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Quantum reference frames for general symmetry groups

Anne-Catherine de la Hamette, Thomas D. Galley

arXiv:2004.14292v3quant-ph

TL;DR

Relational quantum theory needs an account of how descriptions change between quantum reference frames. The paper develops group-based classical and quantum transformations and proves when the reversible quantum transformation is unitary. It extends the framework to imperfect frames and applies explicit frame changes to Wigner’s friend.

  • Problem

    Relational quantum theory describes systems relative to other systems, making it necessary to specify how to change between reference-system descriptions.

    Source: §2 · p. 2, §1 · p. 2

  • Method

    The paper develops a group-based relational formalism for classical and quantum frame changes, including a truncation approach for imperfect frames and an explicit transformation to the friend’s perspective.

    Source: §1 · p. 2, §4.5.1 · p. 9, §9 · p. 16

  • Results

    The framework extends reversible changes beyond translation and Euclidean groups to arbitrary groups, with unitary changes requiring regular representations; quantum entanglement and superposition depend on the reference frame.

    Source: §9 · p. 16

  • Takeaways & Limitations

    In Wigner’s friend, the explicit change of frame yields the relational conclusion that Wigner can say only that the friend is correlated with the measurement outcome.

    Source: §9 · p. 16

    The imperfect-frame treatment covers group-subgroup pairs with a normal subgroup; a full account would need to address non-normal subgroups.

    Source: §8.2.2 · p. 16

Abstract

from arXiv · show

A fully relational quantum theory necessarily requires an account of changes of quantum reference frames, where quantum reference frames are quantum systems relative to which other systems are described. By introducing a relational formalism which identifies coordinate systems with elements of a symmetry group $G$, we define a general operator for reversibly changing between quantum reference frames associated to a group $G$. This generalises the known operator for translations and boosts to arbitrary finite and locally compact groups, including non-Abelian groups. We show under which conditions one can uniquely assign coordinate choices to physical systems (to form reference frames) and how to reversibly transform between them, providing transformations between coordinate systems which are `in a superposition' of other coordinate systems. We obtain the change of quantum reference frame from the principles of relational physics and of coherent change of reference frame. We prove a theorem stating that the change of quantum reference frame consistent with these principles is unitary if and only if the reference systems carry the left and right regular representations of $G$. We also define irreversible changes of reference frame for classical and quantum systems in the case where the symmetry group $G$ is a semi-direct product $G = N \rtimes P$ or a direct product $G = N \times P$, providing multiple examples of both reversible and irreversible changes of quantum reference system along the way. Finally, we apply the relational formalism and changes of reference frame developed in this work to the Wigner's friend scenario, finding similar conclusions to those in relational quantum mechanics using an explicit change of reference frame as opposed to indirect reasoning using measurement operators.

1 Introduction

The paper motivates changes of quantum reference frame as necessary for relational quantum theory, then generalizes relational approaches from translations and rotations to arbitrary finite and locally compact groups.

1 Introduction

  • Quantum states are described relative to measurement devices, which must themselves be treated quantum mechanically if quantum mechanics applies at all scales.Such systems are called quantum reference frames.Source: §1 · p. 1
  • The work develops an account of changing quantum reference frames within relational quantum mechanics, where physically meaningful quantities are relative to a chosen system.The need for such changes is motivated by the role of reference-frame changes in relativity.Source: §1 · p. 1
  • The formalism emphasizes relations between subsystem-based descriptions and extends prior translation- and rotation-group approaches to arbitrary finite and locally compact groups, including non-Abelian groups.The paper outlines classical and quantum transformations, irreversible changes for product and semidirect-product groups, and an application to Wigner’s friend.Source: §1 · pp. 1–2

2 Relational approach to quantum theory

The paper describes states relative to physical systems and asks how to transform between those descriptions, illustrating the relational and coherent-change principles with classical and quantum examples.

