A Reference Frame Becomes Classical by Publishing Its Orientation

A quantum compass has not become classical merely because it points somewhere. It becomes public only when many physically separate fragments of its surroundings let many observers recover the same direction, within the same declared error, without returning to the compass.

That sentence is a research proposal, not a reported discovery. The calculation that would justify it has not been completed. The companion paper directory contains a model specification, a reference-verification starter ledger, covariance and limiting-case tests, and a simulation scaffold; it does not contain a finished rotor calculation or a publication-ready PDF. This article exists to make the proposed object precise enough to fail before a result is allowed to exist.

The object joins two established research programs. Quantum-reference-frame theory treats rods, clocks, phases, and orientations as physical quantum resources rather than invisible classical scaffolding. Quantum Darwinism treats an environment not only as a sink that decoheres a system, but as a collection of fragments through which separate observers can learn the same preferred property ^{[2, 12]}. The proposed synthesis asks a specific question neither label answers alone: when does an environment publish a continuous, group-valued frame parameter redundantly?

A binary pointer record can say left rather than right. An orientation lives on a curved space such as the sphere of directions or the rotation group. Its quality cannot be stated without an angular tolerance, a confidence level, an estimator, a prior or worst-case convention, and a declaration of which frame supplies the coordinates. Counting bits or finding a mutual-information plateau is therefore not enough.

I call the proposed quantity group-valued record redundancy. For accuracy $\epsilon$ and failure probability $\delta$, it counts the largest number of disjoint environment fragments from which covariant measurements can independently estimate the same relational orientation within $\epsilon$, with failure no greater than $\delta$. “Covariant” means a change of reference frame rotates the state, the measurement outcomes, and the reported estimate together; it cannot change whether the estimate was successful.

The provocation is narrow. A reference frame becomes operationally classical when its orientation is redundantly recoverable in that sense. The claim does not solve the measurement problem, create an absolute direction, or turn quantum Fisher information into substance. If the count changes under a legitimate quantum-reference-frame transformation, or if a large count fails to predict genuinely separate recoveries, the proposal is dead.

A compass direction is not a classical label

A laboratory description often begins with an unmentioned gift: an external frame. A Hamiltonian contains $J_z$; a detector sits at angle $\theta$; a photon is called horizontally polarized. Those symbols presuppose an axis relative to which “z,” “angle,” and “horizontal” have meaning. Quantum-reference-frame research asks what happens when the object providing that axis is finite and quantum.

Bartlett, Rudolph, and Spekkens reviewed how lacking a shared phase, direction, or Cartesian frame restricts which preparations and operations parties can perform. They described orientation as “unspeakable” information: unlike an abstract bit string, it cannot be faithfully communicated independently of the physical degrees of freedom that carry it ^[2]. A list of numbers saying “rotate 0.3 radians” is useless unless sender and receiver already agree about the axis and sign convention those numbers reference.

A finite frame also changes when it is used. Bartlett and colleagues modeled a spin-$j$ directional reference and found that repeated measurements degrade its usefulness, with a special quadratic longevity scaling for unpolarized test spins in their setting ^[4]. Poulin and Yard analyzed a quantum gyroscope subjected to sequential angular-momentum measurements and showed that its back-action dynamics depend on the incoming ensemble; generically the reference thermalizes on a time scale linear in its size, while symmetry explains the special unpolarized case ^[3]. The point is not which scaling wins universally. There is no universal scaling without a use protocol.

This immediately separates a quantum frame from the fixed parameter used in many decoherence examples. If photons or probe qubits learn a rotor’s orientation, conservation laws and back-action may reduce the orientation resource left in the rotor. The first fragment and the hundredth need not be identically distributed. Records can become more numerous while their source becomes less sharp.

Resource theory formalizes another part of that intuition. Gour and Spekkens treated states that break a symmetry as resources for parties restricted to symmetric operations and constructed quantities that cannot increase under the allowed transformations ^[7]. Marvian and Spekkens later developed asymmetry measures as constraints under symmetric dynamics, extending the informational consequences of symmetry beyond ordinary expectation-value conservation ^[8]. An oriented gyroscope is asymmetric under rotations. A perfectly rotation-invariant state is not a compass waiting to be read; relative to symmetric operations, it carries no preferred direction.

