Quantum reference frames for general symmetry groups

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.


I. INTRODUCTION
In quantum mechanics, physical systems are implicitly described relative to some set of measurement devices. When writing down the quantum state of a system of interest, say a spin-1/2 system in the state |↑ z , we mean that the state of the system is 'up' relative to a specified directionẑ in the laboratory. In practice, this direction will be associated to a macroscopic physical system in the lab. If we assume that quantum mechanics is a universal theory and therefore applicable at all scales, the systems we make reference to to describe quantum systems should eventually be treated quantum mechanically as well. Reference systems that are themselves treated as quantum systems are referred to as quantum reference frames. Following the success of Einstein's theory of relativity and its inherently relational nature, one may seek to adopt a relational approach to quantum theory as well. In such an approach, most physically meaningful quantities are relational, i.e. they only take on well defined values once we agree on the reference system (or the observer) relative to which they are described. In his papers [1,2], Rovelli suggested that quantum mechanics is a complete theory about the description of physical systems relative to other physical systems. In his Relational Quantum Mechanics (RQM) he rejected the idea of observer-independent states of systems and values of observables. The importance of changes of reference frame * anned@ethz.ch † tgalley1@perimeterinstitute.ca in special and general relativity suggests the development of an account of changes of quantum reference frame in RQM. Such an account is given in the present work.
Recently, there has been an increased interest in analysing spatial and temporal quantum reference frames and in establishing a formalism that allows to switch between different perspectives [3][4][5][6][7]. The present work is partially based on these approaches which define changes of quantum reference frames for systems that transform under the the translation group in space and time. As opposed to other more standard approaches [8][9][10][11], this formalism stresses the lack of an external reference frame from the outset and defines states of subsystems relative to another subsystem. In standard approaches to quantum reference frames [9,10], one often starts from a description relative to an external reference frame and removes any dependence on this reference frame by applying a G twirl (a group averaging over all possible configurations of the external reference frame). In some cases, one can refactor the total Hilbert space into global and relational subsystems and trace out the global degrees of freedom [9,10,12,13]. The main emphasis of these standard approaches is often to obtain the physically meaningful (or reference frame independent) quantities, in a similar fashion to identifying noise free subsystems in error correction. In the work of [3,6,7] however emphasis is given on the relational nature of the description (always starting from a state that is given from the viewpoint of one of the subsystems) and the main object of study is the relation between different accounts. We make a similar emphasis in the present work. We abstract the formalism of [6] and introduce an approach which makes heavy use of the inherently group theoretic nature of quantum reference frames. This allows us to generalise the known results beyond the translation group to arbitrary finite and locally compact groups (including non-Abelian groups).
In Section II we outline the relational approach to quantum theory embraced in the present work as well as give a simple example of a change of reference frame for classical bits and an example of a change of quantum reference frame for qubits. In Section III we define the notion of a reference frame in terms of reference systems and coordinate systems, as well as give a full account of active and passive transformations as left and right regular group actions. Combining these we define changes of reference frame under a group G for classical systems with configuration space X ∼ = G. In Section IV we extend the classical change of reference frame to quantum systems L 2 (G) following the principle of coherent change of reference system; and define a general unitary operator which implements this change of reference system. We prove a theorem stating that only systems carrying a regular representation of G can serve as reference frame, subject to the principle of coherent change of reference system. Following this we extend the change of quantum reference frame operator between L 2 (G) systems describing systems which do not carry the right regular representation of G. In Section V we define irreversible changes of reference frames for groups G = N P and G = N ×P via a truncation procedure. In Section VI we extend this change of reference frame to quantum reference frames using the principle of coherent change of reference system once more. In Section VII we apply the tools developed in the preceding sections to the Wigner's friend thought experiment, providing an explicit change of reference frame from Wigner's description to the friend's. We discuss related work in Section VIII and discuss implications of the present work as well as suggestions for future work. In Section IX we give some concluding remarks.

II. RELATIONAL APPROACH TO QUANTUM THEORY
In the construction of a relational formalism of quantum mechanics, an essential task is to write quantum states of systems relative to a specified reference system. We introduce the following notation: |ψ A B indicates the state of system B relative to system A. In contrast to the approach of [6], we assign a Hilbert space to the system whose perspective is adopted and assign to it the trivial state, corresponding to the identity element of the group. Hence, by convention, system A is in a default 'zero-state' relative to itself. Once we introduce the notion of symmetry groups and how they enter into the formalism, we will see that this default zero-state corresponds to the identity element of the group that describes the transformations of the system. Thus, to be more precise, one can write The upper index refers to the system relative to which the state is given while the lower index refers to the system that is being described (similarly to the perspectival approach of [14]). This description does not make use of any external abstract reference frame nor does it assume the existence of absolute, observer-dependent values of physical observables. We observe that since system A can only ever assign itself a single state there are no state self-assignment paradoxes [15,16].
A natural question to address on the relational approach to quantum theory is how to change reference systems. Namely if the state of B relative to A is |ψ This is the problem which will be addressed in the present work.
Before introducing the general framework we will be using we give two simple examples of changes of reference frame for relational states. The first is classical and the second its quantum generalisation. These should hopefully provide the reader with an intuitive picture of the general mechanisms at play.
Example 1 (Z 2 change of classical reference frame) Let us consider the case where systems can be in two states ↑ or ↓. Every system considers themselves to be in the state ↑ (for example an observer free floating in empty space would always consider the up direction to be aligned from their feet to their head). Consider classical systems where the state relative to then since A views B in the ↓ state, this implies that B views things 'upside down' relative to A. The change of perspective would give ↑ B B ↓ B A ↑ B C . In the next example we give a quantum generalisation of the Z 2 change of reference frame. This is a specific instance of the general changes of quantum reference frame defined in this work.
Example 2 (Z 2 change of quantum reference frame) Let us consider the case with quantum systems C 2 with basis {|↑ , |↓ }. Every system considers themselves to be in the state |↑ . By embedding the classical scenario above with the map ↑ → |↑ and ↓ → |↓ we can reconstruct the classical example: if the state relative to A is |ψ The above example made use of the two guiding principles of this work: the principle of relational physics and the principle of coherent change of reference system. These are defined in Section IV C.
Whenever we use phrases such as 'from the viewpoint of' or 'from the perspective of', we simply mean 'relative to'. Although these expressions might imply that the state |ψ A B indicates how system A perceives system B, we do not make this interpretation here. System A acts as the observer in this description but we should note that there is nothing special about an observer system. No interpretation is made as to what the system sees. Rather a change of reference system A → B is a change of description from one where A is at the origin to one where B is at the origin.

