Quantum Reference Frames in Quantum Mechanics

• Quantum reference frames are quantum systems that replace classical frames by using symmetry group averaging to encode relational descriptions of physical states.
• They facilitate operational transformations between frames through relational measurements that reencode systems while introducing decoherence due to finite resource limits.
• This framework has significant implications for quantum gravity and quantum relativity, offering insights into intrinsic decoherence and the emergence of classicality.

Quantum reference frames (QRFs) are quantum systems that serve as the basis for describing the properties and states of other quantum systems, replacing the classical, external frames conventionally employed in quantum theory. In a fully quantum mechanical or relational picture, every frame—including the observer's laboratory or measurement apparatus—must be treated as a finite, physical system whose own quantum degrees of freedom participate in the global system’s description. Transformations between QRFs, and the physical consequences of such transformations, are central to the pursuit of a generalized relativity principle adequate for quantum physics and quantum gravity, and yield deep operational, mathematical, and interpretational implications.

1. Encoding States and Relational Descriptions

The foundational operation in the QRF paradigm is the relational encoding of physical states. Given a system $S$ and a reference frame $A$—both realized as quantum systems in finite-dimensional Hilbert spaces $\mathcal{H}_S$ and $\mathcal{H}_A$—the encoding of $S$ with respect to $A$ is accomplished by "twirling" over the symmetry group $G$ associated with the relevant physical degree of freedom (e.g., $U(1)$ for phase, $SU(2)$ for orientation): $$\mathcal{E}_{\rho_A}(\rho_S) = \mathcal{G}_{SA}(\rho_S \otimes \rho_A) ,$$ where the $G$-twirl is

$$\mathcal{G}(\rho) = \int d\mu(g) U(g) \rho U^\dagger(g) ,$$

with $U(g)$ a representation of $g \in G$ and $d\mu(g)$ the Haar measure. This averaging eliminates absolute (external) reference to any background, yielding a state that captures only the relation between $S$ and $A$. As a result, observables and states become explicitly relational, an essential prerequisite for background-independent theories and formulations where "absolute" structure (such as fixed space, time, or axes) is not physical.

2. Operational Procedure for Changing Quantum Reference Frames

Changing from an initial QRF $A$ to a new QRF $B$ (both quantum systems, and typically uncorrelated at the start) is an operationally well-defined process that generically entails coupling the two reference frames by a relational measurement—for example, measuring their relative phase or orientation by implementing a POVM constructed to be fully group-covariant. Using projectors of the form $$\Pi_{AB}^{(g,h)} = U_A(g) \otimes U_B(gh)|e_A\rangle \otimes |h_B\rangle$$, followed by integration over $G$ and tracing out $A$, the new encoded state of system $S$ with respect to frame $B$ is given by: $$\rho_S' = \mathcal{F}_S^{(A)} \circ \mathcal{U}_S(a^{-1})[\rho_S],$$ where:

• $\mathcal{U}_S(a^{-1})$ is a group transformation that "undoes" the initial orientation $a$ of $A$,
• $\mathcal{F}_S^{(A)}$ is a completely positive (CP) map determined by the convex mixture of group transformations weighted by the overlaps (fuzziness) of the finite QRF.

Explicitly,

$$\mathcal{F}_S^{(A)} = D_{s_A} \int d\mu(g)\, |\langle g \psi(e)_A|^2\, U_S(g^{-1}),$$

where $|\psi(e)_A\rangle$ is the fiducial “zero-orientation” state. This map $\mathcal{F}_S^{(A)}$—a mixture of group unitaries with a non-localized weight due to finite frame resources—is responsible for the central physical effect arising in QRF changes.

3. Decoherence Induced by Finite Quantum Reference Frames

A consequential and ineliminable feature of the change-of-QRF map is intrinsic decoherence: because finite-dimensional quantum frames cannot perfectly label a group’s continuous parameter, the overlap between reference frame states is broadened, not sharply peaked (unlike the ideal pointer basis limit). Therefore,

• If the Hilbert space dimension ("size") $s_A$ of the QRF $A$ is finite, $\mathcal{F}_S^{(A)}$ is not the identity but a convex mixture, and decoherence is induced on $S$.
• In the classical limit ($s_A \rightarrow \infty$), the overlaps approach delta functions, the CP map becomes the identity, and no decoherence arises.

This decoherence is not caused by the system’s interaction with an external environment but rather by the quantum limitations (resource finiteness) of the reference frame itself. The process is thus operationally intrinsic, and in certain quantum gravity proposals it is conjectured to be a model of fundamental (as opposed to environmentally-induced) decoherence at the Planck scale.