But asymmetry is not yet publicity. A single exquisitely oriented spin ensemble may be a powerful private frame. Group-valued redundancy asks whether the environment has divided useful evidence about that orientation into many separately accessible packages.

The frame in the formula is relational

Let $G$ be a compact group. For a full spatial orientation one may choose $SO(3)$; for a single direction carried by a spin-coherent state the parameter space is the sphere $S^2$, equivalently the quotient $SO(3)/SO(2)$. That distinction is not cosmetic. A spin pointing along $\mathbf n$ does not encode rotation about $\mathbf n$, so it cannot specify a full triad.

Choose a declared laboratory frame $L$ only as a temporary coordinate system. Let $g_{RL}$ be the rotor’s orientation relative to $L$. A covariant family of rotor states can be written

This is a definition. $e$ is the identity orientation, $U_R$ is a unitary representation of $G$, and $\rho_R(e)$ is the chosen fiducial state. Nothing in the equation makes $g$ absolute. Changing the reference frame by $h$ changes the coordinate label and representation together.

Giacomini, Castro-Ruiz, and Brukner constructed transformations between quantum reference frames in which states, observables, and dynamical descriptions all transform, while observed probabilities remain invariant ^[1]. Their result supplies a hard design constraint here: a redundancy score cannot be attached to a coordinate label alone. It must be built from probabilities of relational measurement outcomes.

Let the rotor interact with an environment $E=E_1\otimes\cdots\otimes E_N$ through a channel $\mathcal U$. For a candidate fragment $F$, the conditional fragment state is

This equation defines what the observer receives. $\bar F$ is everything except fragment $F$, including the remaining environment; $\mathcal U(\rho)=U\rho U^\dagger$ in a closed unitary model. The trace is not an assertion that the discarded systems cease to exist. It says the fragment observer has no access to them.

If $\rho_F(g)$ is the same for every $g$, no measurement on $F$ can reveal orientation. If the states differ, a measurement may estimate $g$, but its quality depends on the cost function and on whether one is solving a local or global estimation problem.

A successful estimate needs a metric and a blind-guess control

For $SO(3)$, a coordinate-free angular distance between two rotations can be defined by

with values from zero to $\pi$. Radians are dimensionless in SI, but $d$ still has the operational meaning of an angular error. A different group requires a declared invariant metric appropriate to that group. A coordinate chart’s Euclidean distance is not an acceptable substitute near its singularities.

An estimator is generated by a positive-operator-valued measure $M_F(d\hat g)$ on the fragment. Covariance requires that rotating the encoded state by $h$ rotates the distribution of estimates by the same group action. In schematic form,

Group-covariant estimation is established machinery. Bagan, Baig, and Muñoz-Tapia derived optimal strategies for transmitting and estimating Cartesian frames with finite spin systems ^[5]. Chiribella, D’Ariano, and Sacchi treated optimal estimation of group transformations under a class of invariant cost functions and connected the construction directly to reference-frame transmission ^[6]. The novelty proposed here is not the covariant POVM. It is using an operational estimation threshold to define environmental record redundancy.

For a Haar-uniform prior, define the fragment success probability

This is again a definition. The Haar measure $dg$ is normalized to one. A frequentist worst-case version could replace the average over $g$ with an infimum; the two must not be mixed after seeing results.

There is a trap. Even an orientation-blind fragment can sometimes satisfy a loose threshold by guessing from the prior. For uniform $SO(3)$, the probability that a blind estimate lies within geodesic angle $\epsilon$ of the true rotation is

This expression follows from the Haar distribution of rotation angle, whose density on $[0,\pi]$ is $2\sin^2(\theta/2)/\pi$. It passes the limiting checks: at $\epsilon=0$, blind success is zero; at $\epsilon=\pi$, it is one. The redundancy definition is informative only when $1-\delta>p_{\mathrm{blind}}(\epsilon)$. Otherwise an empty box can qualify as a witness.

That blind-guess clause is a necessary addition to the original architectural sketch. Without it, the declared zero-coupling limit would be false for lax $\epsilon$ and $\delta$. Finding and repairing that defect before simulation is what a paper scaffold is for.