2 Relational approach to quantum theory

  • The notation |ψ⟩^A_B denotes the state of B relative to A, with A assigned the identity or default zero-state relative to itself.The description uses no external abstract reference frame or absolute observer-dependent values.Source: §2 · p. 2
  • A central question is how to change from the state of B relative to A to the state of A relative to B.The section introduces simple classical-bit and qubit examples to illustrate the transformation.Source: §2 · p. 2
  • The quantum example assumes coherent changes of reference frame, applying the superposition principle and producing an entangled state of A and C.The examples use the two guiding principles of relational physics and coherent change of reference system.Source: §2 · p. 3
  • A change from A to B is a change of description from A at the origin to B at the origin, not a claim about what either system perceives.The paper uses “viewpoint” and “perspective” to mean “relative to.”Source: §2 · p. 3

3 Classical changes of reference frames associated to symmetry groups

Classical reference-frame changes are formulated using group elements as relative coordinates; reversible changes rely on a unique group transformation between configurations and are implemented through the right regular action.

3.1 Reference systems, coordinate choices and changes of reference frame

  • A reference frame combines a physical reference system with coordinates placing it at the origin; coordinate systems need not generally be associated with physical systems.The section considers classical systems whose configuration space is a group, before treating cases where it differs from the group.Source: §3.1 · p. 3
  • For three particles on a line, relative coordinates uniquely identify translations between systems, and changing from A to B applies the right regular action associated with their relative displacement.The configuration space and translation group are equivalent; the paper identifies this equivalence as essential to a well-defined reversible quantum change of frame.Source: §3.1 · p. 3

3.2 General treatment of reversible changes of reference frame

  • A reversible group-based framework requires a free and transitive action: a unique group element relates every pair of configurations, so X is isomorphic to G.The paper calls this structure a G-torsor and assumes G is locally compact.Source: §3.2 · p. 4
  • An active transformation changes the object and acts on relative locations from the left, whereas a passive transformation changes the observer and acts from the right.The two regular actions commute; for a change between relative states, the right regular action is required and the left action cannot achieve it unless G is Abelian.Source: §3.2 · pp. 4–5
  • Each system’s relative state records the group transformations from that system to the others, with the system itself assigned the identity.The figure illustrates composition of relative states, including g0_2 g1_0 = g1_2.Source: §3.2 · p. 5, Figure 1 · p. 5

4 Quantum reference frames associated to symmetry groups

This section develops changes of quantum reference frame for systems associated with a symmetry group, extending the translation-group construction to general groups. It derives the transformations from relational and coherent-change principles and identifies when they are unitary.

4 Quantum reference frames associated to symmetry groups

  • The framework defines quantum reference-frame changes for n identical L2(G) systems and extends the known translation-group construction beyond one-parameter subgroups of the Galilean group.For finite groups, L2(G) is replaced by C[G], the space freely generated by group elements.Source: §4 · p. 5, §4.1 · p. 5

4.2 The example of L2(R) ⊗L2(R) ⊗L2(R)

  • For three particles on a line, changing to a superposed reference system translates by each possible reference position, entangling the other systems; the operator matches.The state of the former reference system is mapped to the inverse of the new reference system's old group element, while the remaining system is shifted accordingly.Source: §4.2 · pp. 5–6, Figure 2 · p. 6

4.3 n identical systems L2(G)

  • Relational physics takes each state to describe the other systems relative to one system, while coherent change requires superpositions to transform linearly.For identical L2(G) systems, the resulting change operator is stated to be unitary.Source: §4.3 · pp. 6–7

4.3.1 Change of reference frame for observables

  • The transformation also changes observables: an observable described from system 0 is conjugated by the frame-change operators to obtain its description from system i.The passage gives the operator relation Zi = U 0→i Z0 U i→0.Source: §4.3.1 · p. 7

