Quantum Reference Frame Transformations

• Quantum reference frame transformations are defined as unitary maps or quantum channels that shift the description of a system from one quantum reference frame to another, accounting for inherent uncertainty and superposition.
• They reveal that observer-dependent properties such as entanglement and coherence can be redistributed, preserving overall quantum invariants despite frame shifts.
• Multiple frameworks—perspective-neutral, operational, and extra-particle—offer complementary methods to implement these transformations, balancing ideal reversibility with real-world mixed dynamics.

Quantum reference frame transformations are fundamental operations in quantum theory that translate the description of physical systems between different quantum reference frames (QRFs), which themselves are quantum systems subject to quantum mechanics. Unlike classical reference frame transformations—typically described by group actions with sharp parameters—quantum reference frame transformations account for quantum uncertainty, superposition, and entanglement of the frame itself, leading to nontrivial observer-dependence of physical properties such as entanglement and coherence. Multiple distinct theoretical frameworks exist for defining and implementing these transformations, each reflecting different assumptions about global symmetries, operational accessibility, and the nature of physical observables.

1. Definitions and Core Concepts

A quantum reference frame (QRF) is a physical quantum system that serves as a basis relative to which the states and observables of other systems are described. A quantum reference frame transformation is then a map—often implemented as a unitary or quantum channel—that changes the description of a given system from one QRF to another. Unlike classical transformations, this involves extending transformation parameters (e.g., rotation angles, translations) from real numbers to operators acting on the Hilbert space of the QRF, thereby capturing nonclassical phenomena such as "superpositions of transformations".

Key features of quantum reference frame transformations:

• Relational Viewpoint: Transformations are defined solely in terms of relations between quantum systems, eschewing any fixed background or external "absolute" frame (Giacomini et al., 2017, Vanrietvelde et al., 2018, Hamette et al., 2020, Zelezny, 2021).
• Frame-Dependent Resources: Properties such as entanglement and coherence are observer-dependent and may be interconverted under a change of QRF (Giacomini et al., 2017, Cepollaro et al., 27 Jun 2024).
• Non-Classical Group Structure: The set of QRF transformations often forms a Lie group or quantum group, but when parameters are noncommutative or the transformation is mixed (not pure), the structure may only be a semigroup (Ballesteros et al., 2020, Ballesteros et al., 1 Apr 2025, Fiore et al., 8 Jul 2025).

2. Mathematical Frameworks for Quantum Reference Frame Transformations

Several mathematically rigorous frameworks have been proposed, each motivated by distinct physical or operational considerations.

2.1 Perspective-Neutral and Relational Approaches

The "perspective-neutral" or Dirac quantization approach encodes all possible reference frame perspectives simultaneously in a larger Hilbert space, constrained by symmetry (e.g., total momentum constraint for translations). Internal (operational) perspectives are obtained by a process of gauge fixing or symmetry reduction, which both in classical and quantum theory corresponds to selecting coordinates (Darboux variables) on the constraint surface. Transformations between QRFs are then implemented by canonical or unitary maps (e.g.,

$$\mathcal{S}_{A \to C} = \mathcal{P}_{CA} e^{i \hat{q}_C \hat{p}_B}$$

in the translation case) (Vanrietvelde et al., 2018, Zelezny, 2021).

2.2 Operational and Invariant Subsystem Approaches

Operationally, one defines equivalence classes of states and observables according to what is accessible via symmetry-invariant (or "relative") observables (Carette et al., 2023). A QRF is described by a covariant POVM, and the transformation between QRFs is a map between operational equivalence classes of states. Transformations are typically invertible exactly when both QRFs admit highly localized states with respect to their frame observables; otherwise, information may be lost (irreversibility).

2.3 Extra-Particle and Algebraic Factorization

Certain frameworks highlight the necessity of enlarging the Hilbert space to include "extra particle" degrees of freedom—additional subsystems or algebraic factors that encode global charge, symmetry sector, or non-accessible information (Castro-Ruiz et al., 2021). Changing QRFs then involves a refactorization of the Hilbert space and transformation operators that act both on the system and the extra degrees of freedom, ensuring unitarity in subsystems beyond the original frame and system.

3. Unitary, Irreversible, and Mixed Transformations

The majority of theoretical developments focus on reversible (unitary) QRF transformations under idealized conditions, characterized by operators that implement symmetry transformations either as left/right regular representations (e.g., for group G on

$L^2(G)$

) or by canonical transformations in phase space (Hamette et al., 2020). A key result is that unitarity is achieved if and only if the QRF's representation space supports both the left and right regular actions of the group in question.