Redundancy counts disjoint competent witnesses

Now define

subject to $1-\delta>p_{\mathrm{blind}}(\epsilon)$ and to a fixed rule for which fragment partitions are admissible. The displayed $g$ records the physical parameter of interest; under the Haar-average convention the resulting scalar is independent of its coordinate label. A worst-case definition would retain explicit dependence until the infimum is taken.

The fragments must be physically disjoint, not merely different data columns copied from one detector. This follows the environment-as-witness program, in which independent observers learn about a system by intercepting different environmental fractions ^[12]. Zurek’s quantum Darwinism made redundancy the bridge from fragile quantum alternatives to robust public records ^[11]. Riedel and Zurek showed how scattered photons can carry enormous redundancy about a macroscopic object’s pointer position under appropriate illumination ^[13]. Those results are inherited. They do not yet provide an operational angular estimator for a finite quantum frame.

Disjointness is necessary but may not be sufficient. Two fragments can be conditionally correlated because both descend from one relay or because sequential probes remain entangled through the rotor. If every observer’s estimate succeeds only because their fragments share a hidden common subsystem, counting them as independent witnesses exaggerates publicity.

Spectrum-broadcast-structure research sharpens this issue. Korbicz, Horodecki, and Horodecki characterized states in which classical information is broadcast into distinguishable environmental records even with noise ^[15]. Le and Olaya-Castro related strong quantum Darwinism and conditional independence to spectrum broadcast structure, while also distinguishing the information criterion from the stronger structural condition ^[16]. Brandão, Piani, and Horodecki proved that observers accessing many channels are generically restricted to classical information about one effective measurement, establishing an important form of agreement without promising perfect information in every fragment ^[14].

The proposed calculation will therefore report two numbers. The first is raw estimation redundancy under disjoint partitioning. The second is an independence-screened redundancy that rejects fragment collections whose conditional mutual information or shared-relay structure exceeds a preregistered tolerance. The exact independence functional remains part of the paper’s unfinished derivation. It cannot be selected after a desired curve appears.

A frame change must leave success unchanged

The central mathematical test is simple to state. Apply a common frame transformation $h$ to the physical description. Covariance sends $g$ and $\hat g$ to $hg$ and $h\hat g$, or to the appropriate left/right action fixed by convention. A left-invariant distance satisfies

The Haar measure is invariant, and the transformed POVM produces the correspondingly transformed outcome probabilities. Therefore the integration region defining success is merely relabeled. The value of $p_\epsilon(F)$, and hence $R_{\epsilon,\delta}$, should not change.

This is a derivation from the covariance assumptions, not evidence that a particular numerical implementation respects them. Euler-angle grids can break the result through nonuniform sampling, coordinate singularities, or an incorrectly normalized measure. The code must repeat each calculation after randomly chosen left and right group actions and compare the score to numerical tolerance.

Reparametrization is a separate test. Replacing Euler angles with quaternions or axis-angle coordinates cannot change the physical metric ball, fragment state, or success probability. It can change the components and apparent units of a Fisher matrix. That is why the headline quantity is a probability of falling inside an invariant error ball, not a component-wise variance in a convenient chart.

Giacomini and colleagues’ QRF transformations are more demanding than a classical coordinate substitution because the system chosen as the frame can itself be quantum and entangled ^[1]. The first paper stage will test ordinary group covariance. It may not advertise full quantum-frame covariance until the relational subsystems and transformation map are explicitly implemented. A metric that passes global rotation tests in a fixed laboratory description has cleared a necessary test, not the final one.

The kill criterion follows directly. If two legitimate quantum-reference-frame descriptions of the same relational experiment yield different redundancy after state, channel, fragments, POVMs, and metric are transformed consistently, the proposed quantity is not physical enough to carry the title.