III. CLASSICAL CHANGES OF REFERENCE FRAMES ASSOCIATED TO SYMMETRY GROUPS
A. Reference systems, coordinate choices and changes of reference frame A coordinate system is a purely mathematical object, and need not in general be associated to a physical system. A reference frame consists of a physical system (known as a reference system), and a choice of coordinates such that the reference system is at the origin in that coordinate system. For full definitions we refer the reader to Appendix A. In this section we define changes of reference frames for classical systems where the configuration space is itself a group G. In Section V we will consider cases where this is no longer holds.
We begin by a simple example which illustrates changes of reference frames and the use of group element for relative coordinates.
Example 3 (Three particles on a line) Consider three classical particles A, B and C on a line, with state s = (x A , x B , x C ) in some Cartesian coordinate system (here we omit the velocities since we are just interested in translations in space). The coordinate system x such that x A = 0 is said to be associated to A. In this coordinate system the particles have state . We observe here that the relative coordinates (to A) x A , x B and x C uniquely identify the translation which maps system A to systems A, B and C. Namely the relative distance The relative coordinates x A , x B and x C correspond to the symmetry group transformations relating A, B and C to A. If we label a translation of distance d by These two relative states are themselves related by the transformation s B = s A − (x B − x A ). We define the right regular action of the translation group T R (d) = x − d. The change of reference frame A to B is given by the right regular action In the above example the configuration space R and the symmetry group T = (R, +) acting on it are equivalent as manifolds. This equivalence is essential for the existence of a well defined reversible change of quantum reference frame. In Sections V and VI we study scenarios where this is no longer the case, and the changes of reference frame are irreversible. Since the results in this paper also apply to finite groups we cover a simple example.
Example 4 (Z 2 ) Let us consider systems with configuration space X = {↑, ↓}. The symmetry group G = {I, F } = Z 2 consisting of the identity I(↑) = ↑ and the flip F (↑) = ↓ is the symmetry group of X. A state of 4 systems of the form s =↑, ↑, ↓, ↓ can be expressed as s = I(↑), I(↑), F (↑), F (↑). By considering ↑ as the 'coordinate system' we have that the state s has coefficients (I, I, F, F ). In the 'coordinate system' ↓ the state s would have coefficients (F, F, I, I).
We observe that in the above examples group elements of the global symmetry groups serve as relative coordinates. In the next subsection we make this link more explicit. We also note the importance of the one to one correspondence between states and coordinate transformations. We observe that there is always some conventionality in changes of coordinates: one considers only translations on R for instance, and not all diffeomorphisms of R as relating different coordinates. In Appendix A we give explicit definitions of coordinate systems on a manifold X, emphasising that a coordinate system is different to a coordinate chart (typically used in general relativity), a distinction made in [17]. Roughly speaking a coordinate system on a manifold X is an isomorphism f : X → Y (where Y a known mathematical object used to describe X), whereas a coordinate chart is a map from the mathematical object Y to the physical object X which need not be an isomorphism (for instance multiple charts R n are used to describe a curved n-dimensional manifold X in general relativity but there is no isomorphism from X to R n ).

B. General treatment of reversible changes of reference frame
Let us extract the general features of the above scenario which allow for well defined reference frames and reversible changes of reference frame. Consider a configuration space X (which is typically a set with a manifold structure) and a group G acting on X such that there is a unique transformation g ∈ G relating any pair of points (the action is transitive and free). This implies X ∼ = G (as sets/manifolds), and the action of G on X ∼ = G is the group multiplication on itself: G × G → G. This space is a principle homogeneous space for G, sometimes called a G torsor. We assume G locally compact and thus equipped with a Haar measure, denoted dg. We observe that X ∼ = G follows from the requirement that there exists a unique transformation g ∈ G relating any two points in X.
We use the following example from [18] to introduce active and passive transformations on G torsors.
Example 5 (Single observer and system on X ∼ = G) Consider an observer 0 at location x 0 on X ∼ = G and an object 1 at location x 1 . Then the unique transformation g such that gx 0 = x 1 is the relative location of 1 relative to 0.
An active transformation is a transformation on the object 1. A transformation h on the object 1 is given by the left regular action x 1 → x 1 = hx 1 . The relative location is now k where kx 0 = x 1 . Using gx 0 = x 1 and hx 1 = x 1 we find that k = hg: hgx 0 = hx 1 = x 1 . Therefore an active transformation by h corresponds to the left regular action of h on the relative location g: g → hg.
A passive transformation is a transformation on the observer x 0 → hx 0 . This induces a transformation on the relative location of 1 to 0 which we now outline. Consider the case where gx 0 = x 1 and a passive transformation h on 0 is applied while 1 is left unchanged (at position x 1 ).
We have x 0 → x 0 = hx 0 . Then the relative location of 1 to 0 is k where kx 0 = x 1 . Substituting in gx 0 = x 1 and hx 0 = x 0 gives khx 0 = gx 0 implying that that kh = g, and therefore k = gh −1 (where we remember that since X ∼ = G there is always a unique g ∈ G mapping a pair of points in X). Writing in full gh −1 hx 0 = gh −1 x 0 = x 1 and so the relative location of 1 relative to 0 is now gh −1 . A passive transformation h on 0 corresponds to the right regular action of h on the relative location g: g → gh −1 .
The left regular action and right regular action on G ∼ = X are defined as follows: Both are defined using the group multiplication, where x ∈ G. These two actions naturally commute, and hence X ∼ = G carries an action of G × G, with one factor typically being understood as the active and the second as the passive transformations [19]. Although φ R acts 'to the right' it is a left group action: Here we take φ L as active and φ R as passive. 1 A given state of n systems is s = (x 0 , x 1 , ..., x n−1 ), where we omit the velocitiesẋ i since we are defining changes of reference frame for the 'translation' group G on X ∼ = G. This can be expressed as: where g i j is the unique g ∈ G such that g i j x i = x j , and e = g i i the identity element. We observe that g j k g i j = g i k and (g j i ) −1 = g i j . Then the state s 0 of the n systems relative to system 0 is: The state relative to the system i is: We observe that we can also describe the state relative to hypothetical systems (i.e. relative to a point x ∈ X which is not occupied by a system). For instance in the above consider a point x n ∈ X such that x i = x n ∀i ∈ {0, ..., n − 1}. Then we can write: s n = g n 0 , g n 1 , g n 2 , ..., g n n−1 As in the examples given above we see that a relative state s i is given by all the symmetry transformations g i j from system i to system j for all j ∈ [0, ..., n − 1]. There is a unique relative state s i , which is such that particle i is in state e. A change of reference frame from system 0 to system i is a map s 0 → s i . Let us extend the left and right regular actions of G on itself to states: The transformation s 0 → s i is given by the right regular action of g 0 i .
We observe that this transformation cannot be achieved using the left regular action: there is no elements g ∈ G such that φ L (g, s 0 ) = s i (unless G is Abelian). The transformation s 0 → s i is a passive transformation.

