Relative subsystems and quantum reference frame transformations

TL;DR

The work develops a local, principled framework for quantum reference frame (QRF) transformations by deriving them from standard quantum theory and an incoherent group-twirl over unimodular symmetry groups $G$. It introduces an invariant subsystem structure that includes a gauge part and a physically meaningful 'extra particle' to ensure unitary transformations between QRF perspectives, applicable to a broad class of groups including the centrally extended Galilei group. The formalism recasts QRF changes as refactorisations of the invariant subsystem, linking Alice's and Bob's viewpoints through a unitary map that is block-diagonal in charge sectors and valid for subsystems rather than the whole universe. In the centrally extended Galilei case, the regular representation corresponds to a two-particle QRF (one for position, one for velocity) with explicit generators and a mass-sector interpretation, and it reconciles with prior theories under specific limits. Overall, the framework clarifies the relativity of subsystems and entanglement under QRF transformations and points toward broader applications in relativistic quantum theory and quantum gravity.

Abstract

Recently there has been much effort in developing a quantum generalisation of reference frame transformations. Despite important progress, a complete understanding of their principles is still lacking. In particular, we argue that previous proposals could yield reversible transformations between arbitrary quantum reference frames only when applied to the whole universe. In contrast, here we derive quantum reference frame transformations from first principles, using only standard quantum theory. Our framework, naturally based on incoherent rather than coherent group averaging, yields reversible transformations that only depend on the reference frames and system of interest. We find more general transformations than those studied so far, which are valid only in a restricted subspace. Importantly, our framework contains additional degrees of freedom in the form of an "extra particle," which carries information about the quantum features of reference frame states. Our formalism is valid for a broad range of symmetry groups. We study the centrally extended Galilei group specifically, highlighting key differences from previous proposals.

Figures (3)

• Figure 1: In our setup, an observer, Alice, has only access to degrees of freedom that are invariant under the action of the group $G$. The latter is assumed defined relative to some external observer. As we show in Section \ref{['exttoint']}, the invariant degrees of freedom are independent of any external observer or reference frame. These invariant degrees of freedom include, in particular, degrees of freedom of the system defined relative to the reference frame $A$. The latter are described by operators that from an algebra, called $\sf{S \vert A}$. Importantly, Alice's apparatus, by means of which these degrees of freedom are accessed, are not part of the quantum system under consideration. They lie on the "other side" of Heisenberg's cut.
• Figure 2: The full invariant system can be decomposed in a way that is natural to $\sf{A}$ (vertical, orange "threads") and in a way that is natural to $\sf{B}$ (horizontal, green threads). A QRF is a preferred factorisation of the invariant system, and a QRF transformation is a change from one preferred factorisation to another. In this illustration, when 2 different subsystems overlap it means that their corresponding operators don't commute in general. In this way, when $\sf{A}$ refers to "the system," she is actually referring to the subsystem $\sf{A \vert B}$, which overlaps with $\sf{S \vert B}$ and $\sf{A \vert B}$ from the point of view of $\sf{B}$. Importantly, the inclusion of the subsystems $\overline{\sf{S B \vert A}}$ and $\overline{\sf{S A \vert B}}$ is essential to find a unitary relation between $\sf{A}$ and $\sf{B}$'s tensor product factorisations.
• Figure 3: Physical interpretation of the regular representation of the centrally extended Galilei Group. For a given mass sector $m$, the regular representation can be seen as a system of 2 particles. Here we depict the case where each particle has a mass $m/2$. In this interpretation, the left regular representation corresponds to the degrees of freedom of the centre of mass, $\sf{CM}$, of the 2-particle system. The right regular representation corresponds to the distance of any of the 2 particles to the centre of mass, or half their relative distance, $\sf{REL}$. Imagine that $\sf{A}$ describes an operation on $\sf{S}$ using the regular representation of the centrally extended Galillei Group as a reference. We can then ask how this operation "looks like" from the standard partition viewpoint. Roughly speaking, in this viewpoint, $\sf{A}$ uses one of the particles, $\sf{A}_{m_1}$, as a reference frame for position, and uses the other particle, $\sf{A}_{m_2}$ as a reference frame for velocity (see Eqs. (\ref{['2particles']})).