4.3.2 L2(U(1)) ⊗L2(U(1)) ⊗L2(U(1)

  • For three particles on a circle, changing from A to B inverts B's old group element, shifts C, and swaps the labels of A and B.Figure 3 identifies example basis states for L2(U(1)); Figure 4 depicts three particles on a circle.Source: §4.3.2 · p. 7, Figure 3 · p. 7, Figure 4 · p. 7

4.4 Unitarity of change of reference frame operator

  • The change is unitary if and only if the left and right regular representations act on states forming an orthonormal basis of the Hilbert space or a subspace.The theorem concerns operators obeying the principle of coherent change of reference system.Source: §4.4 · p. 8

4.5 m L2(G) systems describing n−m systems

  • For m L2(G) reference systems describing n−m other systems, the construction requires an injection of group elements into states of the other systems; a qubit representation of SU(2) lacks such an injection.The section then treats two L2(U(1)) systems describing a qubit that carries a U(1) representation and admits an injection.Source: §4.5 · p. 9, §4.5.1 · p. 9

4.6 Changes of reference frame for arbitrary identical systems

  • For arbitrary identical systems, a group is available for finite configuration spaces and countable sets, while uncountable cases depend on the axiom of choice; added manifold structure may prevent a manifold-isomorphic group.Source: §4.6 · p. 9

5 Irreversible changes of classical reference frame

Coarse-grained reference systems cannot distinguish every element of a symmetry group, so changing frames can become irreversible. The section shows how coordinate choices can be made well-defined, gives truncation examples, and identifies a limitation of truncation for relative states.

5.1 E+(3) ∼= R3 ⋊SO(3)

  • A particle at a point in R3 does not determine a unique E+(3) coordinate system, because rotations about that point also assign it the origin.Solid bodies with orientation instead have configuration space E+(3), allowing a unique coordinate assignment to each state.Source: §5.1 · p. 10, §5.1.1 · p. 10

5.1.2 Representative element of each equivalence class

  • Choosing one representative coordinate system per point can work if the representatives share an orientation, making their transformation group T(R3).More generally, the transformations relating the chosen representatives must form a group closed under its action.Source: Figure 6 · p. 10, §5.1.1 · p. 10, §5.1.2 · p. 10

5.2 G = N ⋊P and G = N × P: truncation

  • When multiple states of a G reference system map to the same N state, changing reference frame can be irreversible.The construction considers G = N ⋊ P or G = N × P, with a unique decomposition g = np.Source: §5.2 · p. 11
  • For modular truncation of R to Z, positions sharing a truncated label are indistinguishable to a reference system with configuration space Z.The paper describes this as coarse-graining from finer to coarser resolution.Source: §5.2 · p. 11
  • The R3-to-R2 truncation identifies reference frames along one dimension and can be understood as projecting points onto the x-y plane.The example identifies frames along the z-axis.Source: §5.2 · p. 12

5.3 Inconsistency using the truncation method for G = N ⋊P

  • Truncating before changing reference systems can yield a different relative state than changing first and then truncating, so the method may depend on the chosen state commitment and interpretation.The theorem states that the two procedures agree for systems with configuration space N, but not always for systems with configuration space G; averaging or invariants are alternatives.Source: §5.3 · p. 12

6 Irreversible changes of quantum reference frame

The paper extends irreversible frame changes to quantum systems for G = N ⋊ P or G = N × P. In this setting, changing perspective can entangle a superposition, but truncation can remove that entanglement when the positions fall within one resolution interval.