IV. QUANTUM REFERENCE FRAMES ASSOCIATED TO SYMMETRY GROUPS
We begin this section by reviewing changes of quantum reference frame for three particles on the line. We then define a quantum change of reference frame operator for n identical systems L 2 (G) for arbitrary G. This generalises the change of reference frame in [6] beyond one parameter subgroups of the Galilean group. Furthermore we show that it is only the L 2 (G) system described so far for which a unitary reversible change of reference system is possible. Finally, we define a change of reference frame operator for m identical L 2 (G) systems serving as reference frames and n − m systems which are not. Diagram capturing the relational states between 3 systems. Each system i assigns the relative state along the arrow point from i to j to system j (and the identity to themselves). For instance system 0 assigns state e, g 0 1 , g 0 2 . By the right regular action of g 0 1 we obtain g 1 0 , e, g 0 2 g 1 0 = g 1 2 which is the relative state assigned by system 1.
A. Comment on finite groups and notation All our results apply for finite groups. In this case L 2 (G) should be replaced by C[G] ∼ = C |G| and integrals g∈G |g g| dg by is the vector space freely generated by the elements of G, i.e. for which the elements of G form a basis.
In the cases where it is clear which system is the reference system we sometimes omit the top label for ease of reading. For instance the state |0 Let us first rephrase the known case of the translation group acting on three particles on the line [6] in the formalism outlined above.
Take the translation group T = (R, +) and three systems A, B and C whose joint state space is L 2 (R)⊗L 2 (R)⊗ L 2 (R). Let us for instance consider the state which is the state of three perfectly localised systems, described using a coordinate system centred on system A. In standard quantum mechanics, when changing from a classical, highly localised reference frame at the position of A to another classical reference frame localised at B translated by an amount x 1 , one simply applies the translation operatorT (−x 1 ) = e ix1(p A +p B +p C ) to the state of the three systems, wherep A is the momentum operator for system A and similarly forp B ,p C and systems B and C. This shifts the state to: In the previous language we have g A B = x 1 and g A C = x 3 . The action ofT (−x 1 ) corresponds to the right action of g A B = x 1 . The next step is to begin with a state of the following form: which is described by a coordinate system localised at A. What is the change of reference frame A → B in this case? How can one describe classical coordinates which assign state |0 B , when B is not localised relative to A? A standard translation of all states will not work. Following the reasoning presented in [6] we assume that the change of perspective obeys the principle of superposition. Namely the state of Equation (13) is an equally weighted superposition of the states |0 A |x 1 B |x 3 C and |0 A |x 2 B |x 3 C . The change of reference frame for each of these states individually is obtained by translating by −x 1 and −x 2 respectively.
Assuming that the superposition principle applies to changes of reference systems, the state described in coordinates 'localised' at system B is just the superposition of the classical states obtained by translation by −x 1 and −x 2 . This leads to the following state of the joint system from the viewpoint of B: When changing between the viewpoints of quantum systems, we apply a weighted translation of the states of systems, dependent on the state of the new reference frame whose viewpoint we are adopting. For the state given above, this means applying a translation for the state of B being |x 1 and one for it being |x 2 . The states of A and C become entangled relative to B. We see that to perform this change of reference frame, the state of B is mapped to the inverse of the group element associated with the old state of B. Also for each state of B, the state of C is shifted respectively. Hence, for the translation group on the real line, the reference frame change operator is This operator performs exactly the same reference frame change as the operator given in [6]: whereP AB is the so-called parity-swap operator. It acts asP AB ψ B (x) = ψ A (−x). The proof of this is given in Appendix F.
Consider a configuration space X ∼ = G and n systems each with associated Hilbert space The left and right action of G onto itself induces the left regular and right regular representation of G on each H i . For a given H i this representation acts on the basis {|g } as: An arbitrary basis state of the n systems is: Following the classical case, the choice of coordinates on G associated to H 0 is: where g i j g i = g j . For general H i it is: The change of coordinate system |ψ 0 → |ψ i is given by U R (g 0 i ) ⊗n , when considering orthogonal basis states alone.
Let us observe that the left regular representation on the space of wave functions acts as ψ(x) → ψ(g −1 x) and the right regular representation as ψ(x) → ψ(xg). This follows from x∈G ψ(x) |gx = x∈G ψ(g −1 x) |x and x∈G ψ(x) xg −1 = x∈G ψ(xg) |x . We note that for Lie groups G the objects |g are not in L 2 (G) and one should typically prefer the representation acting on the wavefunctions. In the following however we consider the representation acting on the elements |g in order to describe the continuous and discrete case simultaneously. Now following the fully relational account embraced in this work we do not assign a global state |ψ ∈ H and then work out its expression relative to a certain system. Rather, we begin from a state relative to a system and define changes of reference frame to other systems. We formalise this in the following principle: Principle 1 (Relational physics) Given n systems, states are defined to be relative to one of the systems. A state relative to system i is a description of the other n − 1 systems, relative to i.
We observe that this principle does not preclude the existence of a well defined global state of the n systems.
A superposition state (relative to 0 in the G product basis) is of the form: Since the state is in general not a basis state there is no a priori well defined change given by an operator of the form U R (g 0 i ) ⊗n . However, following the example of [6] one can define a coherent change of reference frame operator. We explicitly state this as a principle: This implies that |ψ 0 defined above changes to: The operator which implements the coherent change of reference systems 0 → i is: The following lemmas are proven in Appendix B.

