The Perspectives of Non-Ideal Quantum Reference Frames

TL;DR

The paper develops a general framework to define and transform the perspective of quantum reference frames (QRFs), extending prior ideal-frame results to non-ideal QRFs with finite resources. It introduces two guiding principles and an incoherent G-twirl construction to ensure unitary QRF transformations, yielding a factorized Hilbert-space structure where relational observables act on minimal subsystems and an extra-particle sector encodes invariants. A central result guarantees a unitary jump into a QRF's perspective, with non-ideal frames exhibiting superselection of observed systems and back-reaction on the frame itself; these effects become explicit in Abelian symmetry cases, where detailed decompositions are given and exemplified by a qubit-qutrit setup. The findings show that finite resources fundamentally alter the QRF perspective beyond mere blurring of the ideal case, with measurable implications for relational measurements and entanglement structure, and they pave the way for exploring non-Abelian and gravitationally relevant symmetry groups. Overall, the work clarifies how to consistently describe physics relative to non-ideal QRFs and connects to broader perspective-neutral and operational approaches, suggesting avenues for experimental tests of QRF back-reaction and extensions to more complex symmetry groups.

Abstract

We define the perspective of any quantum reference frame (QRF) and construct reversible transformations between different perspectives. This extends the framework of [arXiv:2110.13199] to non-ideal QRFs with finite resources such as energy or angular momentum. We derive a QRF's perspective starting from two physically motivated principles, leading to an incoherent group averaging approach. The perspective of a non-ideal QRF deviates significantly from that of a more intuitive ideal frame with infinite resources: Firstly, systems described relative to the QRF appear superselected. Secondly, the structure of the QRF perspective attests that successive operations on a system relative to the QRF leads to back-reaction onto the QRF due to its non-ideality.

Figures (4)

• Figure 1: The relational POVM is obtained from the underlying POVM by compensating the latter with the orientation of $A$ relative to $L$ to remove all traces of orientation relative to $L$. More precisely, measuring $\{\hat{M}_{AS}^{|L}(i)\}_{i\in I}$ can be understood as first measuring the orientation $\{\hat{\gamma}_A^{|L}(g)\}_{g\in G}$ of $A$ relative to $L$, obtaining $g \in G$, followed by measuring the orientation-compensated POVM $\{\mathsf U_S(g)[\hat{m}_S^{|L}(i)]\}_{i\in I}$, obtaining $i \in I$, and finally forgetting $g$ while keeping $i$.
• Figure 2: Salecker-Wigner measurement of the relative position of $S$ with respect to $A$Salecker1958. The position QRF $A$ consists of the centre of mass degrees of freedom (depicted by the blue atom) of a device capable of emitting and detecting a light ray. This device can be used to measure the position of $S$ relative to $A$ by bouncing a light ray off $S$. As the light bounces off $S$ and arrives back at the box, the measurement of a time POVM $\{\hat{T}(t)\}_{t\in\mathbb R}$ on the clock determines the light travel time. This gives an estimate for the relative position of $S$ with respect to $A$. If $A$ and the clock are ideal, this measurement corresponds to the PVM measurement of $\hat{X}_{AS}^{|A}$ in example \ref{['exa:regular_position']}.
• Figure 3: Take example \ref{['exa:Galilei']}, additionally assuming rank-one orientation POVM elements $\hat{\gamma}^{|L}_A(g) = \left|{g}\right\rangle\!\left\langle{g}\right|$, $g = (a,v)$, given by minimal-uncertainty Gaussian states centred around $g$ in phase-space. If $\hat{\rho}_A = \frac{1}{\sqrt{2}} \bigl(\left|g\right\rangle + \left|{g'}\right\rangle\bigr) \times (\text{h.c.})$ is a superposition of two orientation POVM states then we observe two peaks of width $\propto 1/\sqrt{m}$ as well as wave-like interference between the peaks in the probability distribution $p(g)$Garmier2023. Figure adapted from Garmier2023, see appendix \ref{['app:Galilei']} for details.
• Figure 4: The set $\boldsymbol{\kappa}(r)$ as a function of total angular momentum $r$ for two example cases (a) and (b): a point at vertical position $r$ and horizontal position $z$ indicates that $z \in \boldsymbol{\kappa}(r)$. Simultaneous eigenspaces of $\mathcal{Z}^{S:A|A}$ are labelled by such sets and indicated as blue dashed blocks; the representation charges contained in the block can be read off the vertical axis. E.g., the central block in (a) spans three values of $r$, i.e. the corresponding simultaneous eigenspace of the centre carries these three values of angular momentum.

Theorems & Definitions (31)

• theorem 1
• proposition 1
• proposition 2
• proof
• proposition 3
• proposition 4
• lemma 1
• proof : Proof of lemma \ref{['lem:minimal_projectors']}
• proof : Proof of proposition \ref{['prop:algebra_decomposition_1']}
• proposition 5