Quantum reference frames for general symmetry groups

Anne-Catherine de la Hamette1,2 and Thomas D. Galley2

1Institute for Theoretical Physics, ETH Zurich,¨ Wolfgang-Pauli-Str. 27, 8093 Zurich,¨ Switzerland 2Perimeter Institute for Theoretical Physics, 31 Caroline St. N, Waterloo, Ontario, N2L 2Y5, Canada 23 November 2020

A fully relational quantum theory necessar- 1 Introduction ily requires an account of changes of quan- tum reference frames, where quantum refer- In quantum mechanics, physical systems are implic- ence frames are quantum systems relative to itly described relative to some set of measurement de- which other systems are described. By in- vices. When writing down the quantum state of a troducing a relational formalism which iden- system of interest, say a spin-1/2 system in the state tifies coordinate systems with elements of a |↑zi, we mean that the state of the system is ‘up’ rel- symmetry group G, we define a general op- ative to a specified direction zˆ in the laboratory. In erator for reversibly changing between quan- practice, this direction will be associated to a macro- tum reference frames associated to a group G. scopic physical system in the lab. If we assume that This generalises the known operator for trans- quantum mechanics is a universal theory and there- lations and boosts to arbitrary finite and lo- fore applicable at all scales, the systems we make ref- cally compact groups, including non-Abelian erence to to describe quantum systems should eventu- groups. We show under which conditions one ally be treated quantum mechanically as well. Refer- can uniquely assign coordinate choices to phys- ence systems that are themselves treated as quantum ical systems (to form reference frames) and systems are referred to as quantum reference frames. how to reversibly transform between them, Following the success of Einstein’s theory of relativ- providing transformations between coordinate ity and its inherently relational nature, one may seek systems which are ‘in a superposition’ of other to adopt a relational approach to quantum theory as coordinate systems. We obtain the change of well. In such an approach, most physically mean- quantum reference frame from the principles ingful quantities are relational, i.e. they only take of relational physics and of coherent change of on well defined values once we agree on the reference reference frame. We prove a theorem stating system (or the observer) relative to which they are that the change of quantum reference frame described. In his papers [1,2], Rovelli suggested that consistent with these principles is unitary if quantum mechanics is a complete theory about the and only if the reference systems carry the left description of physical systems relative to other phys- and right regular representations of G. We ical systems. In his Relational Quantum Mechanics also define irreversible changes of reference (RQM) he rejected the idea of observer-independent frame for classical and quantum systems in the states of systems and values of observables. The im- case where the symmetry group G is a semi- portance of changes of reference frame in special and direct product G = N o P or a direct prod- general relativity suggests the development of an ac- uct G = N × P , providing multiple examples count of changes of quantum reference frame in RQM. of both reversible and irreversible changes of Such an account is given in the present work. quantum reference system along the way. Fi- Recently, there has been an increased interest in nally, we apply the relational formalism and analysing spatial and temporal quantum reference changes of reference frame developed in this frames and in establishing a formalism that allows work to the Wigner’s friend scenario, finding arXiv:2004.14292v3 [quant-ph] 23 Nov 2020 to switch between different perspectives [3–7]. The similar conclusions to those in relational quan- present work is partially based on these approaches tum mechanics using an explicit change of ref- which define changes of quantum reference frames for erence frame as opposed to indirect reasoning systems that transform under the translation group using measurement operators. (in space and time) and the rotation group in three di- mensional space. As opposed to other more standard approaches [8–11], this formalism stresses the lack of an external reference frame from the outset and de- fines states of subsystems relative to another subsys- Anne-Catherine de la Hamette: tem. In standard approaches to quantum reference [email protected] frames [9, 10], one often starts from a description rel- Thomas D. Galley: [email protected] ative to an external reference frame and removes any

Accepted in Quantum 2020-11-16, click title to verify. Published under CC-BY 4.0. 1 dependence on this reference frame by applying a G 2 Relational approach to quantum twirl (a group averaging over all possible configura- theory tions of the external reference frame). In some cases, one can refactor the total Hilbert space into global In the construction of a relational formalism of quan- and relational subsystems and trace out the global tum mechanics, an essential task is to write quantum degrees of freedom [9, 10, 12, 13]. The main empha- states of systems relative to a specified reference sys- sis of these standard approaches is often to obtain A tem. We introduce the following notation: |ψiB indi- the physically meaningful (or reference frame inde- cates the state of system B relative to system A. In pendent) quantities, in a similar fashion to identifying contrast to the approach of [6], we assign a Hilbert noise free subsystems in error correction. In the work space to the system whose perspective is adopted and of [3–7, 14, 15] however emphasis is given on the rela- assign to it the trivial state, corresponding to the tional nature of the description (always starting from identity element of the group. Hence, by convention, a state that is given from the viewpoint of one of the system A is in a default ‘zero-state’ relative to itself. subsystems) and the main object of study is the re- Once we introduce the notion of symmetry groups and lation between different accounts. We make a similar how they enter into the formalism, we will see that emphasis in the present work. We abstract the for- this default zero-state corresponds to the identity ele- malism of [6] and introduce an approach which makes ment of the group that describes the transformations heavy use of the inherently group theoretic nature of of the system. Thus, to be more precise, one can write quantum reference frames. This allows us to gener- A A alise the known results beyond the translation and ro- |0iA ⊗ |ψiB . (1) tation groups to arbitrary finite and locally compact groups (including non-Abelian groups). 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 per- In Section2 we outline the relational approach to spectival approach of [16]). This description does not quantum theory embraced in the present work as well make use of any external abstract reference frame nor as give a simple example of a change of reference frame does it assume the existence of absolute, observer- for classical bits and an example of a change of quan- dependent values of physical observables. We ob- tum reference frame for qubits. In Section3 we define serve that since system A can only ever assign itself the notion of a reference frame in terms of reference a single state there are no state self-assignment para- systems and coordinate systems, as well as give a full doxes [17, 18]. account of active and passive transformations as left A natural question to address on the relational ap- and right regular group actions. Combining these we proach to quantum theory is how to change reference define changes of reference frame under a group G systems. Namely if the state of B relative to A is ∼ A A A B for classical systems with configuration space X = G. |ψiB = |0iA ⊗ |ψiB, what is the state |ψiA of A rela- In Section4 we extend the classical change of refer- tive to B? This is the problem which will be addressed ence frame to quantum systems L2(G) following the in the present work. principle of coherent change of reference system; and Before introducing the general framework we will define a general unitary operator which implements be using, we give two simple examples of changes of this change of reference system. We prove a theorem reference frame for relational states. The first is classi- stating that only systems carrying a regular repre- cal and the second its quantum generalisation. These sentation of G can serve as reference frame, subject should hopefully provide the reader with an intuitive to the principle of coherent change of reference sys- picture of the general mechanisms at play. tem. Following this we extend the change of quantum Example 1 (Z change of classical reference frame). reference frame operator between L2(G) systems de- 2 Let us consider the case where systems can be in two scribing systems which do not carry the right regular states ↑ or ↓. Every system considers themselves to be representation of G. In Section5 we define irreversible in the state ↑ (for example an observer free floating changes of reference frames for groups G = N P o in empty space would always consider the up direc- and G = N × P via a truncation procedure. In Sec- tion to be aligned from their feet to their head). Con- tion6 we extend this change of reference frame to sider classical systems where the state relative to A is quantum reference frames using the principle of co- ↑A↑A↓A. Since A sees B in the state ↑ relative to itself, herent change of reference system once more. In Sec- A B C B also sees A in the state ↑ relative to itself. The state tion7 we apply the tools developed in the preceding relative to B is ↑B↑B↓B. If the state relative to A was sections to the Wigner’s friend thought experiment, B A C instead ↑A↓A↓A then since A views B in the ↓ state, providing an explicit change of reference frame from A B C this implies that B views things ‘upside down’ relative Wigner’s description to the friend’s. We discuss re- to A. The change of perspective would give ↑B↓B↑B. lated work in Section8 and discuss implications of the B A C present work as well as suggestions for future work. In the next example we give a quantum generali- In Section9 we give some concluding remarks. sation of the Z2 change of reference frame. This is