Change of reference frame for observables
The change of reference frame operator also allows us to transform between observables. Namely if system 0 describes an observable of systems 1, ..., n − 1 as Z 0 0,1,...,n−1 = 1 0 ⊗ Z 0 1,...,n−1 then system i describes the observable as To illustrate the changes of reference frame described previously we will give an example. Let us consider the symmetry group U(1) and three particles A, B, C on a circle with associated Hilbert space L 2 (U(1)) ⊗ L 2 (U(1)) ⊗ L 2 (U(1)). In this case, the map from group elements to elements of the Hilbert space is 3. Basis states |0 , |θ1 and |θ2 of the state space for U (1) [0, 2π[} are states at all angular positions of the unit circle and form a basis of the Hilbert space L 2 (U(1)). A system consisting of three particles on a circle could for instance be in the product state |0 A π 2 B |π C relative to A. An arbitrary state of the joint system relative to particle A is given by where |e = |0 is the state associated to the identity element of U(1). As a specific example, take the state Relative to particle B, the state assigned to the joint system would be We see that the state of B is mapped to the state corresponding to the inverse group element assigned to the old state of B and the state of C is shifted respectively. In the end, the labels of A and B are swapped. The operator that performs this reference frame change is The change of reference frame defined previously is highly constrained: it applies only to systems L 2 (G) with symmetry group G. One could ask whether one could define similar changes of reference frame for systems H ∼ = L 2 (G) with states |ψ(g) carrying two representations: First consider the symmetry group U(1). Our results show that for a change of reference frame to recreate our classical intuitions one needs systems L 2 (U(1)) which carry the right regular representation. However one may wonder whether one could use qubits with states along the X − Y plane |θ = cos(θ/2) |0 + sin(θ/2) |1 as reference systems which transform in a manner which obeys the classical change of reference frame.
We first provide an example to show that this breaks linearity of the change of reference frame operator for the case of rebits before proving a general result.
Example 6 (Three qubits with U(1) group action) Consider three qubits with states restricted to real valued superpositions: |θ = cos(θ/2) |0 + sin(θ/2) |1 (sometimes known as rebits). The space of pure states of the three systems is U(1) ⊗ U(1) ⊗ U(1). We apply our classical intuition of how a reference frame change should act. Let the initial state of the three systems relative to A be The map ψ takes the group element θ of U(1) to the state in the two-dimensional Hilbert space: We want this state to be mapped to the final state relative to B This corresponds to our intuition of what should happen when one changes from the viewpoint of A to the viewpoint of B. Writing out this transformation in the rebit basis corresponds to the map On the other hand, considering the basis states of the joint Hilbert spaces and assuming the map is linear, the following should hold When comparing the coefficients in the map (34), one sees that the reference frame change cannot be linear. This means that the operator describing the change from one rebit reference system to another one is non-linear. As this non-linearity causes issues concerning the invariance of probabilities under reference frame change we conclude that rebits cannot serve as reference frames that allow to reversibly transform between each other.
Given n classical systems with configuration space X ∼ = G acted on by a symmetry group G we have shown how to define states relative to these systems, and to transform between them using the left and right regular action of G on X.
The case L 2 (G) is a very specific 'encoding' of G into a quantum system. It is a natural choice, in that the classical states are embedded into orthogonal states of the quantum system. However one could have an injection G → H, with g → |ψ(g) such that the states |ψ(g) are not all mutually orthogonal and ask whether a change of reference system can be defined. We require H to carry two unitary representations U L and U R , corresponding to active and passive transformations, such that U L (h) |ψ(g) = |ψ(hg) and Although one would naturally desire them to commute (since active and passive transformations as usually defined act on different spaces and therefore trivially commute), we do not impose this here. The following theorem tells us that the change of reference frame which acts as expected on product states |ψ(e) i ψ(g i 0 ) 0 ... ψ(g i j ) j ... ψ(g i n−1 ) n−1 and obeys the principle of coherent change of reference frame is unitary exactly if the states |ψ(g) it acts on form an orthonormal basis of the Hilbert space.
Theorem 1 Take n identical systems with associated Hilbert spaces H i each carrying two representations of G: U L and U R such that U L (h) |ψ(g) = |ψ(hg) and Then any operator U which performs the change |ψ(e) i ψ( and obeys the principle of coherent change of reference system is unitary if and only if the representations U L and U R are the left and right regular representations acting on states |ψ(g) which form an orthonormal basis of H i (or a subspace thereof ).
The proof can be found in Appendix C.
Let us consider the case where reference systems L 2 (G) describe systems of a different type. The total Hilbert space is L 2 (G) ⊗m ⊗ H ⊗n−m where for simplicity we have assumed the n − m systems to be of the same type (but not L 2 (G)). The systems H are such that there exists an injection φ: and two representations V L and V R such that: To change from reference system 0 to i, where both systems are assumed to be of the type L 2 (G), we apply the operator: where U R is the right regular representation acting on the first m L 2 (G) systems. We observe that not all systems H which carry a representation of G will be such that there exists an injective map φ : g → |ψ(g) . For instance the qubit carries a representation of SU(2) but there is no injection of φ : SU(2) → PC 2 (where here we emphasise that the pure states of a C 2 system form PC 2 the projective space of rays). Observe that for U(1) there is an injection φ : U(1) → PC 2 . We explore the example of two L 2 (U(1)) systems describing a system C 2 carrying a representation of U(1).
Let us adapt the previous example of three particles on a circle to the case in which the third system is a qubit H C ∼ = C 2 giving a total Hilbert space of the joint system L 2 (U(1)) ⊗ L 2 (U(1)) ⊗ C 2 . H C carries a representation of U(1) and an injection φ : U(1) → C 2 .
in the {|0 , |1 } basis by matrix multiplication from the left. The injection is a map The operator that maps the state relative to A to the state relative to B is As a specific example, consider the state . From the viewpoint of system B, the state is F. Changes of reference system for arbitrary identical systems The above treatment shows that for any group G one can define a change of quantum reference frame between n identical systems. However given n identical systems can one always find a group allowing for a reversible change of quantum reference frame? Namely for a configuration space X, can one always find a binary operation turning it into a group G ∼ = X? In the case of finite systems C d one can pick an orthonormal basis |x , x ∈ {0, ..., d−1} and choose the cyclic group Z d acting on {0, ..., d − 1}. In the case where X is a countable set one has the group Z. In the case where X is uncountable the existence of a group G such that G ∼ = X is equivalent to the axiom of choice [20]. We observe that if X has some additional structure (such as being a manifold), then one may not be able to find a group which is isomorphic as a manifold.

V. IRREVERSIBLE CHANGES OF CLASSICAL REFERENCE FRAME
For a given configuration space X all changes of coordinates are related by a transformation g ∈ G. In the case X ∼ = G there is a one to one correspondence between points in X and coordinate systems. As such one can identify a coordinate system as systems with configuration space X. Namely if system i is in state x i ∈ X then one assigns it the unique coordinate system x i which maps x i → 0. However typically one has a symmetry group G which is larger than X. We consider an example of such a case. A.
Let us consider nparticles in R 3 , with each particle i having state (x i , y i , z i ) expressed in Cartesian coordinates (x, y, z). The set of Cartesian coordinates is acted on by the Euclidean group E + (3) = R 3 SO(3). A choice of coordinates (x , y , z ) such that (x i , y i , z i ) = (0, 0, 0) is said to be associated to particle i if and only if it is the unique set of coordinates with this property. There are infinitely many such coordinate choices (for instance all coordinate systems which are rotated relative to (x , y , z ) will also assign state (0, 0, 0) to particle i). In this case there is no obvious unique manner of associating a coordinate system to a particle.
All Cartesian coordinate systems for R 3 are related by an element g ∈ E + (3) where E + (3) is the Euclidean group. The action of E + (3) on the set of Cartesian coordinates is transitive and free. To put it visually every element in E + (3) can be considered as a translation followed by a rotation. Every choice of Cartesian coordinates is associated to a set of orthogonal axes located at some point r ∈ R 3 with a given orientation. These are all related to the Cartesian coordinates at (0, 0, 0) in a given orientation by a rotation followed by a translation. We cannot assign a unique coordinate system to each point in R 3 . For instance take a coordinate system centred at the origin; then any other coordinate system obtained by rotation about the origin will also assign the state (0, 0, 0) to the origin. See Figure 5.