6 Irreversible changes of quantum reference frame

  • Irreversible quantum frame changes are defined for G = N ⋊ P and G = N × P.The construction extends the classical irreversible change by applying coherent change of reference frame.Source: §6 · p. 12

6.1 G = N ⋊P and G = N × P

  • Changing to a reference frame in which a system is in a superposition can make the state entangled relative to that frame.The section describes a truncation map from L2(G) to L2(N) and the corresponding change-of-frame operator.Source: §6.1 · pp. 12–13
  • The entanglement vanishes when the superposed positions lie in the same interval of a reference system with configuration space LZ.The paper observes entanglement only when the position uncertainty exceeds that system’s resolution L.Source: §6.1 · p. 13
  • The R3-to-R2 quantum example implements a projection from a three-dimensional configuration space onto the x-y plane.The described change is from a perspective with configuration space R3 to one with R2.Source: §6.1 · p. 13

7 Wigner’s friend experiment

The Wigner’s friend scenario contrasts the friend’s outcome-specific state with Wigner’s description of the friend and system. Applying the explicit change of reference frame gives Wigner’s inference about the friend’s assignments, without supplying the outcome the friend actually observed.

7 Wigner’s friend experiment

  • The friend’s post-measurement state assigns the system a definite basis state, whereas Wigner describes the friend and system as entangled.These are states of different systems, S and S+F, so the paper notes that they are not a logical contradiction within quantum theory.Source: §7 · p. 13
  • Changing from Wigner’s perspective gives the state Wigner infers the friend would assign, not the friend’s actual outcome.A unitary change of reference frame cannot add information about which outcome the friend observed.Source: §7 · p. 14
  • The authors conclude that Wigner’s inference follows from the explicit change-of-frame operator, rather than assuming consistency condition C.Their conclusion matches the correlated-state conclusion in Rovelli’s relational treatment, reached here by an explicit frame change rather than measurement-operator reasoning.Source: §7 · p. 14

8 Discussion

The discussion situates the explicitly relational, group-theoretic approach among prior quantum-reference-frame formalisms and applies it to relational quantum mechanics. It also highlights observer-dependent symmetry and limits the imperfect-reference-frame treatment to particular group extensions.

8.1 Related work

  • The paper’s group-theoretic relational treatment differs from approaches that derive relational states from an external description, and whether that external-to-relational account is consistent is not proven here.The authors note that extending a perspective-neutral framework to their general cases could be future work.Source: §8.1 · pp. 14–15
  • The link between this approach and quantum reference frames studied as resources for measurement, communication, and thermodynamic exchanges is not fully clear to the authors.They note that perfect reference frames of the form L2(G) also feature in their treatment, while the link between imperfect frames and their truncation approach remains to be worked out.Source: §8.1 · p. 15

8.2.1 Relational quantum mechanics

  • The formalism reaches the same conclusion for Wigner’s friend as Rovelli’s relational treatment: Wigner can say only that the friend is correlated with the measurement outcome.It obtains the friend’s perspective through an explicit change of reference frame rather than indirect reasoning with measurement operators.Source: §8.2.1 · p. 15

8.2.2 Observer dependence of the symmetry group

  • Only systems carrying a regular representation of G can describe other systems as transforming under G; two qubits describing a third use Z2 rather than full SU(2).Although a qubit carries an SU(2) representation, two qubits can carry only the regular representation of Z2 in this describing role.Source: §8.2.2 · pp. 15–16
  • The imperfect-reference-frame treatment covers group–subgroup pairs (G, N) when N is normal, using extensions G = N ⋊ P or G = N × P.A full account would require relaxing the normality assumption and investigating whether meaningful changes of reference frame can then be defined.Source: §8.2.2 · p. 16

9 Conclusion

The conclusion presents a relational formalism for describing states from different systems’ perspectives and extends changes of reference frame to arbitrary groups. It also discusses imperfect frames, Wigner’s friend, and the coordinate-system structure used in the formalism.

