Ćwiek and Korbicz (arXiv:2506.05545, 2025) showed that a Cartesian reference frame — information carrying a continuous gauge group, unlike the simple observable values objectivized previously — can be made approximately objective in quantum mechanics: embedding the optimal covariant frame-transmission protocols into a spectrum-broadcast-structure-like state on the left-regular representation of SU(2), they proved that multiple observers asymptotically reach covariant agreement (each decodes a different group element, yet all align to the same frame), but also that the required measurement provably disturbs the state, so only partial objectivity is achieved. The open problem posed by that work is to complete this objectivization of group-valued information: find encoding/decoding (or dynamical) schemes that achieve covariant agreement together with a meaningful non-disturbance condition (weakened or approximate, but explicitly quantified, not left qualitative) — or prove a fundamental tradeoff or no-go — while resolving the square-root ambiguity of the multi-observer decoding, determining whether such states can arise from realistic open-system evolutions despite requiring a left-regular-representation central system, clarifying the operational link to frame-inclusion constructions, and extending the framework beyond SO(3)/SU(2) to arbitrary Lie groups.

Background

Whether the objective classical appearance of the quantum world can be explained as a property of quantum states is studied in the frameworks of quantum Darwinism and spectrum broadcast structures (SBS). Quantum Darwinism describes objectivity as the redundant proliferation of information about a system into fragments of its environment; the SBS approach makes the corresponding quantum-state structure explicit, and the two notions of objectivity have been shown equivalent in their strong forms by Le and Olaya-Castro. An SBS state has the form ρSBS=∑mp(m)∣m⟩⟨m∣⊗⨂i=1kρmi\rho_{SBS}=\sum_m p(m)|m\rangle\langle m|\otimes\bigotimes_{i=1}^{k}\rho_m^i with the local states ρmi\rho_m^i having mutually orthogonal supports for different mm; it operationalizes two assumptions: agreement (all kk observers measure the same value mm) and non-disturbance (the observations disturb neither the system nor the other observers), as reviewed by Korbicz.

All objectivization results obtained so far concern simple information — the value of an observable such as position or spin. A Cartesian reference frame is a fundamentally different carrier of information: it is unspeakable, meaning it cannot be transmitted as a classical bit string but only by sending a physical system, and its possible values form a continuous group (SO(3) rotations, lifted to SU(2)) on which a gauge freedom acts; Bartlett, Rudolph and Spekkens review the general theory of such quantum reference frames.

Quantum transmission of a Cartesian frame was optimized by Bagan, Baig and Muñoz-Tapia and by Chiribella, D'Ariano, Perinotti and Sacchi: Alice encodes her frame into a state of NN spins-1/21/2, and Bob decodes it with a covariant POVM (a positive-operator-valued measure, i.e., a generalized quantum measurement whose effects transform covariantly under the group action), with estimation error scaling as 1/N21/N^2. Ćwiek and Korbicz embedded this protocol into an objectivity scenario with kk observers, each holding NN spins. Using the Hilbert space of the left-regular representation of SU(2) (the representation of the group on square-integrable functions on the group itself, with generalized basis states ∣g⟩|g\rangle), they constructed the SBS-like state ρ=∫dg p0(R(g))∣g⟩⟨g∣⊗⨂i=1k∣A(ggi)⟩⟨A(ggi)∣\rho=\int\mathrm{d}g\,p_0(\mathcal{R}(g))|g\rangle\langle g|\otimes\bigotimes_{i=1}^{k}|A(gg_i)\rangle\langle A(gg_i)|, where p0p_0 is a probability distribution over Alice's possible frames, R\mathcal{R} is the SU(2)-to-SO(3) cover, and ∣A(ggi)⟩|A(gg_i)\rangle are the optimal codewords. They proved that in the limit N→∞N\to\infty the joint decoding statistics take the covariant-agreement form ∫dR p0(R)∏iδ(RRi,Ri′)\int\mathrm{d}R\,p_0(R)\prod_i\delta(RR_i,R_i'): each observer decodes a different group element RRiRR_i, but each of these rotations aligns his frame with Alice's — the quantum analogue of Klein's Erlangen program, in which one geometric object is seen differently but objectively from different reference frames.