Enlarging the space of states of the reference systems
There are multiple ways of addressing this issue. One can say that systems with configuration spaces R 3 (i.e. particles) are not good reference systems for E + (3). Rather one should choose systems with a larger configuration space. This is what is typically done, where we choose solid bodies in R 3 as reference systems. Since solid bodies have an orientation (unlike points), which is to say that rotating a solid body changes its state, they have configuration space E + (3). One can assign a unique coordinate system to every state x ∈ X ∼ = E + (3) of a solid body. An example of a solid body would be three physical orthogonal axes in R 3 labelled 1, 2 and 3. For a given state x of these three physical axes one can associate the coordinate system which assigns +x, +y and +z to the axes 1, 2 and 3. Using this approach would allow us to make use of the results of the previous section. However one could also keep the reference systems as having configuration space X but rather assign to each state x the equivalence class of coordinate systems centred on x. One can either choose a representative member of the equivalence class (in the above case one can fix all coordinate systems to have a given orientation as in Figure 6) or one could average over the possible elements of the equivalence class.

Representative element of each equivalence class
In the case where all the reference systems have configuration space R 3 , it makes sense to assign to each point x ∈ R 3 a unique coordinate system centred on that point (from the equivalence class of coordinate systems centred on that point).
Take K to be the subset of transformations which relates these coordinate systems. We require that K is a group in order for us to have a well defined change of coordinate system. If K is not a group, then by composing different elements in K we can obtain a group G (which is larger than K as a set). This symmetry group will take coordinate systems we have selected to coordinate systems which we have not chosen.
In order for all representative members (i.e. the coordinate system we chose to be associated to each point) to be related by a group K ⊂ E + (3) and for the representative members to be closed under the action of K one can choose them to all have the same orientation (i.e. be related by just translations). In this case the symmetry group relating coordinate choices becomes K ∼ = R 3 once more. B. G = N P and G = N × P : truncation Let us consider the case where there are systems with configuration space G and systems with configuration space N , where G = N P or G = N × P . In both cases N is normal, and for every g ∈ G there is a unique n ∈ N and p ∈ P such that g = np.
The configuration space N is embedded in G via an embedding map E : N → G, E : n → np C for some constant p C , where the choice of p C is conventional and is typically chosen to be the identity. For a choice p C the points np C ∀n ∈ N are related by transformations n ∈ N (acting to the left). As such the symmetry group of E(N ) is K ∼ = N . If the map did not fix a unique convention (for instance np → np 0 (n)) where the image depends on which equivalence class is chosen, then the set K of transformations between the images E(n) would typically not be a group, and its closure would not be isomorphic to N (in some cases it would be the full group G).
Take m systems G and n − m systems N . A general state of the n systems is: s = ((g 0 , g 1 , ..., g m−1 ), (g m , ..., g n−1 )) where g i ∈ G for i ∈ {0, . . . , m − 1} and g j ∈ N for j ∈ {m, . . . , n − 1}. Moreover there is a unique n i ∈ N and p i ∈ P such that g i = n i p i . Here p j = e for systems j ∈ {m, . . . , n − 1}. The description relative to the first m systems and the transformations between them is just the case described in Section III. In the following we describe how to change reference systems from a system with configuration space G to a system with configuration space N . The embedding of N ⊂ G is given by n → ne. We define the truncation map: and the map R i G : R i G : (g 0 , ..., g n−1 ) → (g i 0 , ..., g i n−1 ) .
Then, the relative state s 0 = R 0 G (s) is: where n j i p j i = g j i . Observe that for j ∈ {m, . . . , n − 1} we have g 0 j g 0 = n j which implies g 0 j n 0 p 0 = n j ∈ N . Now using the fact that gng −1 ∈ N for all g ∈ G and n ∈ N and that for every g there is a unique decomposition into g = np we have that g 0 j = m 0 j p −1 0 (for j ∈ {m, . . . , n−1}). Therefore where p C = p −1 0 . For particle j with configuration space N , the state s j is: which is obtained from s 0 by the map (Γ R (m 0 j ) • T). Here Γ R (m) is just shorthand for the right regular action: Γ R (m)(g 0 , ..., g n−1 ) = φ R (m, (g 0 , ..., g n−1 )) = (g 0 m −1 , ..., g n−1 m −1 ). We have m 0 k m j 0 = m j k . We observe that all states s 0 of the form (n 0 0 p 0 0 , n 0 1 p 0 1 , ..., n 0 m−1 p 0 m−1 ), (n 0 m p 0 m , ..., n 0 n−1 p 0 n−1 ) for all p 0 i ∈ P give the same s j . The change of reference frame s 0 → s j is an irreversible change of reference frame.

R ∼ = Z U(1)
An example of such a truncation is the 'modular truncation' of the translation group: R ∼ = Z U(1). Instead of distinguishing all position states on the real line, we consider a reference frame that essentially consists of a classical ruler. All points that lie within an interval of length L are mapped to the same point on the ruler (for simplicity, one can choose L = 1). Hence, the map identifies a subset of elements of G = R with the same element of N = Z: Consider now three particles on the real line at positions x 1 1 , x 1 2 and x 1 3 relative to particle 1, where particle 1 and 2 have configuration space R and particle 3 has configuration space Z. Let us write the state s 1 = (0, n 1 2 L + p 1 2 , n 1 3 L + p 1 3 ), then the state s 3 is obtained by Λ 1→3 T (s 0 ) = Λ 1→3 (0, n 1 2 L, n 1 3 L) = (−n 2 3 L, (n 1 2 − n 2 3 )L, 0). If for instance T (x 1 1 ) = T (x 1 2 ), i.e. n 1 = n 2 , particle 3 assigns the same state to particles 1 and 2.

R 3 ∼ = R 2 × R
Another example is the truncation map R 3 → R 2 . If we have a symmetry group R 2 in a three-dimensional space, all reference frames along a one-dimensional line are identified with each other. In this case, when applying the truncation map, all group elements in R 3 are projected to the associated group elements in R 2 . Consider three systems of which the first two have configuration space R 3 and the last has configuration space R 2 . A general state of the systems is s = (g 0 , g 1 , g 2 ) where g 0 , g 1 ∈ R 3 and g 2 ∈ R 2 . Relative to particle 0, the state is s 0 = (e, g 0 1 , g 0 2 ) = (e, n 0 1 p 0 1 , n 0 2 p C ) where p 0 1 , p C ∈ R and n 0 1 , n 0 2 ∈ R 2 . First, we truncate the state: s = (e = n 0 0 , n 0 1 , n 0 2 ). Then, we change to the state relative to system 2: s 2 = (n 2 0 , n 2 1 , e). This can be understood as projecting the points in R 3 to a plane in R 2 . If all reference frames along the z-axis are identified with each other, this corresponds to projecting on the x-y−plane.
C. Inconsistency with an observer independent description for G = N P In Appendix D we prove that the state s i for a system i with configuration space N obtained from state s 0 by truncating and changing reference system is not equivalent to the states i obtained from the observer independent state s by first truncating to obtains, then findings 0 and then changing reference system to obtaiñ s i . Let us write the change of reference system 0 → i for a group G as Λ 0→i G s 0 = s i : where Γ R (g 0 i )s 0 is φ R (g 0 i , s 0 ), i.e. the right regular action of g 0 i on s 0 .
Then The above theorem shows that depending on our prior commitment to a well defined observer independent state, and our interpretation of relative states s k the truncation method of obtaining relative states may not be desirable. One may prefer an averaging procedure or a method based on finding invariants. These two approaches are described in Appendix D using the example of SO(3) T (R 3 ). We make use of the truncating procedure in the present work since it can easily be extended to a quantum version using the principle of coherent change of reference system. We describe this quantum generalisation in the next section.