4. Explicit Examples: Phase and Orientation Reference Frames

Phase Reference (U(1))

For the group $U(1)$, describing phase, the QRF can be realized as a phase eigenstate,

$$|s; g\rangle = N_s^{-1/2} \sum_{k=0}^s e^{i k g} |k\rangle$$

with $N_s = s + 1$. The induced decoherence map is

$$\mathcal{F}_S^{(A)} = \frac{1}{s_A + 1} \int \frac{dg}{2\pi} \frac{1 - \cos[(s_A + 1) g]}{1 - \cos g} U_S(-g).$$

When coherent states are used, an analogous decoherence map results, determined by the spread associated with the finite coherent amplitude.

Orientation Reference (SU(2))

For spatial orientation (SO(3), or more generally SU(2)), the QRF is constructed of appropriately superposed spin states (up to cutoff $s$), and the decoherence map reads,

$$\mathcal{F}_S^{(A)} = \text{const} \int d\mu(g)\, w(g) U_S(g^{-1})$$

with $w(g)$ explicitly constructed from the overlap of rotated and unrotated QRF states, involving SU(2) characters and the allowed representations up to $j \leq s_A$. In the case of axis-indicating frames (SU(2)/U(1) coherent states), the decoherence carries an additional "dephasing" due to the stabilizer subgroup.

These explicit constructions confirm that decoherence is directly set by the frame’s resource limit—both qualitatively and quantitatively.

5. Mathematical Structure and Equivalence to Recovery-Reencoding

The "change of frame" process is mathematically equivalent to a recovery-reencoding protocol: first, recover the original system state $\rho_S$ in an idealized background frame (via a recovery map $\mathcal{R}$), then reencode it relationally with respect to the new QRF $B$. The recovery map, in turn, incorporates the decohering CP map $\mathcal{F}_S^{(A)}$ and the corrective group unitary: $$\mathcal{R} \circ \mathcal{E}_{\psi(a)}(\rho_S) \equiv \mathcal{F}_S^{(A)} \circ \mathcal{U}_S(a^{-1})[\rho_S].$$ This operational equivalence demonstrates that the act of switching between quantum frames is not merely a basis transformation but involves physically modifying the encoded state—and quantifies precisely how.

6. Implications for a Quantum Relativity Principle

When all degrees of freedom, including reference frames, are treated quantum mechanically, the act of "changing observer" is described not by a unitary alone but by a CP (decohering) map. This stands in contrast to the classical relativity principle, where coordinate changes are perfect, invertible, and do not affect the state’s purity. In the quantum domain, the unavoidable "fuzziness" of finite frames means that transformations between observers (each with their own QRF) inevitably introduce noise on the systems being described. This feature must be incorporated in any attempt to formulate a fully quantum relativity principle, and implies that phenomena such as relational decoherence or intrinsic noise are generic, even in the absence of external environments. The analysis thus supports the operational basis for proposals of intrinsic gravitational decoherence and illustrates the need to account for quantum limitations of reference systems in any fundamental physical theory.

7. Summary Table: Decoherence Maps for U(1) and SU(2) Reference Frames

Symmetry Group	Reference State	Decoherence Map $\mathcal{F}_S^{(A)}$
U(1)	Phase eigenstate $|s;g\rangle$	$$\frac{1}{s_A+1} \int \frac{dg}{2\pi} \frac{1-\cos[(s_A+1)g]}{1-\cos g} U_S(-g)$$
U(1)	Coherent state	Analogous; weight determined by finite amplitude $s_A$
SU(2)	Cartesian frame, eq. (7)/(8)	Weighted average over rotations given by overlap and SU(2) characters (eq. (9))

These decoherence maps are determined by the “spread” in overlap of quantum states associated to different orientations. As $s_A \rightarrow \infty$, the weights become sharply peaked and the decoherence vanishes.

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A quantum reference frame is thus a specific quantum system state in a Hilbert space, encoding an orientation (phase, axis, etc.) according to the action of a symmetry group. Changing such a frame is a physical operation—implemented by a relational measurement—which, due to the finite resources available to any quantum system, induces necessarily a decohering channel on the state of the system being described. This phenomenon not only has operational significance for tasks involving limited reference resources (quantum communication, clock synchronization, quantum thermodynamics) but also has foundational implications for understanding the emergence of classicality, the structure of quantum gravity, and the possible existence of fundamental decoherence mechanisms driven by the quantum nature of frames themselves (Palmer et al., 2013).