Fisher information is a bound, not a public compass

Quantum Fisher information measures the local statistical distinguishability of neighboring states. Braunstein and Caves derived the quantum statistical metric by optimizing distinguishability over measurements ^[9]. For a single parameter $\theta$ and $\nu$ independent repetitions, the quantum Cramér–Rao relation is commonly written

under the relevant regularity and unbiasedness conditions. The variance has squared-coordinate units, so $F_Q$ has inverse-squared-coordinate units. Paris’s review emphasizes that quantum estimation is an optimization problem over measurements and explains the symmetric-logarithmic-derivative construction used to calculate the bound ^[10].

For orientation, several parameters and noncommuting rotation generators enter. The best measurement for one component may be incompatible with the best measurement for another. A local Fisher matrix near a known orientation does not automatically yield a globally covariant estimator over $SO(3)$, and a bound need not be attainable by each finite fragment. Multiplying fragment QFI by a count can also double-count correlations if the fragments are not independent.

The simulation will therefore use QFI as a diagnostic, not as the definition of publicity. For each fragment it will calculate a local quantum Fisher matrix where feasible, optimize or approximate a covariant measurement for the global loss, and directly estimate $p_\epsilon(F)$. A fragment with high local QFI but multiple globally ambiguous orientations may fail the operational threshold. Conversely, a fragment can pass a coarse angular threshold without saturating the local bound.

This distinction is the paper’s main methodological wager. Ordinary discrete redundancy asks how much information about a pointer observable a fragment contains. Group-valued redundancy asks whether an observer with an explicitly declared measurement can make a sufficiently accurate relational estimate. Information remains central, but competence is operationalized as a task.

The first numerical object is a direction, not a full triad

The initial simulation uses a spin-$j$ coherent state as a finite directional reference and $N$ sequential spin-one-half probes. That choice intentionally restricts the first result to $S^2$. Calling it a full $SO(3)$ frame would add an unencoded rotation about the spin axis. A later quantum-rotor or asymmetric-top model can carry a triad after the directional pipeline passes its tests.

Let $\mathbf J$ be the spin-$j$ angular-momentum operator and $\boldsymbol\sigma^{(n)}$ the Pauli vector for probe $n$. A candidate rotationally invariant interaction is

where $\Omega$ has units of inverse seconds, interaction time $\tau$ has seconds, and $\kappa=\Omega\tau$ is dimensionless. The factor $2j$ is a declared normalization choice, not an inherited law. Alternative normalizations must be compared because they change the large-$j$ limit if coupling strength is held fixed under different conventions.

The probes begin in a stated fiducial ensemble. Maximally mixed probes avoid importing a directional reference in their preparation, but a rotationally invariant interaction can still transfer asymmetry from an oriented rotor into them; “isotropic input” does not by itself imply “no record.” Polarized probes supply a laboratory axis and test a different relational task. Both cases belong in the grid, and their meanings must remain separate.

After sequential interactions, fragments are blocks of probe qubits selected under a fixed partition rule. The program will vary $j$, $N$, $\kappa$, fragment size, probe polarization, rotor purity, probe loss, and access geometry. It will calculate conditional fragment density matrices, direct estimation success, QFI diagnostics, back-action on the rotor, and conditional correlations between proposed witnesses.

No map over those parameters has been produced. The paper scaffold includes only unit-level checks of the group metric and blind-guess formula. It deliberately refuses to print a redundancy curve until the quantum state propagation and estimator have run.

Reproduction will require more than releasing a final array. Each numerical record must carry the spin size, Hilbert-space ordering, probe preparation, coupling normalization, interaction sequence, fragment partition, orientation samples, Haar weights, optimizer settings, random seed, and software versions. Density matrices will be checked for unit trace, Hermiticity, and nonnegative spectrum within a declared floating-point tolerance after every partial trace. The sequential channel will be compared against direct full-unitary propagation for small $j$ and $N$, where both are computationally affordable. Only after those checks agree may tensor-network or symmetry-reduced methods replace the direct calculation at larger size.

The estimator needs its own audit. A numerical POVM can appear successful because the orientation grid is too coarse, because training and evaluation reuse the same samples, or because Euler-angle quadrature overweights a coordinate pole. The registered protocol therefore separates optimization orientations from a fresh Haar-distributed evaluation set, repeats the result in quaternion coordinates, and includes an orientation-independent fragment as a blinded negative control. Uncertainty on $p_\epsilon(F)$ will be reported before the integer threshold converts it into redundancy. Otherwise a probability sitting one Monte Carlo error bar either side of $1-\delta$ could make the public witness count jump by several units without the underlying physics changing.

