the 'doubly quantum mechanical interpretation' rather than disguisedly assuming the conclusion. Self-citations, notably [53] for the quantum-angle interpretation and Gaussian states, are contextual: the discretized angles follow from the representation (18), and the Gaussian states are here derived as approximations to the probability eigenstates, so the cited work is not load-bearing for the central claim. Overall the derivation is self-contained modulo the explicitly stated probability-measurement postulate, with only minor self-citation; score 2.

Assumptions & free parameters 2 free parameters · 8 assumptions · 2 invented entities

The central claim rests on the quantization of the spin, Stern-Gerlach, and relative-orientation coefficients as operators in SU_q(2), plus a new operational axiom that probability itself is an observable. The deformation parameter q and the shape parameters mu are free inputs, not derived from data. The model also leans on standard SU_q(2) representation theory and on the authors' earlier interpretation of these representations as quantum angles.

free parameters (2)
  • q = 0.99 (illustrative; physical value not specified)
    Deformation parameter of SU_q(2), real in (0,1); all results reduce to SU(2) as q tends to 1. The numerical examples set q=0.99.
  • mu_theta = 7 for theta approximately pi/2 at q=0.99
    Labels the semi-classical probability eigenstates |mu,omega>; chosen so the peak of |f(mu,n)|^2 matches the quantum angle closest to the classical angle theta. This is a hand-fitted relation used in Appendix B.
assumptions (8)
  • domain assumption Axiom 0: directions and relative orientations are described by geometry states in H_SUq(2), not by classical angles.
    Introduced in Section 4.1; this is the defining quantization step of the framework.
  • domain assumption Axiom 1: coefficients of spin states are elements of C(SU_q(2)) satisfying the algebra (15).
    Section 4.1, Eqs. (19)-(22).
  • domain assumption Axiom 2: observables are q-deformed Pauli matrices built from a second copy of SU_q(2).
    Section 4.1, Eq. (25).
  • domain assumption Axiom 3: expectation values map to operator-valued probabilities; P(up sigma) and P(down sigma) are positive self-adjoint operators summing to 1 tensor 1.
    Section 4.1, Eqs. (34)-(37).
  • domain assumption Axiom 4: measurement projects onto |up sigma> or |down sigma> and updates the geometry state.
    Section 4.1.
  • ad hoc to paper Operational interpretation: probability is an observable; the joint density of k spin-up outcomes in N trials is the binomial distribution convolved with the spectral distribution f(p).
    Section 4.3 and Appendix C.3, Eq. (57); not derived from standard quantum measurement theory.
  • ad hoc to paper Semi-classical conditions (51)-(52) select geometry states whose expectation values reproduce classical QM up to O(1-q) and variances up to O(1-q).
    Section 5.1; used to define the class of states for the numerical protocol.
  • standard math Representation theory of the SU_q(2) algebra: H_SUq(2) = H_pi plus H_rho with actions (17)-(18).
    Standard results from the quantum group literature [78,80], used throughout.
invented entities (2)
  • Geometry states |Phi> in H_SUq(2)
    purpose: Quantum description of the orientation of spins, measurement apparata, and reference frames
    Postulated in Axiom 0; their physical meaning is defined only through the model's own measurement postulates, with no independent experimental signature given.
  • Operator-valued probability P(up sigma)
    purpose: Promotes Born-rule probabilities to self-adjoint operators on geometry states; its spectral distribution f(p) is treated as a measurable probability-of-probability
    Defined in Section 4.1 Axiom 3 and interpreted in Section 4.3; no external benchmark or prediction independent of the framework.

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Pith. "Pith review of Doubly Quantum Mechanics." pith.science (2026). https://pith.science/paper/NWR6XNUQ

@misc{pith2026241205997,
  author       = {Pith},
  title        = {Pith review of: Doubly Quantum Mechanics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NWR6XNUQ}},
  note         = {Machine review of arXiv:2412.05997}
}
abstract

Motivated by the expectation that relativistic symmetries might acquire quantum features in Quantum Gravity, we take the first steps towards a theory of ''Doubly'' Quantum Mechanics, a modification of Quantum Mechanics in which the geometrical configurations of physical systems, measurement apparata, and reference frame transformations are themselves quantized and described by ''geometry'' states in a Hilbert space. We develop the formalism for spin-$\frac{1}{2}$ measurements by promoting the group of spatial rotations $SU(2)$ to the quantum group $SU_q(2)$ and generalizing the axioms of Quantum Theory in a covariant way. As a consequence of our axioms, the notion of probability becomes a self-adjoint operator acting on the Hilbert space of geometry states, hence acquiring novel non-classical features. After introducing a suitable class of semi-classical geometry states, which describe near-to-classical geometrical configurations of physical systems, we find that probability measurements are affected, in these configurations, by intrinsic uncertainties stemming from the quantum properties of $SU_q(2)$. This feature translates into an unavoidable fuzziness for observers attempting to align their reference frames by exchanging qubits, even when the number of exchanged qubits approaches infinity, contrary to the standard $SU(2)$ case.

Figures

Figures reproduced from arXiv: 2412.05997 by the authors.

Figure 1
Figure 1. Schematic description of the alignment protocol. Alice sends spin systems aligned along her axes and [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. The doubly quantum version of the protocol in Figure [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Distribution f(µ, n) with n ∈ [0, 130]∩N0 for two different values of µ, µ = 7 (blue plot) and µ = 17 (green plot), corresponding to distributions centred in n = 34 and n = 96 (or θ(n) ≈ π 2 and θ(n) ≈ π 4 ), respectively, for q = 0.99. These distributions are well approximated by the Gaussians that we used in [53]. Notice also that the variance increases as the value of n on which they are peaked increases, in the … view at source ↗

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Forward citations

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Reference graph

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Pith tools

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