VI. IRREVERSIBLE CHANGES OF QUANTUM REFERENCE FRAME
The irreversible change of reference frame of the previous section can be extended to the quantum case by applying the principle of coherent change of reference frame, as in the reversible case. We describe this in more detail in the following. We note that we only define this change of reference frame for the specific cases G = N P and G = N × P .
A. G = N P and G = N × P Let us consider the change of reference frame from system i with Hilbert space L 2 (G) to a system j with Hilbert space L 2 (N ). Let us consider the first m systems as being L 2 (G) and the last n − m systems as being L 2 (N ).
A generic product basis state with m i j ∈ N , q k l ∈ P and p 0 k = p 0 j for all j, k ∈ {m, ..., n− 1} (and similarly for q 0 k ) maps to: First define the truncation map T : L 2 (G) → L 2 (N ).
and the map: The change of reference frame is given by: Consider again the example of the modular truncation of the translation group: R ∼ = Z U(1). Take three (quantum) systems A, B and C where A has configuration space R but states relative to C are encoded in LZ. Changing reference frame from A to C requires The truncation operator is: and the change of reference frame operator for L 2 (N ) is: The following operator performs the change A → C: When applying the change of reference frame from A to C to a state in which system C is in a superposition state relative to A, the state becomes entangled relative to C: However, if x 1 and x 1 are located in the same interval relative to C (with configuration space LZ), the entanglement vanishes. Here, we recognize the dependence of entanglement on the reference frame relative to which it is described. This was already pointed out in [6]. Note that we observe entanglement only if the uncertainty in the position of C with respect to A (i.e. the difference between x 1 and x 1 ) is larger than the resolution L of the configuration space of reference system C.
As mentioned before, the truncation map R 3 → R 2 essentially consists of a projection from a three-dimensional configuration space to a two-dimensional one. Here, we project all points in R 3 onto the x-y-plane.
On the level of Hilbert spaces, we assign the state P relative to a three-dimensional configuration space while | p denotes a state in H ∼ = R 2 . Hence, to change from the state relative to A with configuration space R 3 to the state relative to C with configuration space R 2 , we apply the operator where Q := (x , y , z ) → q := (x , y ).

VII. WIGNER'S FRIEND EXPERIMENT
In this section we consider the Wigner's friend experiment, introduced by E. Wigner [21] in 1961. The thought experiment consists of two observers, Wigner and his friend, and a two-level quantum system. While Wigner is an external observer of the experiment, his friend is located inside an isolated box, together with the system S in the quantum state |ψ . Once the experiment is initiated, the friend measures the system in a certain basis. According to the projection postulate of standard quantum theory, the state of the system collapses to one of the eigenstates of the measurement operator. On the other hand, Wigner describes this process from the outside and would assign a unitary evolution. After the measurement of the system by the friend, the outcome of which Wigner does not know, Wigner would assign an entangled state to the joint system of the friend and the system. These two seemingly contradicting prescriptions of standard quantum mechanics are at the core of the so-called Wigner's friend paradox.
Before proceeding we observe that there is no logical contradiction in the above two descriptions. The state after applying the projection postulate is a state of the system S alone, of the form |ψ S . The entangled state assigned by Wigner is a state on S + F (where F is the friend) of the form |φ SF . Since these are states of two different objects (S versus S + F) there is no logical contradiction within the postulates of quantum theory.
We will now apply the relational formalism introduced previously to this apparent paradox. Let us consider W, short for Wigner, describing his friend F who measures the system S in the {|↑ , |↓ } basis. The friend is located inside a perfectly isolated box. We model everything using two-dimensional systems, since we are interested in two degrees of freedom alone. The change of reference frame will therefore be for Z 2 , first described in Example 2. We consider the ready state of the friend to just be |↑ and the state of the system before the measurement to be |ψ = α |↑ + β |↓ . We first describe the measurement interaction from the point of view of W starting in the case where S is in an eigenstate of the measurement operator: The state of the friend depends on the measurement outcome, hence the record of the outcome can be seen as being stored in the state of the friend. The state |↑ F is the state 'the friend sees up', and similarly for |↓ F and 'the friend sees down'. The change of reference frame W → F for the final states gives: where we observe that in both cases the state of S relative to F is |↑ F S . This state encodes the fact that the friend and the system are perfectly correlated in both cases; |↑ F S tells us that the state of the friend and the system are related by the identity element. For an arbitrary superposition state of the system the unitary measurement interaction gives the following evolution: (69) Now, we want to apply the change of reference frame to switch to the perspective of F. If we apply the operator U W→F to the initial state, using the principle of coherent change of reference system, we obtain: If we apply the change of reference frame U W→F to the final state we get: If we interpret this result as the state seen by the friend, this would imply that the friend always sees the system as correlated with herself. This is consistent with the initial description from Wigner's perspective, where the state of the friend and system are correlated in both terms of the entangled state α |↑ . Now, we want to compare this result to the state we get if we simply start from the perspective of the friend. She describes the initial state as: where Wigner is in the ready state. Applying the projection postulate to the system gives: Whatever outcome the friend observes, she would always describe the system definitely being in either one of the basis states and Wigner in the state |↑ F W . This seems very different to the description we obtain by taking Wigner's perspective and changing to the friend's reference frame. Note however that this does not necessarily imply a contradiction. More precisely, one should be careful when interpreting the state in Equation (71) as the final state seen by the friend. Rather, one should interpret it as the state Wigner infers (concludes) the friend would see. In fact, starting from the state from Wigner's perspective, we only take into account the information that Wigner has at his disposal. Simply changing reference frames by applying a unitary operator can by no means introduce new information, such as the actual outcome observed by the friend.
Hence, it is not a trivial assumption that the state actually seen by the friend should be the same as the state of the system relative to the friend, obtained by changing perspective from Wigner's viewpoint. In fact, this is assumption C (consistency assumption) of the Frauchiger-Renner no-go theorem [22]. In the framework of relational quantum mechanics, one should not assume the consistency condition to hold a priori. Indeed in this work we show that what Wigner infers about the friend's state assignments is given by the change of reference frame U W→F and not the seemingly straightforward assumption C.
We observe that the conclusion that what Wigner can infer about the state of the system relative to the friend is that they are correlated is the same as Rovelli's treatment of the measurement process in relational quantum mechanics [2]. Here instead of reasoning using Wigner's measurement operators we used an explicit change of reference frame from Wigner to the friend.
One may think that introducing an additional reference system R into the box containing F and S could help resolve the issue with Wigner's friend. We show in appendix E that this is not the case.