However, they also proved that their scheme is necessarily disturbed: the trace distance (a standard distinguishability measure for quantum states) between the averaged pre- and post-measurement states obeys 1−λk≤lim⁡N→∞12∥ρ−ρ′∥1≤1−λk1-\lambda^k\le\lim_{N\to\infty}\frac{1}{2}\|\rho-\rho'\|_1\le\sqrt{1-\lambda^k} with λ≈0.236\lambda\approx0.236, so even a single observer induces substantial disturbance, and the distance approaches 1 as the number of observers grows. The disturbance is local to each observer, but by the SBS criteria only partial objectivity is achieved. The decoding step itself contains an unresolved ambiguity: taking the square root of the multi-observer POVM effects is defined only up to unitary rotations that must satisfy a hierarchy of consistency conditions from partial traces.

Several adjacent results delimit the question without answering it. The no-broadcasting theorems for asymmetry of Marvian and Spekkens and of Lostaglio and Müller show that symmetry-breaking (asymmetry) cannot be broadcast to further systems by dynamics respecting the symmetry, in finite dimensions — and a shared reference frame is precisely distributed asymmetry — but they do not cover the unbounded, asymptotic-NN, measurement-based setting of the objectivization scheme. Frame-inclusion (G-twirling) constructions of Bartlett, Rudolph and Spekkens, and the quantum-reference-frame formalism of Giacomini, Castro-Ruiz and Brukner, produce group-invariant states by averaging over the group, but an SBS-like objectivization state necessarily carries the prior p0p_0, and no nontrivial distribution on a group is translation-invariant, so the two constructions cannot coincide and their operational relation is unknown; Le, Mironowicz and Horodecki showed that quantum-Darwinist objectivity is itself frame-dependent, and Vanrietvelde sharpened the operational meaning of quantum reference frames, without building this bridge. Dynamical derivations of SBS states from open-system evolution exist only for simple observables — quantum Brownian motion (Tuziemski and Korbicz), von-Neumann-type interaction Hamiltonians including continuous variables (Acevedo, Wehr and Korbicz), and the measurement-theoretic construction of Korbicz, Aguilar, Ćwikliński and Horodecki — and the universality result of Scandolo, Salazar, Korbicz and Horodecki shows the SBS structure is robust across general probabilistic theories. Asadian, Gams and Sponar recently formulated a covariant correlation-disturbance tradeoff for measurements without shared frames. Group-valued information has not otherwise been addressed in the objectivization sense in the literature.

Problem Statement

Complete the objectivization of reference-frame (group-valued) information begun by Ćwiek and Korbicz. The central unresolved question — the disturbance drawback — is: can one design encoding/decoding schemes for group-valued information that achieve covariant agreement while also satisfying a meaningful non-disturbance condition — meaningful in that the weakened or approximate alternative must be explicitly quantified (e.g. as a bound on the disturbance measured by the trace distance 12∥ρ−ρ′∥1\frac{1}{2}\|\rho-\rho'\|_{1} between the averaged pre- and post-measurement states, or a comparable operational measure) rather than left qualitative — or is there a fundamental agreement-disturbance tradeoff or no-go, and if so what is its quantitative content? One technical facet sharpens this core: the decoding step takes square roots of the multi-observer POVM effects, a choice defined only up to unitary rotations constrained by a hierarchy of partial-trace consistency conditions, and it is open whether any consistent choice can reduce disturbance while preserving decoding performance across observers. Three further facets, posed by the same source as directions of the same overarching objective, complete the program: (a) dynamical origin — do the SBS-like states arise approximately from realistic physical evolutions such as decoherence and environment-induced dynamics, as happens for ordinary SBS states, given that the central system would need to realize the left-regular representation of SU(2), or can an equivalent objectivization be dynamically grounded? (b) physical realizability and interpretation — how can the constructed SBS-like state, which breaks group invariance through the nontrivial prior p0p_0, be reconciled and connected with frame-inclusion/frame-internalization constructions that emphasize frame-independence, and can a clear operational link be established? (c) generalization — can the covariant objectivization and the quantitative disturbance analysis be extended from SO(3)/SU(2) Cartesian frames to arbitrary Lie groups and more general geometric structures, and what are the resource-versus-disturbance tradeoffs in those cases? A complete answer resolves the central achievability-or-impossibility core, including the square-root/POVM-consistency facet, with quantitative statements, and resolves each of the three program facets (a)–(c) or proves it is not achievable; these parts are all required, and resolving the core alone does not constitute a complete answer.

References

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