Accepted in Quantum 2020-11-16, click title to verify. Published under CC-BY 4.0. 2 a specific instance of the general changes of quantum We begin by a simple example which illustrates reference frame defined in this work. changes of reference frames and the use of group ele- ments for relative coordinates. Example 2 (Z2 change of quantum reference frame). Let us consider the case with quantum systems C2 with Example 3 (Three particles on a line). Consider basis {|↑i , |↓i}. Every system considers themselves to three classical particles A, B and C on a line, with be in the state |↑i. By embedding the classical sce- state s = (xA, xB, xC) in some Cartesian coordinate nario above with the map ↑ 7→ |↑i and ↓ 7→ |↓i we system (here we omit the velocities since we are just can reconstruct the classical example: if the state rel- interested in translations in space). The coordinate A A A A system x0 such that x0 = 0 is said to be associated to ative to A is |ψiBC = |↑iA |↑iB |↓iC then the state rel- A B B B B A. In this coordinate system the particles have state ative to B is |ψiAC = |↑iB |↑iA |↓iC. If the state was 0 0 0 A A A A s = (xA = 0, xB = xB −xA, xC = xC −xA). We observe |φiBC = |↑iA |↓iB |↓iC then the change of perspective 0 0 0 B B B B here that the relative coordinates (to A) xA, xB and xC would give |φiAC = |↑iB |↓iA |↑iC. uniquely identify the translation which maps system A Let us move to the quantum case with an ini- to systems A, B and C. Namely the relative distance A A  A A A 0 tial state |τiBC = |↑iA |↑iB + |↓iB |↓iC. What is xB = xB−xA is the distance needed to translate A to B. The relative coordinates x0 , x0 and x0 correspond to the state |τiB ? First let us observe that |τiA = A B C AC BC the symmetry group transformations relating A, B and |ψiA + |φiA , and let us assume that changes of BC BC C to A. If we label a translation of distance d by tL(d), quantum reference frame are coherent (they observe B where tL(d)x = d + x, we have the state relative to A the superposition principle). Then the state |τi = A AC as s = (tL(0), tL(xB − xA), tL(xC − xA)). The state B  B B B B B |↑iB |↑iA |↓iC + |↓iA |↑iC which is an entangled state relative to B is s = (tL(xA −xB), tL(0), tL(xC −xB)). of A and C. These two relative states are themselves related by B A the transformation s = s − (xB − xA). We de- The above example made use of the two guiding fine the right regular action of the translation group principles of this work: the principle of relational TR(d) = x − d. The change of reference frame A to physics and the principle of coherent change of ref- B is given by the right regular action TR(xB − xA) of erence system. These are defined in Section 4.3. the group element xB − xA mapping A to B. Whenever we use phrases such as ‘from the view- In the above example the configuration space R and point of’ or ‘from the perspective of’, we simply mean the symmetry group T = (R, +) acting on it are equiv- ‘relative to’. Although these expressions might imply alent as manifolds. This equivalence is essential for that the state |ψiA indicates how system A perceives B the existence of a well defined reversible change of system B, we do not make this interpretation here. quantum reference frame. In Sections5 and6 we System A acts as the observer in this description but study scenarios where this is no longer the case, and we should note that there is nothing special about an the changes of reference frame are irreversible. Since observer system. No interpretation is made as to what the results in this paper also apply to finite groups we the system sees. Rather a change of reference system cover a simple example. A → B is a change of description from one where A is at the origin to one where B is at the origin. Example 4 (Z2). Let us consider systems with con- figuration space X = {↑, ↓}. The symmetry group G = {I,F } = Z2 consisting of the identity I(↑) = ↑ 3 Classical changes of reference and the flip F (↑) = ↓ is the symmetry group of X.A state of four systems of the form s =↑, ↑, ↓, ↓ can be frames associated to symmetry groups expressed as s = I(↑),I(↑),F (↑),F (↑). By consider- ing ↑ as the ‘coordinate system’ we have that the state 3.1 Reference systems, coordinate choices and s has coefficients (I,I,F,F ). In the ‘coordinate sys- changes of reference frame tem’ ↓ the state s would have coefficients (F,F,I,I).

A coordinate system is a purely mathematical object, We observe that in the above examples group el- and need not in general be associated to a physical ements of the global symmetry groups serve as rel- system. A reference frame consists of a physical sys- ative coordinates. In the next subsection we make tem (known as a reference system), and a choice of this link more explicit. We also note the importance coordinates such that the reference system is at the of the one to one correspondence between states and origin in that coordinate system. For full definitions coordinate transformations. We observe that there we refer the reader to AppendixA. In this section is always some conventionality in changes of coordi- we define changes of reference frames for classical sys- nates: one considers only translations on R for in- tems where the configuration space is itself a group stance, and not all diffeomorphisms of R as relating G. In Section5 we will consider cases where this is different coordinates. In AppendixA we give explicit no longer holds. definitions of coordinate systems on a manifold X,

Accepted in Quantum 2020-11-16, click title to verify. Published under CC-BY 4.0. 3 emphasising that a coordinate system is different to Consider the case where gx0 = x1 and a passive trans- a coordinate chart (typically used in general relativ- formation h on 0 is applied while 1 is left unchanged 0 ity), a distinction made in [19]. Roughly speaking a (at position x1). We have x0 7→ x0 = hx0. Then the 0 coordinate system on a manifold X is an isomorphism relative location of 1 to 0 is k where kx0 = x1. Sub- 0 f : X → Y (where Y a known mathematical object stituting in gx0 = x1 and hx0 = x0 gives khx0 = gx0 used to describe X), whereas a coordinate chart is a implying that that kh = g, and therefore k = gh−1 map from the mathematical object Y to the physi- (where we remember that since X =∼ G there is al- cal object X which need not be an isomorphism (for ways a unique g ∈ G mapping a pair of points in n −1 −1 0 instance multiple charts R are used to describe a X). Writing in full gh hx0 = gh x0 = x1 and so curved n-dimensional manifold X in general relativ- the relative location of 1 relative to 0 is now gh−1. ity but there is no isomorphism from X to Rn). A passive transformation h on 0 corresponds to the right regular action of h on the relative location g: g 7→ gh−1. 3.2 General treatment of reversible changes of reference frame The left regular action and right regular action on G =∼ X are defined as follows: Let us extract the general features of the above sce- nario which allow for well defined reference frames φL(g, x) = gx , (2) and reversible changes of reference frame. Consider a −1 φR(g, x) = xg . (3) configuration space X (which is typically a set with a manifold structure) and a group G acting on X such Both are defined using the group multiplication, that there is a unique transformation g ∈ G relat- where x ∈ G. These two actions naturally commute, ing any pair of points (the action is transitive and and hence X =∼ G carries an action of G × G, with ∼ free). This implies X = G (as sets/manifolds), and one factor typically being understood as the active ∼ the action of G on X = G is the group multiplica- and the second as the passive transformations [21]. tion on itself: G × G → G. This space is a principle Although φR acts ‘to the right’, it is a left group ac- homogeneous space for G, sometimes called a G tor- tion: φ(gh, x) = x(gh)−1 = xh−1g−1 = φ(h, x)g−1 = sor. We assume G locally compact and thus equipped φ(g, φ(h, x)). Here we take φL as active and φR as with a left Haar measure, denoted dg. Many groups passive.1 of interest in physics, such as the Poincar´egroup, A given state of n systems is s = (x0, x1, ..., xn−1), the symmetric group, SU(d) and SO(d) are locally where we omit the velocities x˙i since we are defining compact (compact and finite groups are instances of changes of reference frame for the ‘translation’ group locally compact groups). One exception is the dif- G on X =∼ G. This can be expressed as: feomorphism group of some space-time manifold M, which in general is not locally compact. We observe 0 0 0 0
s = g0x0, g1x0, g2x0, ..., gn−1x0 , (4) that X =∼ G follows from the requirement that there i i exists a unique transformation g ∈ G relating any two where gj is the unique g ∈ G such that gjxi = xj, i points in X. Note that we will later go beyond such and e = gi the identity element. We observe that perfect reference frames and consider cases for which j i i j −1 i 0 gkgj = gk and (gi ) = gj. Then the state s of the the configuration space X differs from the group G. n systems relative to system 0 is: We use the following example from [20] to introduce 0 0 0 0 0
active and passive transformations on G torsors. s = g0, g1, g2, ..., gn−1 . (5)

Example 5 (Single observer and system on X =∼ G). The state relative to the system i is: ∼ Consider an observer 0 at location x0 on X = G and i i i i i
an object 1 at location x1. Then the unique transfor- s = g0, g1, g2, ..., gn−1 . (6) mation g such that gx0 = x1 is the relative location of 1 relative to 0. We observe that we can also describe the state relative An active transformation is a transformation on the to hypothetical systems (i.e. relative to a point x ∈ X object 1. A transformation h on the object 1 is given which is not occupied by a system). For instance in 0 the above consider a point xn ∈ X such that xi 6= by the left regular action x1 7→ x1 = hx1. The relative 0 xn ∀i ∈ {0, ..., n − 1}. Then we can write: location is now k where kx0 = x1. Using gx0 = x1 and hx = x0 we find that k = hg: hgx = hx = x0 . 1 1 0 1 1 sn = gn, gn, gn, ..., gn
. (7) Therefore an active transformation by h corresponds 0 1 2 n−1 to the left regular action of h on the relative location 1 g: g 7→ hg. Active and passive transformations are typically defined as either left actions on different spaces (states and coordinates) A passive transformation is a transformation on the or a left and a right action on the same space (typically coordi- ∼ observer x0 7→ hx0. This induces a transformation on nates). In this case (X = G) they can be defined as left actions the relative location of 1 to 0 which we now outline. on the same space.