A. Related work
The underlying approach of this work is based on the relational quantum mechanics of [2]. Other work formulating a fully relational quantum theory include the toy model of [23] and the systematic treatment of quantum reference frames in [11,24]. The main difference between [11,24] and the present treatment is that we begin from an explicitly relational state and emphasise the notion of changing between reference frames as opposed to deriving relational states from an external nonrelational state. Related approaches to relational quantum mechanics include the perspectival quantum mechanics of [14] and the Ithaca interpretation [25].
The notion of quantum reference frames first appeared as part of the debate on the existence of charge superselection rules [26]. Later, Aharanov and Kaufherr gave the first explicit study of quantum reference frames [8]. Typically a description of a quantum system is given relative to a classical measuring device of infinite mass. It was shown that there is a consistent description of quantum systems relative to quantum reference frames of finite mass. This addresses the issue of universality of quantum theory; namely quantum systems are usually described relative to an implicit classical reference frame. If quantum theory is universal, then one would expect that reference frames should correspond to quantum systems.
Given a quantum reference frame R and a quantum system S the relational observables are observables of R + S which are invariant under the symmetry group. Depending on the state of R one may recover a relational description of S which is equivalent to the standard abso-lute description of S (i.e. relative to an implicit classical reference frame). Equivalently, for any absolute description of the state or the observables on S, one can find a system R such that there is a gauge invariant description on S + R. This observation was important for resolving the 'optical coherence controversy' [27]. Explicit maps between the absolute and relativised descriptions can be found in [9,11], where an emphasis on the localisation of the reference system is placed in [11,24].
In [10,28] a description of quantum systems relative to other quantum systems is given, in particular for the translation group. The initial description of R and S is given relative to an external classical reference frame (in the position basis), and the description relative to R is obtained by refactoring into center of mass and relative position. Tracing out the COM partition then allows to remove all global degrees of freedom and leaves us with a relational description of the systems. Moreover the description of systems is shown to be dependent on the mass of the quantum reference system. Note that such global degrees of freedom never enter into our formalism in the first place. Instead, we give the states of systems relative to a specified system from the outset.
The change of reference frame in this work generalises the known changes of reference frame L 2 (R) and L 2 (R 3 ) for spatial position of [6,29], and L 2 (R) for temporal degrees of freedom of [4,5,30]. Reference frames for rotational degrees of freedom are studied in the perspective neutral approach in [29]. Brukner and Mikusch are independently working on a treatment of rotational degrees of freedom of quantum reference frames using large spin coherent states.
We observe that in both [6] and the present work the consistency of the fully relational account with a description which is initially given externally (and from which the relative description is then obtained) is not proven. However in [3] it is shown that there is an external 'perspective-neutral' framework which encompasses all perspectives for [6]. Extending this perspective neutral approach to the general cases in the present paper could constitute an interesting direction for future work.
In [31] an emphasis is placed on the notion that a reference system in a superposition relative to another system gives new coordinates, which are not related by a classical coordinate transformation to the initial ones. This is conceptually closer to the perspective in the present work than that of [6]. Namely we do not make the claim that the relative descriptions are 'operational' as in [6].
Prior work with an emphasis on changing perspectives between quantum reference frames is found in [32] where this change of reference frame is mediated by an external description, and uses tools such as G-twirling.
In recent years there has been a significant interest in the study of quantum reference frames as resources for measurements, communication tasks and thermodynamic exchanges amongst others [33][34][35][36][37][38][39]. This approach is not motivated by the same considerations as the present paper and the link between the two approaches is not fully clear to the authors. It is possible that the present exposition, carried out explicitly in a group theoretic language, could be used to relate the two. For instance the 'perfect reference frames' of [9] are of the form L 2 (G), which also plays a prominent role here. The link between imperfect reference frames as standardly defined [9,36,38] and the truncation based approach of the present work also remains to be worked out.
B. General comments

Relational quantum mechanics
We have provided a formalism for relational quantum mechanics which captures its spirit and recovers the same conclusions for Wigner's friend and the Frauchiger-Renner theorem. Future work could involve applying the formalism developed in this work to other thought experiments, in order to provide an explicit account of how relational quantum mechanics addresses various apparent paradoxes.

Observer dependence of the symmetry group
One further contribution of this work is that symmetries are not only relative to the system being described, but also to the system doing the describing. A system with Hilbert space H has a symmetry G if it carries a representation of G, however it can only be described as transforming under G by a system which carries a regular representation of G. For example although a qubit carries a representation of SU(2), two qubits 'describing' each other and a third qubit would use the subgroup Z 2 , since they can only carry a regular representation of Z 2 and not the full SU (2). This is reminiscent of the argument by Penrose in [40]. 3. G = N P and G = N × P For the imperfect reference frames we describe a very specific case, though it seems natural since it is the structure of well known space time groups. Group extension tells us that for any group subgroup pair (G, N ) with N normal, the extension is of the form G = N P or G = N × P . Hence our approach fully covers imperfect reference frames for group subgroup pairs (G, N ) with N normal. However for a full account of imperfect reference frames one would need to relax the assumption of N normal and see whether one can define meaningful changes of reference frame.

IX. CONCLUSION
The construction of a relational quantum theory requires a description of systems and states relative to other systems. In this work, we present a formalism that allows to describe the relative states of systems and to change between the descriptions of different reference systems. Starting from the analysis of reference frames in the classical realm, we moved on to the description of quantum states relative to quantum reference frames. Depending on the system whose viewpoint is adopted, the description of physical phenomena changes. We find that quantum properties such as entanglement and superposition are not absolute but depend on the reference frame relative to which they are described. As such the conclusions of [6] are shown to be generic to quantum reference frames for arbitrary groups.
Previous work on quantum reference frames in the relational paradigm restricted itself to the translation group (for time, space and momentum translations). Here we have extended this to arbitrary groups, including finite groups and non-Abelian groups. Moreover we have highlighted the key representation theoretic features needed for reversible changes of quantum reference frame of a group G, namely that the quantum reference systems have Hilbert space L 2 (G). Using this insight we have developed a more abstract formulation directly in terms of the relevant symmetry group.
We also extended the study of reference frames in the relational approach to imperfect reference frames using a 'truncation' approach, in contrast to group averaging approaches more common in quantum information based approaches. Future work involves exploring imperfect reference frames for different cases than (G, N ) with N a normal subgroup. One benefit of our approach is that it formulates the relational approach of [6] in a group theoretic language which is more common in the quantum information literature. As such it may help in developing a fully rigorous account of the link between the two. Recent work [41] in this direction has been exploring links between different approaches in the case of the translation group and would be useful to pursue in this more general case.
Our exploration of the Wigner's friend scenario shows how to apply our relational framework to an important discussion point in the literature. Using our approach we reach the same conclusion as Rovelli in his relational treatment of the measurement process in [2]: all that can be said by Wigner is that the friend is correlated with the measurement outcome. The novel aspect of this work is the use of an explicitly relational formalism (which embodies the philosophical position of relational quantum mechanics) as well as obtaining the friend's perspective through an explicit change of reference frame, as opposed to reasoning indirectly using measurement operators.
We show that U i→j is unitary. To change reference frame between systems i and j, where i and j have state space L 2 (G), and n − 2 other systems carrying the right regular representation k , the general operator is We can show that this operator is unitary: Hence, Here, we used h j i g j i = δ gh , dg i j g i j g i j j = 1 j and U k (e) = 1 k .