These requirements are not numerical housekeeping appended after theory. They determine what the proposed quantity means in finite computation. A published map without them would be a picture of one discretization, not a measurement of frame objectivity.

The numerical result to produce is a four-dimensional family of maps, not one favorable example: accuracy versus fragment size; redundancy versus coupling; residual rotor asymmetry versus number of probes; and raw versus independence-screened redundancy. Every map must carry regions where finite Hilbert-space truncation, optimizer failure, or Monte Carlo uncertainty makes the answer unreliable.

Four limits can reject the construction before interpretation

Zero coupling. At $\kappa=0$, the fragment state is independent of rotor orientation. For any threshold stricter than blind guessing, $R_{\epsilon,\delta}=0$. A nonzero answer exposes leakage from the parameter grid, prior, or estimator.

Orientation-blind access. “An isotropic bath cannot publish orientation” is too broad if it means only that the incoming probes are unpolarized. An asymmetric rotor can transfer orientation to initially isotropic carriers through a covariant interaction. The correct zero-record condition is operational: if the accessible channel satisfies $\rho_F(g)=\rho_F(e)$ for all $g$—for example because a complete group twirl erases directional dependence—then no fragment beats the Haar-prior baseline.

Classical limit. As $j$ grows, a spin-coherent direction becomes sharper. As $N$ grows, more probes become available. It is tempting to write “therefore redundancy diverges,” but back-action, finite coupling, loss, and fragment-size requirements decide the rate. The required limit is more modest: under a scaling that keeps the physical interaction and total disturbance controlled, the quantum calculation should approach the corresponding classical direction-estimation model. Bartlett and colleagues’ finite-frame results and Poulin and Yard’s dynamical analysis provide benchmarks for how use and degradation can scale ^{[4, 3]}.

Coordinate change. Euler angles, rotation matrices, quaternions, and axis-angle coordinates must give the same success probability when they represent the same Haar measure and geodesic ball. Disagreement is a numerical failure, not new physics.

Two nontrivial limits matter beyond those checks. At very weak nonzero coupling, each probe carries little directional evidence, so the minimum competent fragment may consume most of the environment and redundancy should remain small. At very strong sequential coupling, early probes may acquire sharp records while severely disturbing the rotor, causing later fragments to encode a changed direction. More interaction need not monotonically produce more agreement. That possible turnover is a hypothesis to calculate, not a result to hint into existence.

The closest rival says ordinary Darwinism already suffices

The strongest objection is that quantum Darwinism already defines redundancy for arbitrary observables. Choose an orientation POVM as the observable, calculate mutual information between the rotor and fragments, and stop. Brandão and colleagues’ channel result is general; spectrum-broadcast approaches already characterize objective classical information ^{[14, 15]}. Why invent another quantity?

Because continuous group parameters make three choices load-bearing that a discrete pointer label can hide. Resolution determines what counts as the same record. Covariance determines whether the answer is tied to a coordinate convention. A task-level estimator determines whether a finite observer can actually recover the orientation rather than merely share some correlation with it. Group-valued record redundancy earns its name only if those choices change conclusions in cases where a mutual-information plateau is ambiguous.

That condition is a novelty test. If the operational count is always a monotone relabeling of an existing discrete or continuous-variable redundancy measure across the full model grid, the proposed object adds no scientific decision and should be withdrawn. A new symbol is not a contribution.

A second rival is asymmetry resource theory. Perhaps the only quantity needed is how much rotational asymmetry moves from rotor to environment. Gour and Spekkens and Marvian and Spekkens already provide monotonicity tools ^{[7, 8]}. Yet total asymmetry in the environment can reside in one globally accessible entangled state. Public classicality requires useful information in many separate fragments. The calculation must show that total environmental asymmetry and redundant recoverability can diverge; if it cannot, resource theory has already done the work.