Accepted in Quantum 2020-11-16, click title to verify. Published under CC-BY 4.0. 4 As in the examples given above, we see that a relative space freely generated by the elements of G, i.e. for state si is given by all the symmetry transformations which the elements of G form a basis. i gj from system i to system j for all j ∈ {0, ..., n − 1}. In the cases where it is clear which system is the There is a unique relative state si, which is such that reference system we sometimes omit the top label for A A A particle i is in state e. ease of reading. For instance the state |0iA |x1iB |x2iC A change of reference frame from system 0 to sys- is written as |0iA |x1iB |x2iC. tem i is a map s0 → si. Let us extend the left and right regular actions of G on itself to states: 4.2 The example of L2(R) ⊗ L2(R) ⊗ L2(R) φL(g, s) = (gx0, gx1, ..., gxn−1) , (8) φ (g, s) = (x g−1, x g−1, ..., x g−1). (9) Let us first rephrase the known case of the translation R 0 1 n−1 group acting on three particles on the line [6] in the The transformation s0 → si is given by the right formalism outlined above. 0 regular action of gi : Take the translation group T = (R, +) and three 2 0 0 i 0 i 0 i 0 i
systems A, B and C whose joint state space is L (R)⊗ φR(g , s ) = eg , g g , g g , ..., g g i 0 1 0 2 0 n−1 0 L2(R) ⊗ L2(R). Let us for instance consider the state i i i i
i = g0, g1, g2, ..., gn−1 = s . (10)

We observe that this transformation cannot be |0iA |x1iB |x3iC , (11) achieved using the left regular action: there is no el- 0 i ements g ∈ G such that φL(g, s ) = s (unless G is which is the state of three perfectly localised sys- Abelian). The transformation s0 → si is a passive tems, described using a coordinate system centred transformation. on system A. In standard quantum mechanics, when changing from a classical, highly localised reference 1 0 1 g2 =g2 g0 frame at the position of A to another classical refer- 1 2 ence frame localised at B translated by an amount x1, g2=g0g2 0 1 1 0 2 ˆ g1 g0 one simply applies the translation operator T (−x1) = ix1(ˆpA+ˆpB+ˆpC) 1 0 e to the state of the three systems, where g0 g2 pˆA is the momentum operator for system A and simi- 0 larly for pˆB, pˆC and systems B and C. This shifts the state to: Figure 1: Diagram capturing the relational states between three systems. Each system i assigns the relative state along |−x1iA |0iB |x3 − x1iC . (12) the arrow point from i to j to system j (and the identity A A 0 0 In the previous language we have gB = x1 and gC = to themselves). For instance system 0 assigns state e, g1 , g2 . 0 1 0 1 1 x . The action of Tˆ(−x ) corresponds to the right By the right regular action of g1 , we obtain g0 , e, g2 g0 = g2 3 1 A which is the relative state assigned by system 1. action of gB = x1. The next step is to begin with a state of the follow- ing form:

4 Quantum reference frames associ- 1 |0i √ (|x1i + |x2i)B |x3i , (13) ated to symmetry groups A 2 C

We begin this section by reviewing changes of quan- which is described by a coordinate system localised at tum reference frame for three particles on the line. A. What is the change of reference frame A → B in We then define a quantum change of reference frame this case? How can one describe classical coordinates operator for n identical systems L2(G) for arbitrary which assign state |0iB, when B is not localised rel- G. This generalises the change of reference frame ative to A? A standard translation of all states will in [6] beyond one parameter subgroups of the Galilean not work. group. Furthermore we show that it is only the L2(G) Following the reasoning presented in [6] we assume system described so far for which a unitary reversible that the change of perspective obeys the principle of change of reference system is possible. Finally, we de- superposition. Namely the state of Equation (13) fine a change of reference frame operator for m iden- is an equally weighted superposition of the states 2 tical L (G) systems serving as reference frames and |0iA |x1iB |x3iC and |0iA |x2iB |x3iC. The change of n − m systems which are not. reference frame for each of these states individually is obtained by translating by −x1 and −x2 respectively. 4.1 Comment on finite groups and notation Assuming that the superposition principle applies to changes of reference systems, the state described in All our results apply for finite groups. In this case coordinates ‘localised’ at system B is just the super- L2(G) should be replaced by C[G] =∼ C|G| and inte- position of the classical states obtained by translation R P grals g∈G |gihg| dg by i |giihgi|. C[G] is the vector by −x1 and −x2. This leads to the following state of

Accepted in Quantum 2020-11-16, click title to verify. Published under CC-BY 4.0. 5 the joint system from the viewpoint of B: L2(G) is the space of square integrable functions G → C. 1 The left and right action of G onto itself induces |0iB √ (|−x1iA |x3 − x1iC + |−x2iA |x3 − x2iC). 2 the left regular and right regular representation of G (14) on each Hi. For a given Hi this representation acts When changing between the viewpoints of quantum on the basis {|gi} as: systems, we apply a weighted translation of the states of systems, dependent on the state of the new refer- UL(g2): |g1i 7→ |g2g1i , (18) ence frame whose viewpoint we are adopting. For the U (g ): |g i 7→ g g−1 . (19) state given above, this means applying a translation R 2 1 1 2 for the state of B being |x1i and one for it being |x2i. An arbitrary basis state of the n systems is:

|ψi = |g0i0 |g1i1 ... |gn−1in−1 . (20) Following the classical case, the choice of coordinates on G associated to H0 is:

0 0 0 |ψi = |ei0 g1 1 ... gn−1 n−1 , (21)

i where gjgi = gj. For general Hi it is:

i i i i |ψi = g0 0 g1 1 ... |eii ... gn−1 n−1 . (22)

Figure 2: Example for translation group: In the upper sub- The change of coordinate system |ψi0 → |ψii is figure, the state of the three systems is given from the per- given by U (g0)⊗n, when considering orthogonal ba- spective of system A. The lower subfigure shows the state R i relative to B. sis states alone. Let us observe that the left regular represen- tation on the space of wave functions acts as The states of A and C become entangled relative ψ(x) 7→ ψ(g−1x) and the right regular rep- to B. We see that to perform this change of refer- resentation as ψ(x) 7→ ψ(xg). This follows ence frame, the state of A is mapped to the inverse from R ψ(x) |gxi dx = R ψ(g−1x) |xi dx and of the group element associated with the old state of x∈G x∈G R −1 R B. Also for each state of B, the state of C is shifted x∈G ψ(x) xg dx = x∈G ψ(xg) |xi dx. We note 2 respectively. Hence, for the translation group on the that for Lie groups G the objects |gi are not in L (G) real line, the reference frame change operator is and one should typically prefer the representation act- ing on the wavefunctions. In the following however we A→B U =SWAPA,B◦ consider the representation acting on the elements |gi Z in order to describe the continuous and discrete case dxidxj |−xiihxi|B ⊗ 1A ⊗ |xj − xiihxj|C . simultaneously. (15) Unlike some approaches to relational quantum dy- namics [3, 14, 15] we do not assign a global state This operator performs exactly the same reference |ψi ∈ H and then work out its expression relative to frame change as the operator given in [6]: a certain system. Rather, we begin from a state rela- tive to a system and define changes of reference frame A→B i/ xˆB pˆC Sˆ = PˆABe ~ , (16) to other systems. We formalise this in the following principle: where PˆAB is the so-called parity-swap operator. It Principle 1 (Relational physics). Given n systems, acts as PˆABψB(x) = ψA(−x). The proof of this is given in AppendixF. 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. 4.3 n identical systems L2(G) We observe that this principle does not preclude the existence of a well defined global state of the n ∼ Consider a configuration space X = G and n sys- systems. ∼ 2 tems each with associated Hilbert space Hi = L (G) A superposition state (relative to 0 in the G product |G| for G continuous (or C for G finite): basis) is of the form:

2 G → L (G) , 0 0 0 0 0 |ψi = |ei0 g1 1 ... gn−1 n−1 + |ei0 h1 1 ... hn−1 n−1 . gi 7→ |gii . (17) (23)

Accepted in Quantum 2020-11-16, click title to verify. Published under CC-BY 4.0. 6 Since the state is in general not a basis state there is no a priori well defined change given by an operator of 0 ⊗n the form UR(gi ) . However, following the example of [6] one can define a coherent change of reference frame operator. We explicitly state this as a principle: Principle 2 (Coherent change of reference system). If |ψi0 7→ |ψii and |φi0 7→ |φii then α |ψi0 + β |φi0 7→ α |ψii + β |φii, α, β ∈ C. This implies that |ψi0 defined above changes to:

i i 0 i 0 i |ψi = |eii g0 0 g1g0 1 ... gn−1g0 n−1 i 0 i 0 i + |eii h0 0 h1h0 1 ... hn−1h0 n−1 . (24) Figure 3: Basis states |0i , |θ1i and |θ2i of the state space The operator which implements the coherent change for L2(U(1)). of reference systems 0 → i is:

0→i U =SWAP0,i◦ Z where |ei = |0i is the state associated to the identity i 0 0 ⊗n−2 0 element of U(1). As a specific example, take the state g0 gi ⊗ 10 ⊗ UR(gi ) dgi . 0 i gi ∈G (25) ! r1 r2 |0i ⊗ |θ1i + |θ2i ⊗ |θ3i . (28) The following lemmas are proven in AppendixB. A 3 3 C B Lemma 1. U 0→i is unitary. Relative to particle B, the state assigned to the joint † Lemma 2. U 0→i
= U i→0 system would be

i→j k→i k→j Lemma 3. U U = U ! r1 r2 |0iB |−θ1iA ⊗ |θ3 − θ1iC + |−θ2iA ⊗ |θ3 − θ2iC . 4.3.1 Change of reference frame for observables 3 3 (29) The change of reference frame operator also allows us to transform between observables. Namely if system We see that the state of B is mapped to the state 0 describes an observable of systems 1, ..., n − 1 as corresponding to the inverse group element assigned 0 0 to the old state of B and the state of C is shifted Z = 10 ⊗ Z then system i describes 0,1,...,n−1 1,...,n−1 respectively. In the end, the labels of A and B are the observable as Zi = U 0→iZ0 U i→0. 0,1,...,n−1 0,1,...,n−1 swapped. The operator that performs this reference frame change is 4.3.2 L2(U(1)) ⊗ L2(U(1)) ⊗ L2(U(1))