9 Conclusion

  • The formalism extends relational quantum-reference-frame changes from translations and the three-dimensional Euclidean group to arbitrary groups, including finite and non-Abelian groups.Reversible changes require the quantum reference systems to have Hilbert space L2(G).Source: §9 · p. 16
  • The paper treats imperfect reference frames through truncation, while extending the treatment beyond pairs (G, N) with N normal remains future work.The authors also identify rigorous links between this group-theoretic relational approach and quantum-information approaches as a direction to pursue.Source: §9 · p. 16
  • The Wigner’s friend analysis concludes that Wigner can say only that the friend is correlated with the measurement outcome, deriving the friend’s perspective by an explicit reference-frame change.This matches Rovelli’s relational conclusion and differs from indirect reasoning using measurement operators.Source: §9 · p. 16

A.1 Coordinate systems and coordinate charts

  • A coordinate system is an isomorphism from the object of interest to a known object used to describe it, whereas a coordinate chart maps in the opposite direction and need not be an isomorphism.The paper considers coordinate systems of a manifold, not coordinate charts.Source: §A.1 · p. 18

A.2 Coordinate systems on G-torsors

  • For a space X isomorphic to G, each group element determines a coordinate system by selecting an origin, so the set of coordinate choices is also G.The passage describes X as a G-torsor, which has the group structure without a distinguished identity.Source: §A.2 · p. 18

A.3 Reference frames

  • A reference frame pairs a coordinate system with a physical system whose configuration uniquely determines that coordinate system.For X isomorphic to G, the physical system must have configuration space G.Source: §A.3 · p. 18

B.1 Proof of Lemma 1

The appendix establishes unitarity for the reference-frame change operator and describes algebraic steps used in the proof. It also states a composition identity for successive changes of frame.

B.1 Proof of Lemma 1

  • The change-of-reference-frame operator U i→j is shown to be unitary when i and j have state space L2(G) and the other systems carry right regular representations.The proof passage states these system conditions and asserts the operator’s unitarity.Source: §B.1 · p. 18

B.2 Proof of Lemma 2

  • The proof moves a SWAP operator by relabelling partitions and uses equality of the integrals over g0i and gi0.The passage also invokes U† in the subsequent derivation.Source: §B.2 · p. 19

B.3 Proof of Lemma 3

The proof establishes that successive changes of reference frame compose to the direct change between the endpoint frames. It does so by examining the operators’ action on an arbitrary basis state.

B.3 Proof of Lemma 3

  • U i→j U k→i = U k→j, as shown by evaluating the operators on an arbitrary basis state.Source: §B.3 · p. 19

C On the unitarity of the change of reference frame operator

The change-of-reference-frame operator is unitary exactly when the group-labelled states form an orthonormal basis, or a subspace thereof, of the relevant Hilbert space. The argument considers commuting left and right group actions on those states.

C On the unitarity of the change of reference frame operator

D.1 Proof of Theorem 2

The section gives an example where sequential transformations in a semidirect-product setting yield different results, then treats rotational ignorance, Wigner’s friend, and comparison with an earlier real-line operator. It identifies when probabilistic descriptions or invariant quantities are used and reports that adding a reference system does not resolve the friend paradox.

D.1 Proof of Theorem 2

  • For the stated R = Z ⋊ S1 example, the two transformation orders give different results when x1 − x0 < 0.The passage explicitly compares T(s)^2 with T(s′)^2 and states that they are unequal.Source: §D.1 · p. 20

D.2 Averaging and invariants for the example of SO(3) ⋊T(R3)

  • In these cases, integrating over a group action on pure states is a projection, not a representation of ignorance; inferring other particles’ states requires a probabilistic description using measures on X.Alternatively, one can seek quantities invariant under all rotations; any f(|⃗a|) is invariant.Source: §D.2 · p. 21

E Wigner’s friend with additional reference system

  • In the Wigner’s-friend scenario, adding a reference system leaves the system correlated with the friend and entangles Wigner with that reference system, so it does not resolve the paradox.The conclusion follows from the states after the change of perspective to the friend.Source: E Wigner’s friend with additional… · p. 21

F Comparison to RF change operator of [6]

  • On general states of three particles on the real line, the operator in Equation (111) and the operator of act in the same way.The passage identifies this as an equivalence between their actions on the most general such states.Source: F Comparison to RF change operator of… · pp. 21–22
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