A third rival says the estimator imports a classical frame. Any apparatus that reports an angle seems to presuppose the very orientation standard being explained. This is the deepest resistance. The first simulation can only estimate the rotor relative to an explicitly declared laboratory frame. A full relational treatment must model the observer’s token frame or transform between quantum frames and verify that the success event is invariant ^[1]. Until then, “classical frame emergence” is provisional language attached to a fixed-lab calculation.

A fourth rival is more corrosive than the first three because it does not challenge the definitions; it challenges whether they require quantum mechanics at all. Nothing in $\rho_F(g)$, $p_\epsilon(F)$, or $R_{\epsilon,\delta}$ forces the fragments to be quantum-correlated in a way a classical broadcast could not reproduce. Replace the spin-$j$ rotor with a single classical variable drawn once from the same prior and handed to $N$ independent classical sensors, each corrupted by rotation-equivariant noise tuned to match the quantum construction’s per-fragment error rate. That model satisfies the covariance requirement, clears the blind-guess bound, and can be dialed to any $R_{\epsilon,\delta}(g)$ the quantum calculation might report, simply by choosing the classical noise level. If a classical dial can always be tuned to impersonate the quantum score, the score is not evidence that a quantum frame became classical; it is a redundancy count that classical estimation theory already understood, dressed in Hilbert-space notation.

The reply cannot be a difference in the redundancy number, since by construction the two numbers can be made to agree. It must be a difference in what redundancy costs the source. The classical sensors read a fixed value without disturbing it, so an observer could add a thousand-and-first fragment for free. The quantum construction cannot make that promise: every additional fragment is drawn from a probe that has already interacted with the rotor through $U_n$, and the paper’s own back-action results predict that competence gained by later fragments should trade against residual orientation asymmetry in the source ^{[4, 3]}. A classical emulator has no such cost curve to reproduce; its marginal fragment statistics can be copied indefinitely at zero disturbance to the source. This proposes an additional falsifier alongside the frame-covariance test: at every parameter point where $R_{\epsilon,\delta}$ is reported, the calculation must also check whether a non-disturbing classical channel matched to the same fragment marginals reproduces it. Wherever one does, and the rotor shows no measurable degradation as $M$ grows, the reported number is describing an already-classical broadcast rather than a quantum frame becoming one, and should be labeled accordingly rather than counted toward the paper’s central claim.

What can survive a failed paper

The project can fail in several ways. The proposed count may vary under quantum-frame transformations. Independence screening may eliminate every apparent redundancy plateau. The estimator may add no information beyond existing quantum-Darwinism measures. The spin model may publish only a direction and resist extension to a full orientation. A priority review may find the same metric already derived under another name.

If any of those occurs, the inherited physics survives. Finite quantum frames remain dynamical resources whose use creates back-action ^{[4, 3]}. Covariant group estimation remains the right language for communicating directions and rotations ^{[5, 6]}. Quantum Darwinism and spectrum broadcasting remain operational accounts of why many observers can agree about selected classical information ^{[12, 16]}. None of those claims depends on this paper’s metric succeeding.

The blind-guess correction also survives. Any thresholded continuous-parameter redundancy measure must demonstrate that its competence threshold beats the prior volume. Otherwise it counts lucky ignorance as a record. The distinction between a direction on $S^2$ and a full orientation on $SO(3)$ survives as well; it prevents a common conceptual inflation before any simulation runs.

If the project succeeds, it changes one question asked of classical emergence. Instead of asking only how much information a fragment contains, one asks how many separated observers can perform the same covariant estimation task at a declared resolution, and how much the source frame degrades while making them competent. That is a measurable bridge between a private quantum gyroscope and a public classical direction.

The paper’s embarrassment test is now on record. It would be embarrassing if a frame-dependent coordinate artifact produced the headline plateau; if an orientation-blind channel scored above zero; if six detector files copied from one relay were counted as six witnesses; or if local QFI were advertised as a globally attainable orientation. The original environment-as-witness program would already be sufficient if the new score never distinguishes systems it classifies together.

A reference frame does not become classical because its spin is large, its Wigner function looks narrow, or a theorist writes an angle beside it. Those may be useful signs. The operational event is harder: the frame has left enough competent, separate evidence in the world that observers can align without touching the source.

That event has now been defined. It has not yet been found.