A→B To illustrate the changes of reference frame described U =SWAPA,B◦ previously we will give an example. Let us consider Z 0 0 0 the symmetry group U(1) and three particles A, B, C dθdθ |−θihθ|B ⊗ 1A ⊗ |θ − θihθ |C 2 on a circle with associated Hilbert space L (U(1)) ⊗ Z 2 2 0 L (U(1))⊗L (U(1)). In this case, the map from group =SWAPA,B ◦ dθdθ |−θihθ|B ⊗ 1A ⊗ UR(θ)C. elements to elements of the Hilbert space is (30) U(1) → L2(U(1)) ,

θi 7→ |θii , (26) with θi ∈ [0, 2π[ and hθi|θji = δ(θi − θj). The states {|θii | θi ∈ [0, 2π[} are states at all angular positions of the unit circle and form a basis of the Hilbert space L2(U(1)). A system consisting of three particles on a circle could for instance be in the product state π |0iA 2 B |πiC relative to A. An arbitrary state of the joint system relative to particle A is given by Z Figure 4: Example of three particles on a circle. |0iA ⊗ dθidθj ψ(θi, θj) |θiiB ⊗ |θjiC , (27)

Accepted in Quantum 2020-11-16, click title to verify. Published under CC-BY 4.0. 7 4.4 Unitarity of change of reference frame op- ear, the following should hold: erator 0 |000iABC 7→ |000iABC θ = θ = 0 , The change of reference frame defined previously is 0 |001iABC 7→ |001iABC θ = 0, θ = π , highly constrained: it applies only to systems L2(G) |010i 7→ |101i θ = π, θ0 = 0 , with symmetry group G. One could ask whether ABC ABC 0 one could define similar changes of reference frame |011iABC 7→ − |100iABC θ = θ = π. (35) for systems H=∼ 6 L2(G) with states |ψ(g)i carry- When comparing the coefficients in the map (34), ing two representations: U (h) |ψ(g)i = |ψ(hg)i and L one sees that the reference frame change cannot be U (h) |ψ(g)i = ψ(gh−1) . R linear. This means that the operator describing the First consider the symmetry group U(1). Our change from one rebit reference system to another one results show that for a change of reference frame is non-linear. As this non-linearity causes issues con- to recreate our classical intuitions one needs sys- cerning the invariance of probabilities under reference tems L2(U(1)) which carry the right regular repre- frame change we conclude that rebits cannot serve as sentation. However one may wonder whether one reference frames that allow to reversibly transform be- could use qubits with states along the X − Y plane tween each other. |θi = cos(θ/2) |0i + sin(θ/2) |1i as reference systems which transform in a manner which obeys the classical Given n classical systems with configuration space change of reference frame. X =∼ G acted on by a symmetry group G we have We first provide an example to show that this shown how to define states relative to these systems, breaks linearity of the change of reference frame op- and to transform between them using the left and erator for the case of rebits before proving a general right regular action of G on X. result. The case L2(G) is a very specific ‘encoding’ of G into a quantum system. It is a natural choice, in Example 6 (Three qubits with U(1) group action). that the classical states are embedded into orthogo- Consider three qubits with states restricted to real val- nal states of the quantum system. However one could ued superpositions: |θi = cos(θ/2) |0i + sin(θ/2) |1i have an injection G → H, with g 7→ |ψ(g)i such that (sometimes known as rebits). The space of pure states the states |ψ(g)i are not all mutually orthogonal and of the three systems is U(1) ⊗ U(1) ⊗ U(1). We apply ask whether a change of reference system can be de- our classical intuition of how a reference frame change fined. We require H to carry two unitary representa- should act. Let the initial state of the three systems tions UL and UR, corresponding to active and passive relative to A be transformations, such that UL(h) |ψ(g)i = |ψ(hg)i −1 0 and UR(h) |ψ(g)i = ψ(gh ) , where |ψ(g)i = |ψ(0)iA ⊗ |ψ(θ)iB ⊗ |ψ(θ )iC . (31) |ψ(h)i ↔ g = h. Although one would naturally desire them to commute (since active and passive transfor- The map ψ takes the group element θ of U(1) to the mations as usually defined act on different spaces and state in the two-dimensional Hilbert space: therefore trivially commute), we do not impose this here. The following theorem tells us that the change ψ : U(1) → H of reference frame which acts as expected on prod- θ 7→ cos(θ/2) |0i + sin(θ/2) |1i . (32) uct states |ψ(e)i ψ(gi ) ... ψ(gi ) ... ψ(gi ) i 0 0 j j n−1 n−1 and obeys the principle of coherent change of reference We want this state to be mapped to the final state frame is unitary exactly if the states |ψ(g)i it acts on relative to B: form an orthonormal basis of the Hilbert space.

0 |ψ(−θ)iA ⊗ |ψ(0)iB ⊗ |ψ(θ − θ)iC . (33) Theorem 1. Take n identical systems with associated Hilbert spaces Hi each carrying two representations of This corresponds to our intuition of what should hap- G: UL and UR such that UL(h) |ψ(g)i = |ψ(hg)i and −1 pen when one changes from the viewpoint of A to the UR(h) |ψ(g)i = ψ(gh ) , where |ψ(g)i = |ψ(h)i ↔ viewpoint of B. Writing out this transformation in the g = h. Then any operator U which performs the rebit basis corresponds to the map: change |ψ(e)i ψ(gi ) ... ψ(gi ) ... ψ(gi ) 7→ i 0 0 j j n−1 n−1 E E E j j j |ψ(e)ij ψ(g0) ... ψ(gi ) ... ψ(gn−1) and |0iA ⊗ (cos(θ/2) |0i + sin(θ/2) |1i)B 0 i n−1 0 0 obeys the principle of coherent change of reference ⊗ (cos(θ /2) |0i + sin(θ /2) |1i)C system is unitary if and only if the representations U 7→(cos(−θ/2) |0i + sin(−θ/2) |1i) ⊗ |0i L A B and U are the left and right regular representations 0 0 R ⊗ (cos((θ − θ)/2) |0i + sin((θ − θ)/2) |1i)C. (34) acting on states |ψ(g)i which form an orthonormal basis of Hi (or a subspace thereof). On the other hand, considering the basis states of the joint Hilbert spaces and assuming the map is lin- The proof can be found in AppendixC.

Accepted in Quantum 2020-11-16, click title to verify. Published under CC-BY 4.0. 8 4.5 m L2(G) systems describing n−m systems As a specific example, consider the state π E Let us consider the case where reference systems |0iA |πiB ψ( ) (43) L2(G) describe systems of a different type. The total 2 C 2 ⊗m ⊗n−m Hilbert space is L (G) ⊗ H where for sim- π 1 relative to system A, where ψ( ) = √ (|0i + |1i). plicity we have assumed the n − m systems to be of 2 2 the same type (but not L2(G)). The systems H are From the viewpoint of system B, the state is such that there exists an injection φ: π E 1 √ |0iB |πiA ψ(− )) = |0iB |πiA (|0i − |1i)C. φ : G → H , 2 C 2 (44) g 7→ |ψ(g)i , (36) and two representations VL and VR such that: 4.6 Changes of reference frame for arbitrary