Proof of Lemma 2
We show U 0→i † = U i→0 .
From the second to the third line, we commuted the SWAP operator through to the left by changing the labels of partitions 0 and i. From the third to the fourth line, we used the fact that the integral over g 0 i is the same as the integral over
Hence, the operator which performs the required change of reference frame and obeys the principle of coherent change of RF is unitary if and only if the states |ψ(g) form an orthonormal basis for H i or a subspace thereof. where n i k n i = n k and n i k n j i = n j k . Now we computes j = Λ 0→j (T (R 0 G s)) . s 0 = R 0 G s is: where g k i = m k i q k i is the decomposition into N P . Let us consider the elements m 0 j q 0 j for j ∈ {m, . . . , n − 1}. We have that g 0 j g 0 = n j → m 0 j q 0 j n 0 p 0 = n j . Now observe that since m 0 j and n j in N this implies that q 0 j n 0 p 0 = n ∈ N . Now there is a unique q 0 j such that this holds, and observe that p −1 0 n 0 p 0 = m . There is a unique m ∈ N such that mm = n . Therefore q 0 j = mp −1 0 . However since q 0 j ∈ P this implies m = e and therefore q 0 j = p −1 0 . Let us truncate:   which is not equal to n j k in general. For a specific example we can look at R = Z S 1 . Then s = (n 0 L + x 0 , n 1 L + x 1 , n 2 L, n 3 L) , Then T (s) = (n 0 L, n 1 L, n 2 L, n 3 L) and T (s) 2 = (n 0 − n 2 , n 1 − n 2 , 0, n 3 − n 2 )L.
For the other order we get s 0 = (0, n 1 L + x 1 − (n 0 L + x 0 ), n 2 L − (n 0 L + x 0 ), n 3 L − (n 0 L + x 0 )). Let us assume that x 1 − x 0 < 0, then T (s 0 ) = (0, n 1 − n 0 − 1, n 2 − n 0 , n 3 − n 0 )L and T (s 0 ) 2 = (−(n 2 − n 0 ), n 1 − n 0 − 1 − (n 2 − n 0 ), n 2 − n 0 − (n 2 − n 0 ), n 3 − n 0 − (n 2 − n 0 ))L which gives T (s 0 ) 2 = (n 0 −n 2 , n 1 −n 2 −1, 0, n 3 −n 2 )L and does not equal T (s) 2 . Now let us return to the case where the configuration space is R 3 but the transformation group is E + (3) ∼ = T (R 3 ) SO(3) and the case of three particles. Here E + (3) is the special Euclidean group consisting of translations followed by a rotation. Let us say that particle 0 uses a coordinate system (x 0 , y 0 , z 0 ) such that it assigns itself the state (0, 0, 0) = 0. There are infinitely many such coordinate systems corresponding to all reference frames centred on particle 0, related by rotations O ∈ SO(3). We write v 0 i = (x 0 i , y 0 i , z 0 i ) the state of particle i in a reference frame centred at 0. The coordinates corresponding to Cartesian reference frames are in one to one correspondence with group elements in T ( The description of the state of the three particles is s 0 0,1,2 = 0, v 0 1 , v 0 2 . What can particle 0 infer about the description used by particle 1, knowing only that the convention is such that particle 1 uses Cartesian coordinates placed at its position? There are infinitely many such choices of coordinates, related to the coordinates ( for any O ∈ SO(3) correspond to coordinate choices centred on particle 1. In order to describe the ignorance about O one needs a probabilistic representations of states: a state is now a measure on R 3 . The state − a is the Dirac measure δ − a and the description that particle 0 assigns to particle 1 is: where µ S 2 ,| a| is the normalised Haar measure on the sphere S 2 of radius | a| centred at the origin (in particle 1's coordinates). We observe that there is no reversible transformation from s 1,average 0,1 representing particle 0's knowledge of particle 1's description back to the initial description of particle 0. Starting from s 0 and mapping to s 2 one obtains: We leave to future work the development of a full account of changes of reference frames in this probabilistic case (for instance it is not clear that there is a well defined change s 1,av 0,1,2 → s 2,av 0,1,2 ). Here we are interested only in the single change from s 0 → s 1 .
We also note that when describing ignorance one cannot integrate over the pure states, or configurations. In the above example this would result in s 1,average 0,1 being 0, 0 which is nonsensical. This is similar to the case in quantum theory where integrating over a group action on pure states does not represent ignorance (rather it is a projection) but integrating over mixed states corresponds to ignorance. One must move to a probabilistic description of states in terms of measures on X in order to infer states of other particles for these types of cases. Alternatively one could stay at the pure state description, and instead of finding the point of view that particle 0 infers particle 1 holds we can search for what particle 0 knows holds true for all possible descriptions s 1 0,1 = O −a, 0 for all O ∈ SO(3) from the point of view of particle 1. In other words what are the invariants? Here it is straightforward to see that any f (| a|) is an invariant quantity.
Appendix E: Wigner's friend with additional reference system One could argue that introducing an additional reference system R into the box containing F and S might help to resolve the paradox of Wigner's friend. In this case, we can describe the measurement interaction from the point of view of W as follows: The reference system is not affected by the measurement. Now, we want to apply the change of reference frame to switch to the perspective of F. We apply the operator U W→F AS to the initial state: We see that, again, the state of the system is always correlated with the state of the friend. Wigner and the additional reference system end up being entangled. We conclude that introducing an additional reference system into the laboratory does not provide a resolution to the paradox.
In the formalism of [6], the reference system is not explicitly included in the state of the joint system. The equivalent state would be |Ψ BC = dx dyψ(x, y) |x B ⊗ dy |y C (F3) When changing from reference system A to reference system B, we apply the operator of [6]: The final state is then When applying the operator in Equation (F1) to the initial state in Equation (F2), we get the final state U A→B |Ψ ABC = SWAP A,B • |0 A dx dy dxdy ψ(x, y) |−x x | |x |y − x y | |y = |0 B dx dy ψ(x, y) |−x A ⊗ |y − x C (F6) Hence, we see that both operators act in the same way on the most general states of particles on the real line.