However, in realistic settings—where reference frames or transformations are imperfect or performed with uncertainty—the transformation may be inherently mixed (represented by a probability distribution over the group) and yields a completely positive (but not invertible) map: a semigroup structure arises rather than a group. For example, a mixed translation with density

$\tilde{\rho}(a)$

yields on state $|\psi\rangle$:

$$\rho_S' = \int \tilde{\rho}(a) \, |\psi(x+a)\rangle\langle\psi(x+a)| \, da,$$

which may increase the mixture of the system even if it was pure in the original frame (Fiore et al., 8 Jul 2025). Mixed time translations are closely linked to thermalization, with a direct connection between uncertainties in time and the appearance of a Boltzmann distribution in the energy basis.

4. Frame-Dependent Quantum Resources: Entanglement, Coherence, and Nonlocality

Quantum reference frame transformations generically change the way quantum resources are partitioned. For example:

• Changing frames can convert subsystem coherence into bipartite entanglement and vice versa, but certain resource sums remain invariant under ideal QRF transformations, e.g.,

$$\mathcal{C}_e^{(C)} + \mathcal{E}_e^{(C)} = \mathcal{C}_e^{(A)} + \mathcal{E}_e^{(A)},$$

where

$\mathcal{C}_e$

is the entropy of coherence and

$\mathcal{E}_e$

the entanglement entropy (Cepollaro et al., 27 Jun 2024).

• Bell nonlocality (the violation of Bell inequalities) persists in all QRFs, but the quantum resource responsible can shift from state entanglement in one frame to nonlocal observables (i.e., entangled measurement bases) in another (Cepollaro et al., 27 Jun 2024).
• In Wigner’s friend-type scenarios, states that are separable in one QRF may appear entangled in another, but the mathematics of transformation ensures all physical predictions are consistent (Pienaar, 2016, Giacomini et al., 2017, Hamette et al., 2020).

5. Symmetry Groups, Quantum Deformations, and Groupoid Structures

Transformations between quantum reference frames naturally generalize group-theoretic structures:

• For configurations ideally isomorphic to groups (torsors), unitarity and reversibility are ensured.
• For coarse-grained or imperfect frames, one must average or discard some information, typically leading to irreversible (non-invertible) transformations (Hamette et al., 2020).
• Quantum group techniques demonstrate that the full set of QRF transformations may require a quantum deformation of the symmetry group: For the Galilei group, noncommutative group parameters (with deformation parameter

$\alpha$

proportional to inverse frame mass) encode the quantum nature of the reference frame (Ballesteros et al., 1 Apr 2025).

• The Lie algebra of these transformations is richer than that of the classical Galilei group; in the QRF context it typically includes not only translation and boost generators but also dilations and additional rescaling operators, reflecting the nontrivial structure of quantum phase space (Ballesteros et al., 2020).

6. Relativistic and Non-Abelian QRF Transformations

Recent advances have extended QRF transformations to settings with relativistic and non-Abelian symmetry:

• Quantum Lorentz transformations are constructed as unitary operators acting on spacetime wavefunctions, enabling "superpositions of Lorentz boosts." This leads to physically observable consequences such as superpositions of time dilations and length contractions (Apadula et al., 2022).
• In systems with spin degrees of freedom, rotational QRFs are implemented by quantizing the classical rotation group SO(3) with operators replacing Euler angles, allowing for superpositions of orientations and observer-dependent entanglement structure (Mikusch et al., 2021).
• Quantum reference frame transformations have been operationally defined for arbitrary finite and locally compact groups (Hamette et al., 2020, Carette et al., 2023).

7. Framework Comparisons and Operational Meaning

Three major frameworks offer complementary interpretations and mathematical prescriptions for QRF transformations, as illustrated by simple discrete models (e.g., three-qubit systems with

$\mathbb{Z}_2$

symmetry) (Castro-Ruiz et al., 13 Aug 2025):

• Perspective-Neutral: All reference frames are treated equally; transformations are invertible and global state information is preserved.
• Extra-Particle: An additional degree of freedom (encoding global "charge" or symmetry sector) is introduced; transformations are invertible with access to the extra label.
• Operational: Only locally accessible observables are used, leading to transformations defined up to operational equivalence; transformations may not be invertible, and different global states are grouped into equivalence classes, often represented as mixed states.

The operational approach is particularly relevant when experimental access to global state information is restricted, emphasizing the "framed" nature of observables and reinforcing the relational ethos of quantum mechanics.

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In summary, quantum reference frame transformations provide a unifying language for describing how physical properties, information-theoretic resources, and dynamical laws manifest in the absence of absolute frames. Their mathematical structure—ranging from unitary group actions and quantum group deformations to operationally defined semigroups—captures the interplay between quantum uncertainty, global symmetries, and the concrete outcomes of physical measurements. The field is deeply relevant for quantum information theory, foundational work in quantum gravity, and the operational interpretation of quantum mechanics in laboratory settings.