VL(g) |ψ(h)i = |ψ(gh)i , (37) identical systems −1 VR(g) |ψ(h)i = ψ(hg ) . (38) The above treatment shows that for any group G one can define a change of quantum reference frame be- To change from reference system 0 to i, where both tween n identical systems. However given n identical systems are assumed to be of the type L2(G), we ap- systems can one always find a group allowing for a re- ply the operator: versible change of quantum reference frame? Namely 0→i U = SWAP0,i◦ for a configuration space X, can one always find a bi- Z nary operation turning it into a group G ∼ X? In the i 0 0 ⊗m−2 0 ⊗n−m 0 = g g ⊗ 1 ⊗ U (g ) ⊗ V (g ) dg , d 0 i i 0 R i R i i case of finite systems C one can pick an orthonormal g0∈G i basis |xi, x ∈ {0, ..., d−1} and choose the cyclic group (39) Zd acting on {0, ..., d − 1}. In the case where X is a where UR is the right regular representation acting on countable set one has the group Z. In the case where the first m L2(G) systems. X is uncountable, the existence of a group G such We observe that not all systems H which carry a that G =∼ X is equivalent to the axiom of choice [22]. representation of G will be such that there exists an We observe that if X has some additional structure injective map φ : g 7→ |ψ(g)i. For instance the qubit (such as being a manifold), then one may not be able carries a representation of SU(2) but there is no in- to find a group which is isomorphic as a manifold. jection of φ : SU(2) → PC2 (where here we emphasise that the pure states of a C2 system form PC2 the pro- jective space of rays). Observe that for U(1) there is 5 Irreversible changes of classical ref- an injection φ : U(1) → PC2. We explore the exam- erence frame ple of two L2(U(1)) systems describing a system C2 carrying a representation of U(1). In some cases one may not have access to a refer- ence system which can distinguish all elements of the 4.5.1 L2(U(1)) ⊗ L2(U(1)) ⊗ C2 symmetry group. Consider once more the case of par- ticles on R acted on by the translation group. Given Let us adapt the previous example of three particles a ruler with a set of marks corresponding only to on a circle to the case in which the third system is the subset of integers (i.e. with configuration space a qubit H =∼ C2 giving a total Hilbert space of the C X =∼ Z), one would not be able to distinguish all joint system L2(U(1)) ⊗ L2(U(1)) ⊗ C2. H carries a C possible different configurations of the particles and representation of U(1) and an injection φ : U(1) → by extension all possible translations. Such imperfect C2. The representation V is given by: R reference frames, with configuration space X which is a coarse-graining of the group G, will lead to irre-   cos(−θ/2) − sin(−θ/2) versible changes of reference frame as we will see in VR(θ) = (40) sin(−θ/2) cos(−θ/2) this section. in the {|0i , |1i} basis and acts by matrix multipli- For a given configuration space X all changes of co- cation from the left. The injection is a map ordinates are related by a transformation g ∈ G. In the case X =∼ G there is a one to one correspondence 2 φ : U(1) → C , between points in X and coordinate systems. As such θ 7→ cos(θ/2) |0i + sin(θ/2) |1i . (41) one can identify coordinate systems as systems with configuration space X. Namely if system i is in state The operator that maps the state relative to A to xi ∈ X then one assigns it the unique coordinate sys- the state relative to B is 0 Z tem xi which maps xi 7→ 0. However in situations A→B such as the one described previously one has a sym- U = SWAPA,B ◦ dθ |−θihθ|B ⊗ 1A ⊗ VR(θ)C. metry group G which is larger than X and there is (42) no unique element in G mapping a point xi to 0. We

Accepted in Quantum 2020-11-16, click title to verify. Published under CC-BY 4.0. 9 z y

z00 z0 y0

y0 y 00 z x y00

x x y z 00 x0 0 x0

Figure 5: Reference frames associated to the same point in z00 R3. x00 Figure 6: For each point in R3 a representative member of all reference frames centred at the point is chosen. Here the consider an explicit example of this in the following representative member is chosen so that each representative before describing the general case. member has the same orientation. This ensures that the closure of the set of transformations relating the different + ∼ 3 3 5.1 E (3) = R o SO(3) reference frames is T (R ) and not a larger group. Let us consider n particles in R3, with each parti- cle i having state (xi, yi, zi) expressed in Cartesian can assign a unique coordinate system to every state coordinates (x, y, z). The set of Cartesian coordi- x ∈ X =∼ E+(3) of a solid body. An example of a nates is acted on by the Euclidean group E+(3) = solid body would be three physical orthogonal axes in R3 o SO(3). A choice of coordinates (x0, y0, z0) such R3 labelled 1, 2 and 3. For a given state x of these 0 0 0 that (xi, yi, zi) = (0, 0, 0) is said to be associated to three physical axes one can associate the coordinate particle i if and only if it is the unique set of coor- system which assigns +x, +y and +z to the axes 1, 2 dinates with this property. There are infinitely many and 3. Using this approach would allow us to make such coordinate choices (for instance all coordinate use of the results of the previous section. 0 0 0 systems which are rotated relative to (x , y , z ) will However one could also keep the reference systems also assign state (0, 0, 0) to particle i). In this case as having configuration space X but rather assign to there is no obvious unique manner of associating a each state x the equivalence class of coordinate sys- coordinate system to a particle. tems centred on x. One can either choose a represen- 3 All Cartesian coordinate systems for R are related tative member of the equivalence class (in the above + + by an element g ∈ E (3) where E (3) is the Eu- case one can fix all coordinate systems to have a given + clidean group. The action of E (3) on the set of orientation as in Figure6) or one could average over Cartesian coordinates is transitive and free. To put the possible elements of the equivalence class. 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 or- 5.1.2 Representative element of each equivalence class thogonal axes located at some point r ∈ R3 with a In the case where all the reference systems have con- given orientation. These are all related to the Carte- figuration space R3, it makes sense to assign to each sian coordinates at (0, 0, 0) in a given orientation by a point x ∈ R3 a unique coordinate system centred on rotation followed by a translation. We cannot assign a that point (from the equivalence class of coordinate unique coordinate system to each point in R3. For in- systems centred on that point). stance take a coordinate system centred at the origin; then any other coordinate system obtained by rota- Take K to be the subset of transformations which tion about the origin will also assign the state (0, 0, 0) relates these coordinate systems. We require that K is to the origin. See Figure5. a group in order for us to have a well defined change of coordinate system. If K is not a group, then by com- posing different elements in K we can obtain a group 5.1.1 Enlarging the space of states of the reference sys- G0 (which is larger than K as a set). This symmetry tems group will take coordinate systems we have selected There are multiple ways of addressing this issue. One to coordinate systems which we have not chosen. can say that systems with configuration spaces R3 (i.e. In order for all representative members (i.e. the particles) are not good reference systems for E+(3). coordinate system we chose to be associated to each Rather one should choose systems with a larger con- point) to be related by a group K ⊂ E+(3) and for the figuration space. This is what is typically done, where representative members to be closed under the action we choose solid bodies in R3 as reference systems. of K one can choose them to all have the same ori- Since solid bodies have an orientation (unlike points), entation (i.e. be related by just translations). In this which is to say that rotating a solid body changes case the symmetry group relating coordinate choices its state, they have configuration space E+(3). One becomes K =∼ R3 once more.

Accepted in Quantum 2020-11-16, click title to verify. Published under CC-BY 4.0. 10 5.2 G = N o P and G = N × P : truncation Combining these gives: −1 −1 0 0 Let us consider the case where there are systems with njp0 n0 = nj pj . configuration space G and systems with configuration Now let us introduce an identity p p−1 on the RHS: space N, where G = N o P or G = N × P . In both 0 0 cases N is normal, and for every g ∈ G there is a −1 −1 −1 0 0 njp0 n0 p0p0 = nj pj , unique n ∈ N and p ∈ P such that g = np. −1 The configuration space N is embedded in G via an and observe that gng ∈ N for all g ∈ G, which implies that p−1n−1p ∈ N, in turn implying that embedding map E : N → G, E : n 7→ npC for some 0 0 0 n p−1n−1p ∈ N. Since the decomposition of g0 into constant pC , where the choice of pC is conventional j 0 0 0 j 0 0 0 −1 −1 and is typically chosen to be the identity. For a choice nj pj is unique, this implies that nj = njp0 n0 p0 0 −1 pC , the points npC ∀n ∈ N are related by transfor- and pj = p0 for all j ∈ {k, . . . , l − 1}. Therefore mations n ∈ N (acting to the left). As such the sym- 0 0 0 0 0
metry group of E(N) is K =∼ N. If the map did not s = (e, g1, ..., gk−1), (gk, ..., gl−1) 0 0 0 0 0 0
fix a unique convention (for instance np 7→ np0(n)) = (e, n1p1, ..., nk−1pk−1), (nkpC , ..., nl−1pC ) , where the image depends on which equivalence class (49) is chosen, then the set K of transformations between −1 the images E(n) would typically not be a group, and where pC = p0 . its closure would not be isomorphic to N (in some For particle j with configuration space N, the state j cases it would be the full group G). s is: Take k systems G and l − k systems N. A general j  j j j j j  s = (n0, n1, ..., n ), (n , ..., n ) , (50) state of the l systems is: k−1 k l−1 0 0 which is obtained from s by the map (ΓR(nj ) ◦ T). s = ((g0, g1, ..., gk−1), (gk, ..., gl−1)) , (45) Here ΓR(n) is just shorthand for the right regular where g ∈ G for i ∈ {0, . . . , k − 1} and g ∈ N for action: ΓR(n)(g0, ..., gl−1) = φR(n, (g0, ..., gl−1)) = i j −1 −1 0 j j j ∈ {k, . . . , l − 1}. Moreover there is a unique n ∈ N (g0n , ..., gl−1n ). We have ni n0 = ni . i 0 and p ∈ P such that g = n p . Here p = e for We observe that all states s of the form i i i i j 0 0 0 0 0 0 0 0 0 0
systems j ∈ {k, . . . , l − 1}. The description relative to (n0p0, n1p1, ..., nk−1pk−1), (nkpk, ..., nl−1pl−1) 0 j the first k systems and the transformations between for all pi ∈ P give the same s . The change of 0 j them is just the case described in Section3. In the reference frame s 7→ s is an irreversible change of following we describe how to change reference system reference frame. from a system with configuration space G to a system ∼ with configuration space N. 5.2.1 R = Z o U(1) The embedding of N ⊂ G is given by n 7→ ne. We An example of such a truncation is the ‘modular trun- define the truncation map: ∼ cation’ of the translation group: R = Z o U(1). In- stead of distinguishing all position states on the real T: G → N, line, we consider a reference frame that essentially g = np 7→ n , (46) consists of a classical ruler. All points that lie within and the map Ri : an interval of length L are mapped to the same point G on the ruler (for simplicity, one can choose L = 1). i i i Hence, the map identifies a subset of elements of RG :(g0, ..., gl−1) 7→ (g0, ..., gl−1) . (47) G = R with the same element of N = Z: Then, the relative state s0 = R0 (s) is: G T : R → Z , 0 0 0 0 0
s = (e, g1, ..., gk−1), (gk, ..., gl−1) x 7→ n , (51) = (e, n0p0, ..., n0 p0 ), (n0 p0 , ..., n0 p0 )
, 1 1 k−1 k−1 k k l−1 l−1 where x = nL + p, p ∈ [0,L[, n =  x  ∈ Z. (48) L Physically, this truncation can be viewed as coarse- j j j graining; in the sense of resolution, this means we where n p = g . i i i transform from a finer to a coarser resolution. Let us consider the case of j ∈ {k, . . . , l − 1}, i.e. Consider now three particles on the real line at po- where g = n . Then we have that g0 = g g−1 = j j j j 0 sitions x1, x1 and x1 relative to particle 1, where n g−1. Now let us observe that for all g there is a 1 2 3 j 0 particle 1 and 2 have configuration space R and par- unique decomposition into g = np, and so we write 0 0 0 0 0 ticle 3 has configuration space Z. Let us write the gj = nj pj . We now work out this nj pj in terms of nj 1 1 1 1 1 3 state s = (0, n2L + p2, n3L + p3), then the state s and g0 = n0p0 using the two following equalities: 1→3 1
1→3 1 1 is obtained by Λ T (s ) = Λ (0, n2L, n3L) = 1 1 1 1 1 g0 = n g−1 = n p−1n−1 , (−n3L, (n2 − n3)L, 0). If for instance T (x1) = T (x2), j j 0 j 0 0 1 1 0 0 0 i.e. n1 = n2, particle 3 assigns the same state to gj = nj pj . particles 1 and 2.

