The group structure of dynamical transformations between quantum reference frames

Ballesteros, Ángel; Giacomini, Flaminia; Gubitosi, Giulia

doi:10.22331/q-2021-06-08-470

Cited by 32 publications

(28 citation statements)

References 39 publications

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“…• We elucidate why the perspective-neutral approach to quantum frame covariance, further developed here, is not in general equivalent to the purely perspective-dependent approaches [48,[65][66][67][68]71,72]. The latter start with a fixed subsystem Hilbert space in a given internal quantum frame perspective that is not in general compatible with gauge-invariance under the symmetry group.…”

Section: Summary Of the Resultsmentioning

confidence: 98%

“…The isomorphism maps the state |φ(e) R ∈ H i ⊗ H j , which is invariant under ρ i (g)⊗ρ j (g), to an element M φ(e) ∈ Hom( Hj , H i ) which is necessarily invariant under the action of g. This shows that the G-equivariant subspace Hom G ( Hj , H i ) of Hom( Hj , H i ) is non trivial, where the G-equivariant subspace is the space of all maps M ∈ Hom( Hj , H i ) such that ρ i (g)M = M ρj (g) for all g ∈ G. Due to the irreducibility of the representations involved, Schur's lemma entails that H j ∼ = Hi . For an irrep H i ⊗ H j ∼ = H i ⊗ Hi the specific form of |φ(e) i follows directly from the fact that M φ(e) ∈ Hom( Hj , H i ) ∼ = Hom(H i , H i ) is proportional to the identity matrix and applying the explicit isomorphism Hom(H i , H i ) → H i ⊗ Hi yields a state |φ(e) i of the form given in Equation (68). The extension to reducible representations requires Example 1.…”

Section: Left-right Systems Of Coherent Statesmentioning

confidence: 99%

“…The changes of quantum reference frames introduced in [48,65,67,68,72] did not make use of a perspectiveneutral structure (in particular, did not invoke a gauge symmetry principle), but rather mapped directly between what are here reduced descriptions, i.e. frame perspectives.…”

Section: Perspective-neutral Vs Purely Perspective-dependent Approachesmentioning

confidence: 99%

See 2 more Smart Citations

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

Hamette¹,

Galley²,

Hoehn³

et al. 2021

Preprint

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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.

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Section: Summary Of the Resultsmentioning

confidence: 98%

Section: Left-right Systems Of Coherent Statesmentioning

confidence: 99%

See 1 more Smart Citation

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

Hamette¹,

Galley²,

Hoehn³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…If spacetime is defined in terms of such quantum reference frames, then it inherits their quantum properties. While this statement still needs to be made more precise, and work on this is in progress, recent preliminary results showed that the associated symmetry group defining transformations between quantum reference frames is a more general group than that of Galilei symmetries [56].…”

Section: Discussionmentioning

confidence: 99%

Double Quantization

Gubitosi,

Lizzi,

Relancio

et al. 2021

Preprint

Self Cite

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In a quantum gravity theory, it is expected that the classical notion of spacetime disappears, leading to a quantum structure with new properties. A possible way to take into account these quantum effects is through a noncommutativity of spacetime coordinates. In the literature, there is not a clear way to describe at the same time a noncommutativity of spacetime and the phasespace noncommutativity of quantum mechanics. In this paper we address this issue by constructing a Drinfel'd twist in phase space which deals with both quantizations. This method can be applied to a noncommutativity which involves only space, leaving time aside. We apply our construction to the so-called λ-Minkwoski and R 3 λ noncommutative spaces.

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“…[29,30] extend the formalism to the special relativistic regime, applying it to concrete problems in Relativistic Quantum Information. Other works [31,32,33] have focussed on the group properties of QRFs both in discrete and in continuous-variable systems. A time version of QRFs, also related to the Page-Wootters mechanism [34,35], has been introduced in Ref.…”

Section: Introductionmentioning

confidence: 99%

Spacetime Quantum Reference Frames and superpositions of proper times

Giacomini

2021

Quantum

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In general relativity, the description of spacetime relies on idealised rods and clocks, which identify a reference frame. In any concrete scenario, reference frames are associated to physical systems, which are ultimately quantum in nature. A relativistic description of the laws of physics hence needs to take into account such quantum reference frames (QRFs), through which spacetime can be given an operational meaning. Here, we introduce the notion of a spacetime quantum reference frame, associated to a quantum particle in spacetime. Such formulation has the advantage of treating space and time on equal footing, and of allowing us to describe the dynamical evolution of a set of quantum systems from the perspective of another quantum system, where the parameter in which the rest of the physical systems evolves coincides with the proper time of the particle taken as the QRF. Crucially, the proper times in two different QRFs are not related by a standard transformation, but they might be in a quantum superposition one with respect to the other.Concretely, we consider a system of N relativistic quantum particles in a weak gravitational field, and introduce a timeless formulation in which the global state of the N particles appears "frozen", but the dynamical evolution is recovered in terms of relational quantities. The position and momentum Hilbert space of the particles is used to fix the QRF via a transformation to the local frame of the particle such that the metric is locally inertial at the origin of the QRF. The internal Hilbert space corresponds to the clock space, which keeps the proper time in the local frame of the particle. Thanks to this fully relational construction we show how the remaining particles evolve dynamically in the relational variables from the perspective of the QRF. The construction proposed here includes the Page-Wootters mechanism for non interacting clocks when the external degrees of freedom are neglected. Finally, we find that a quantum superposition of gravitational redshifts and a quantum superposition of special-relativistic time dilations can be observed in the QRF.

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The group structure of dynamical transformations between quantum reference frames

Cited by 32 publications

References 39 publications

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

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

Double Quantization

Spacetime Quantum Reference Frames and superpositions of proper times

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