Perspective-neutral approach to quantum frame covariance for general symmetry groups

Abstract

In the absence of external relata, internal quantum reference frames (QRFs) appear widely in the literature on quantum gravity, gauge theories and quantum foundations. Here, we extend the perspectiveneutral approach to QRF covariance to general unimodular Lie groups. This is a framework that links internal QRF perspectives via a manifestly gauge-invariant Hilbert space in the form of "quantum coordinate transformations", and we clarify how it is a quantum extension of special covariance. We model the QRF orientations as coherent states which give rise to a covariant positive operator-valued measure, furnishing a consistent probability interpretation and encompassing non-ideal QRFs whose orientations are not perfectly distinguishable. We generalize the construction of relational observables, establish a variety of their algebraic properties and equip them with a transparent conditional probability interpretation. We import the distinction between gauge transformations and physical symmetries from gauge theories and identify the latter as QRF reorientations. The "quantum coordinate maps" into an internal QRF perspective are constructed via a conditioning on the QRF's orientation, generalizing the Page-Wootters formalism and a symmetry reduction procedure to general groups. We also find two types of QRF transformations: the gauge induced "quantum coordinate transformations" as passive changes of description, which we show are always unitary, and symmetry induced active changes of relational observables from one QRF to another. We then reveal novel physical effects: (i) QRFs whose orientations admit non-trivial isotropy groups can only resolve isotropy-group-invariant properties of other subsystems; (ii) when the QRF does not admit symmetries, its internal perspective Hilbert space is not fixed and "rotates" through the kinematical subsystem Hilbert space as the QRF changes orientation. We also invoke the symmetries to extend a recent observation on the quantum relativity of subsystems to general groups, which explains the QRF dependence of entanglement and other physical properties. Finally, we compare with other approaches to QRF covariance and illustrate our findings in various examples.

Entanglement/Asymmetry correspondence for internal quantum reference frames

“…We will abbreviate |ψ S|R := |ψ(e) S|R , and we will sometimes emphasize the dependence of this state on the choice of |e R by writing |ψ |e S|R . We have [47] |ψ(g…”

“…Recently, it has led to a surge of interest in the quantum foundations community on the behavior of quantum systems under transformations between such "internal" quantum reference frames [30][31][32][33][34][35][36][37][38][39]. Among other results, this has led to proposals for quantum formulations of Einstein's equivalence principle [40][41][42][43][44] and to insights into the relativity of the notion of subsystem [45][46][47].…”

“…It has been shown that several of these approaches can be unified and generalized in a "perspective-neutral" framework [47][48][49][50][51][52][53] for which the Page-Wootters mechanism [52][53][54][55][56] (PWM) is a well-known special case, see Fig. 1: the global system is in a state |ψ RS that is invariant under symmetry transformations, but conditioning on a subsystem R defines a state |ψ S|R of S relative to R that is asymmetric.…”

“…Demanding this on the level of pure states (not of density matrices) can be motivated by preserving entanglement with a purifying system, see [39] for details. We are interested in the quantum state of S conditional on R being "oriented in some direction" g ∈ G. To make precise sense of this intuition, we define a coherent state system [47,60] {|g R } g∈G by choosing some normalized state |e R , where e is the unit element of G, and setting |g R := U g |e R . It follows that U g |g R = |gg R .…”

“…We demand that |e R is chosen such that we obtain a resolution of the identity, G dg |g g| = c • 1 R with c > 0 some constant, where dg denotes the Haar measure on G (it follows that c = 1/d R , with d R = dim R). If g → U g is an irreducible representation (irrep), then this follows automatically from Schur's lemma; otherwise it imposes some conditions on |e R described in [47].…”