Accepted in Quantum 2020-11-16, click title to verify. Published under CC-BY 4.0. 11 interpretation of relative states si, the truncation method of obtaining relative states may not be de- G = R sirable. One may prefer an averaging procedure or a method based on finding invariants. These two ap- proaches are described in AppendixD using the exam- L ple of SO(3) o T (R3). We make use of the truncating N = Z procedure in the present work since it can easily be extended to a quantum version using the principle of Figure 7: Modular encoding of the real line. coherent change of reference system. We describe this quantum generalisation in the next section. 5.2.2 R3 =∼ R2 × R Another example is the truncation map R3 → R2. If 6 Irreversible changes of quantum ref- we have a symmetry group R2 in a three-dimensional erence frame space, all reference frames along a one-dimensional line are identified with each other. In this case, when The irreversible change of reference frame of the pre- applying the truncation map, all group elements in vious section can be extended to the quantum case by R3 are projected to the associated group elements in applying the principle of coherent change of reference R2. Consider three systems of which the first two frame, as in the reversible case. We describe this in have configuration space R3 and the last has config- more detail in the following. We note that we only uration space R2. A general state of the systems is define this change of reference frame for the specific 3 2 s = (g1, g2, g3) where g1, g2 ∈ R and g3 ∈ R . Rel- cases G = N o P and G = N × P . 1 1 1 ative to particle 1, the state is s = (e, g2, g3) = 1 1 1 1 1 1 2 (e, n2p2, n3pC ) where p2, pC ∈ R and n2, n3 ∈ R . 6.1 G = N P and G = N × P 1 1 1 o First, we truncate the state: s˜ = (e = n1, n2, n3). Then, we change to the state relative to system 3: Let us consider the change of reference frame from 2 s3 = (n3, n3, e). This can be understood as project- system i with Hilbert space L (G) to a system j with 1 2 2 ing the points in R3 to a plane in R2. If all reference Hilbert space L (N). Let us consider the first k sys- 2 frames along the z-axis are identified with each other, tems as being L (G) and the last l − k systems as 2 this corresponds to projecting on the x-y−plane. being L (N). 0 A generic product basis state ψ = |ei n0p0 ... n0 p0 n0 p0 ... n0 p0 5.3 Inconsistency using the truncation method 0 1 1 1 k−1 k−1 k−1 k k k l−1 l−1 l−1 (where p0 = e for j ∈ {k, . . . , l − 1}) maps for G = N P j o i i i i to ψ = |ei n0 n1 ... n for i i 0 1 l−1 l−1 In AppendixD we prove that the state s for a system i ∈ {k, . . . , l − 1}. Applying the coherent change 0 i with configuration space N obtained from state s of reference system we have that the action on a by truncating and changing reference system is not superposition state: equivalent to the state s˜i obtained from the state s by first truncating to obtain s˜, then finding s˜0 and 0 O 0 0 O 0 0 ψ = |ei nrpr + |ei msqs , (53) i 0 r 0 s then changing reference system to obtain s˜ . Let us r6=0 s6=0 write the change of reference system 0 → i for a group 0→i 0 i with m0 ∈ N, q0 ∈ P and p0 = p0 for all r, s ∈ G as ΛG s = s : s s r s {k, ..., l − 1} (and similarly for q0) maps to: 0→i 0 0 0 0 0 s ΛG (g0, ..., gl−1) = ΓR(gi )(g0, ..., gl−1) , (52) i O 0 i O 0 i 0 0 0 0 ψ = |eii nrn0 r + |eii msm0 s . (54) where ΓR(gi )s is φR(gi , s ), i.e. the right regular 0 0 r6=i s6=i action of gi on s . 2 Theorem 2. Let s = ((g , g , ..., g ), (g , ..., g )) First define the truncation map T: L (G) → 0 1 k−1 k l−1 L2(N). for G = N o P with gj = nj for j ∈ {k, . . . , l − 1} and gi = nipi otherwise. Then Z Z 0→i 0
0→i 0
T = |nihnp| dpdn , (55) ΛN T (RG(s)) 6= ΛN RN (T (s)) . Let 0→i 0
n∈N p∈P ΛN T (RG(s)) = ((n0, n1, ..., nk−1), (nk, ..., nl−1)) 0→i 0
and ΛN RN (T (s)) = and the map: ((m0, m1, ..., mk−1), (mk, ..., ml−1)), then nj = mj i→j for j ∈ {k, . . . , l − 1}. For j ∈ {0, . . . , k − 1} this is UN =SWAPi,j◦ Z ED not always the case. j i i ⊗l−2 i ni nj ⊗ 1i ⊗ UR(nj) dnj. ni ∈N j The above theorem shows that depending on our j prior commitment to a well defined state s and our (56)

Accepted in Quantum 2020-11-16, click title to verify. Published under CC-BY 4.0. 12 E The change of reference frame is given by: ~ On the level of Hilbert spaces, we assign the state P V i→j = U i→j ◦ T⊗k ⊗ 1⊗l−k
. (57) relative to a three-dimensional configuration space N while |~pi denotes a state in H =∼ R2. Hence, to change from the state relative to A with configuration space 6.1.1 R =∼ Z U(1) o R3 to the state relative to C with configuration space Consider again the example of the modular truncation R2, we apply the operator ∼ of the translation group: R = Z o U(1). Take three ZZ (quantum) systems A, B and C where A has configu- A→C ED ED V = SWAPA,C ◦ dP~ dQ~ −~p P~ ⊗ ~q − ~p Q~ , ration space R but states relative to C are encoded in C B LZ. Changing reference frame from A to C requires (64)

0 0 0 0 0 |0iA |x0iB |x1iC = |0iA |n0L + p0iB |n1LiC where Q~ := (x , y , z ) 7→ ~q := (x , y ).

7→ |0iC |−n1LiA |(n0 − n1)LiB . (58) The truncation operator is: 7 Wigner’s friend experiment X Z T = dp |nLihnL + p| , (59) In this section we consider the Wigner’s friend exper- n∈Z p∈U(1) iment, introduced by E. Wigner [23] in 1961. The thought experiment consists of two observers, Wigner 2 and the change of reference frame operator for L (N) and his friend, and a two-level quantum system. is: While Wigner is an external observer of the exper- Z A→C −1 iment, his friend is located inside an isolated box, to- UN = SWAPA,C ◦ n n C ⊗ 1A ⊗ UR(n)Bdn. n∈N gether with the system S in the quantum state |ψi. (60) Once the experiment is initiated, the friend measures the system in a certain basis. According to the projec- The following operator performs the change A → C: tion postulate of standard quantum theory, the state V A→C = U A→C ◦ (1 ⊗ T ⊗ T ). (61) of the system collapses to one of the eigenstates of the N A B C measurement operator. On the other hand, Wigner When applying the change of reference frame from describes this process from the outside and would A to C to a state in which system C is in a superposi- assign a unitary evolution. After the measurement tion state relative to A, the state becomes entangled of the system by the friend, the outcome of which relative to C: Wigner does not know, Wigner would assign an en-

A→C 0 tangled state to the joint system of the friend and the V |0iA |x0iB (|x1iC + |x1iC) system. These two seemingly contradicting prescrip- A→C 0 =V |0iA |n0L + p0iB (|n1LiC + |n1LiC) tions of standard quantum mechanics are at the core 0 0 of the so-called Wigner’s friend paradox. = |0iC (|−n1LiA |(n0 − n1)LiB + |−n1LiA |(n0 − n1)LiB) . (62) Before proceeding we observe that there is no logi- cal contradiction in the above two descriptions. The 0 However, if x1 and x1 are located in the same in- state after applying the projection postulate is a state terval relative to C (with configuration space LZ), the of the system S alone, of the form |ψiS. The entangled entanglement vanishes. Here, we recognize the depen- state assigned by Wigner is a state on S+F (where F is dence of entanglement on the reference frame relative the friend) of the form |φiSF. Since these are states of to which it is described. This was already pointed out two different objects (S versus S + F) there is no log- in [6]. Note that we observe entanglement only if the ical contradiction within the postulates of quantum uncertainty in the position of C with respect to A (i.e. theory. 0 the difference between x1 and x1) is larger than the We will now apply the relational formalism intro- resolution L of the configuration space of reference duced previously to this apparent paradox. Let us system C. consider W, short for Wigner, describing his friend F who measures the system S in the {|↑i , |↓i} ba- 6.1.2 R3 =∼ R2 × R sis. The friend is located inside a perfectly isolated 3 2 box. We model everything using two-dimensional sys- As mentioned before, the truncation map R → R tems, since we are interested in two degrees of free- essentially consists of a projection from a three- dom alone. The change of reference frame will there- dimensional configuration space to a two-dimensional 3 fore be for Z2, first described in Example2. We con- one. Here, we project all points in R onto the x-y- sider the ready state of the friend to just be |↑i and plane. the state of the system before the measurement to be R3 → R2 |ψi = α |↑i+β |↓i. We first describe the measurement interaction from the point of view of W starting in the ~ P := (x, y, z) 7→ ~p := (x, y) . (63) case where S is in an eigenstate of the measurement

Accepted in Quantum 2020-11-16, click title to verify. Published under CC-BY 4.0. 13 operator: Whatever outcome the friend observes, she would always describe the system definitely being in either |↑iW |↑iW 7→ |↑iW |↑iW , (65) F F S F S one of the basis states and Wigner in the state |↑iW. |↑iW |↓iW 7→ |↓iW |↓iW . (66) This seems very different to the description we obtain F S F S by taking Wigner’s perspective and changing to the The state of the friend depends on the measurement friend’s reference frame. Note however that this does outcome, hence the record of the outcome can be seen not necessarily imply a contradiction. More precisely, as being stored in the state of the friend. The state one should be careful when interpreting the state in Equation (71) as the final state seen by the friend. |↑iF is the state ‘the friend sees up’, and similarly for Rather, one should interpret it as the state Wigner |↓iF and ‘the friend sees down’. The change of reference frame W 7→ F for the final infers (concludes) the friend would see. In fact, start- states gives: ing from the state from Wigner’s perspective, we only take into account the information that Wigner has at W→F W W F F U |↑iF |↑iS = |↑iW |↑iS , (67) his disposal. Simply changing reference frames by ap- U W→F |↓iW |↓iW = |↓iF |↑iF , (68) plying a unitary operator can by no means introduce F S W S new information, such as the actual outcome observed where we observe that in both cases the state of S by the friend. F Hence, it is not a trivial assumption that the state relative to F is |↑iS. This state encodes the fact that the friend and the system are perfectly correlated in actually seen by the friend should be the same as F the state of the system relative to the friend, ob- both cases; |↑iS tells us that the state of the friend and the system are related by the identity element. tained by changing perspective from Wigner’s view- For an arbitrary superposition state of the system the point. In fact, this is assumption C (consistency as- unitary measurement interaction gives the following sumption) of the Frauchiger-Renner no-go theorem evolution: [24]. In the framework of relational quantum mechan- ics, one should not assume the consistency condition W W W W W W W |↑iF (α |↑iS + β |↓iS ) 7→ α |↑iF |↑iS + β |↓iF |↓iS . to hold a priori. Indeed in this work we show that (69) what Wigner infers about the friend’s state assign- Now, we want to apply the change of reference ments is given by the change of reference frame U W→F frame to switch to the perspective of F. If we ap- and not the seemingly straightforward assumption C. ply the operator U W→F to the initial state, using the We observe that the conclusion that what Wigner principle of coherent change of reference system, we can infer about the state of the system relative to the obtain: friend is that they are correlated is the same as Rov- elli’s treatment of the measurement process in rela- W→F W W W F F F U |↑iF (α |↑iS + β |↓iS ) = |↑iW (α |↑iS + β |↓iS) . tional quantum mechanics [2]. Here instead of reason- (70) ing using Wigner’s measurement operators we used an W→F If we apply the change of reference frame U to explicit change of reference frame from Wigner to the the final state we get: friend. One may think that introducing an additional ref- U W→Fα |↑iW |↑iW + β |↓iW |↓iW = (α |↑iF + β |↓iF ) |↑iF . F S F S W W S erence system R into the box containing F and S could (71) help resolve the issue with Wigner’s friend. We show If we interpret this result as the state seen by the in AppendixE that this is not the case. friend, this would imply that the friend always sees the system as correlated with herself. This is consis- tent with the initial description from Wigner’s per- 8 Discussion spective, where the state of the friend and system are correlated in both terms of the entangled state 8.1 Related work W W W W α |↑iF |↑iS + β |↓iF |↓iS . The underlying approach of this work is based on the Now, we want to compare this result to the state relational quantum mechanics of [2]. Other work for- we get if we simply start from the perspective of the mulating a fully relational quantum theory include [3– friend. She describes the initial state as: 7, 14, 15], the toy model of [25] and the systematic treatment of quantum reference frames in [11, 26]. |↑iF (α |↑iF + β |↓iF) , (72) W S S The main difference between [11, 26] and the present where Wigner is in the ready state. Applying the treatment is that we begin from an explicitly rela- projection postulate to the system gives: tional state and emphasise the notion of changing between reference frames as opposed to deriving re- 2 F F lational states from an external non-relational state. p↑ = |α| : |↑iW |↑iS , (73) Related approaches to relational quantum mechanics 2 F F p↓ = |β| : |↑iW |↓iS . (74) include the perspectival quantum mechanics of [16]

Accepted in Quantum 2020-11-16, click title to verify. Published under CC-BY 4.0. 14 and the Ithaca interpretation [27]. a description which is initially given externally (and The notion of quantum reference frames first ap- from which the relative description is then obtained) peared as part of the debate on the existence of is not proven. However in [3] it is shown that there charge superselection rules [28]. Later, Aharonov and is a ‘perspective-neutral’ framework which encom- Kaufherr gave the first explicit study of quantum ref- passes all perspectives for [6]. Note that this frame- erence frames [8]. Typically a description of a quan- work does not describe an external structure as it tum system is given relative to a classical measuring only encodes relative information. Extending this device of infinite mass. It was shown that there is perspective-neutral approach to the general cases in a consistent description of quantum systems relative the present paper could constitute an interesting di- to quantum reference frames of finite mass. This ad- rection for future work. dresses the issue of universality of quantum theory; In [31] an emphasis is placed on the notion that a namely quantum systems are usually described rela- reference system in a superposition relative to another tive to an implicit classical reference frame. If quan- system gives new coordinates, which are not related tum theory is universal, then one would expect that by a classical coordinate transformation to the initial reference frames should correspond to quantum sys- ones. This is conceptually closer to the perspective tems. in the present work than that of [6]. Namely we do Given a quantum reference frame R and a quantum not make the claim that the relative descriptions are system S the relational observables are observables of ‘operational’ as in [6]. R + S which are invariant under the symmetry group. Prior work with an emphasis on changing per- Depending on the state of R one may recover a re- spectives between quantum reference frames is found lational description of S which is equivalent to the in [32] where this change of reference frame is medi- standard absolute description of S (i.e. relative to an ated by an external description, and uses tools such implicit classical reference frame). Equivalently, for as G-twirling. any absolute description of the state or the observ- In recent years there has been a significant interest ables on S, one can find a system R such that there is in the study of quantum reference frames as resources a gauge invariant description on S + R. This observa- for measurements, communication tasks and thermo- tion was important for resolving the ‘optical coherence dynamic exchanges amongst others [33–39]. This ap- controversy’ [29]. Explicit maps between the absolute proach is not motivated by the same considerations and relativised descriptions can be found in [9, 11], as the present paper and the link between the two ap- where an emphasis on the localisation of the reference proaches is not fully clear to the authors. It is possible system is placed in [11, 26]. Other works discussing that the present exposition, carried out explicitly in relational observables include [3–5, 14, 15]. a group theoretic language, could be used to relate In [10, 30] a description of quantum systems rela- the two. For instance the ‘perfect reference frames’ of 2 tive to other quantum systems is given, in particular [9] are of the form L (G), which also plays a promi- for the translation group. The initial description of R nent role here. The link between imperfect reference and S is given relative to an external classical refer- frames as standardly defined [9, 36, 38] and the trun- ence frame (in the position basis), and the description cation based approach of the present work also re- relative to R is obtained by refactoring into center of mains to be worked out. mass and relative position. Tracing out the center of mass partition then allows to remove all global degrees 8.2 General comments of freedom and leaves us with a relational description of the systems. Moreover the description of systems 8.2.1 Relational quantum mechanics is shown to be dependent on the mass of the quan- We have provided a formalism for relational quan- tum reference system. Note that such global degrees tum mechanics which captures its spirit and recov- of freedom never enter into our formalism in the first ers the same conclusions for Wigner’s friend and the place. Instead, we give the states of systems relative Frauchiger-Renner theorem. Future work could in- to a specified system from the outset. volve applying the formalism developed in this work The change of reference frame in this work gen- to other thought experiments, in order to provide an 2 eralises the known changes of reference frame L (R) explicit account of how relational quantum mechanics 2 3 2 and L (R ) for spatial position of [6, 14], and L (R) addresses various apparent paradoxes. for temporal degrees of freedom of [4,5,7, 15]. Ref- erence frames for rotational degrees of freedom are 8.2.2 Observer dependence of the symmetry group studied in the perspective-neutral approach in [14]. Brukner and Mikusch are independently working on One further contribution of this work is that sym- a treatment of rotational degrees of freedom of quan- metries are not only relative to the system being de- tum reference frames using large spin coherent states. scribed, but also to the system doing the describing. We observe that in both [6] and the present work A system with Hilbert space H has a symmetry group the consistency of the fully relational account with G if it carries a representation of G, however it can

Accepted in Quantum 2020-11-16, click title to verify. Published under CC-BY 4.0. 15 only be described as transforming under G by a sys- formation based approaches. Future work involves ex- tem which carries a regular representation of G. For ploring imperfect reference frames for different cases example although a qubit carries a representation of than (G, N) with N a normal subgroup. One bene- SU(2), two qubits ‘describing’ each other and a third fit of our approach is that it formulates the relational qubit would use the subgroup Z2, since they can only approach of [6] in a group theoretic language which is carry a regular representation of Z2 and not the full more common in the quantum information literature. SU(2). This is reminiscent of the argument by Pen- As such it may help in developing a fully rigorous ac- rose in [40]. count of the link between the two. Recent work [15] in this direction has been exploring links between dif- ferent approaches in the case of the translation group 8.2.3 G = N o P and G = N × P and would be useful to pursue in this more general For the imperfect reference frames we describe a very case. specific case, though it seems natural since it is the Our exploration of the Wigner’s friend scenario structure of well known space time groups. Group shows how to apply our relational framework to an extension tells us that for any group subgroup pair important discussion point in the literature. Using (G, N) with N normal, the extension is of the form our approach we reach the same conclusion as Rovelli G = N o P or G = N × P . Hence our approach fully in his relational treatment of the measurement pro- covers imperfect reference frames for group subgroup cess in [2]: all that can be said by Wigner is that the pairs (G, N) with N normal. However for a full ac- friend is correlated with the measurement outcome. count of imperfect reference frames one would need The novel aspect of this work is the use of an explic- to relax the assumption of N normal and see whether itly relational formalism (which embodies the philo- one can define meaningful changes of reference frame. sophical position of relational quantum mechanics) as well as obtaining the friend’s perspective through an explicit change of reference frame, as opposed to rea- 9 Conclusion soning indirectly using measurement operators. The construction of a relational quantum theory re- quires a description of systems and states relative to other systems. In this work, we present a formalism Acknowledgments that allows to describe the relative states of systems ˇ and to change between the descriptions of different The authors thank Caslav Brukner, Esteban Castro- reference systems. Starting from the analysis of refer- Ruiz, Flaminia Giacomini, Matt Leifer, Leon ence frames in the classical realm, we moved on to the Loveridge, Pierre Martin-Dussaud, Carlo Rovelli, description of quantum states relative to quantum ref- David Schmid and Rob Spekkens for helpful dis- erence frames. Depending on the system whose view- cussions. The authors thank Philipp H¨ohn, Leon point is adopted, the description of physical phenom- Loveridge and Markus M¨ullerfor helpful comments ena changes. We find that quantum properties such on a draft of this paper. This research was supported as entanglement and superposition are not absolute by Perimeter Institute for Theoretical Physics. Re- but depend on the reference frame relative to which search at Perimeter Institute is supported in part by they are described. 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Accepted in Quantum 2020-11-16, click title to verify. Published under CC-BY 4.0. 17 a bounded-size quantum reference frame, New case where X =∼ G is a Lie group and we are not con- Journal of Physics 11, 063013 (2009). cerned with the group structure but only the smooth [37] N. Yunger Halpern, P. Faist, J. Oppenheim, differentiable manifold structure one could find other and A. Winter, Microcanonical and resource- coordinate systems via the isomorphism f : x 7→ mx theoretic derivations of the thermal state of a where m is an element of Diff(X), the diffeomorphism quantum system with noncommuting charges, group on X. In the case where X =∼ R this would Nature Communications 7, 12051 (2016). consist of considering coordinate systems which are [38] T. Miyadera, L. Loveridge, and P. Busch, Ap- related by more general transformations than trans- proximating relational observables by absolute lations, for instance re-scaling by a real scalar. quantities: a quantum accuracy-size trade-off, Journal of Physics A Mathematical General 49, A.3 Reference frames 185301 (2016). [39] M. Skotiniotis, W. D¨ur,and P. Sekatski, Macro- A reference frame is a coordinate system together scopic superpositions require tremendous mea- with a physical system whose configuration uniquely surement devices, Quantum 1, 34 (2017). determines the coordinate system. Since every choice [40] R. Penrose, Angular momentum: an approach to of coordinate system is in correspondence with an el- ∼ combinatorial spacetime, Bastin, T. (ed.), Quan- ement g ∈ G for a space X = G, a valid choice of tum Theory and Beyond , 151–180 (1971). physical system used to form a reference frame is a [41] J. Baez, Torsors made easy, John Baez Stuff physical system with configuration space G. (2009). B Proofs of Lemmas1,2 and3

A Background B.1 Proof of Lemma1 We show that U i→j is unitary. A.1 Coordinate systems and coordinate charts To change reference frame between systems i and j, where i and j have state space L2(G), and n − 2 We distinguish two separate notions: coordinate sys- other systems carrying the right regular representa- tems and coordinate charts following the presenta- E E i i i j j tion Uk(gj) ψ(gk) = ψ(gkgi ) = ψ(gk) , the tion in [19]. A coordinate system is an isomorphism k k k f : X → Y where X is the object of interest and Y general operator is is some known object we are using to describe X. In Z ED i→j i j i the terminology of Korzybski, X is the territory and U = SWAPi,j ◦ dgj gi gj ⊗ 1i⊗ gi ∈G j Y is the map. j However it is often the case (as in general relativ- O i Uk(gj). (75) ity) that there is no isomorphism between the object k6=i,j (space-time) and the description of the object (coor- dinate chart). A coordinate chart is a map in the We can show that this operator is unitary: opposite direction: f : Y → X where once more X is Z ED i→j † i i j the object of interest and Y is the known object be- (U ) = dhj hj hi ⊗ 1i⊗ hi ∈G j ing used to describe X, where the map f is no longer j O † i † an isomorphism (for instance for X a curved manifold Uk (hj) ◦ SWAPi,j. (76) n one needs multiple coordinate charts =∼ R to describe k6=i,j X fully). Hence, In the present paper we consider coordinate systems of a manifold X and not coordinate charts. (U i→j)†U i→j Z ED i i j O † i † = dhj hj hi ⊗ 1i ⊗ Uk (hj) ◦ SWAPi,j A.2 Coordinate systems on G-torsors hi ∈G j j k6=i,j Given a space X ∼ G for some group G it is clear that Z ED = i j i O i ◦ SWAPi,j ◦ dgj gi gj ⊗ 1i ⊗ Uk(gj) every g ∈ G gives rise to an isomorphism f : X → Y gi ∈G j j k6=i,j (where Y =∼ G) via the map f : x 7→ gx. While e is Z Z D E the identity element of the group G it is the origin of i i i j j i = dhj dgj hj hi gi gj ⊗ 1i ⊗ i i j the G-torsor. Different isomorphisms f : X → Y cor- hj ∈G gj ∈G respond to different choices of origin. To quote [41]: O U †(hi )U (gi ) ‘A torsor is like a group that has forgotten its iden- k j k j k6=i,j tity.’ Since every g ∈ G leads to a different coordinate Z O system, the set of different choices of coordinate sys- = dgi gi gi ⊗ 1 ⊗ U †(gi )U (gi ) j j j j i k j k j gi ∈G tems is also G. As noted in the main body in the j k6=i,j

Accepted in Quantum 2020-11-16, click title to verify. Published under CC-BY 4.0. 18 O i −1 i i i = 1j ⊗ 1i ⊗ Uk((gj) gj) The change of reference frame A → B: ψ 7→ φ k6=i,j gives: O = 1j ⊗ 1i ⊗ 1k 1 −1 −1 φ = |ψ(e)iB ψ(g1 ) A ψ(g2g1 ) C , (79) k6=i,j φ2 = |ψ(e)i ψ(h−1) ψ(g h−1) . (80) = 1. B 1 A 2 1 C 3 1 2 D E For a superposition state ψ = α ψ + β ψ Here, we used hj gj = δ , R dgi gi gi = 1 i i gh j j j j j (α, β 6= 0) the change of reference system gives: and Uk(e) = 1k. 3 1 2 3 1 2 ψ = α ψ + β ψ 7→ φ = α φ + β φ , B.2 Proof of Lemma2 (81) † We show U 0→i
= U i→0. where 3 −1 −1 φ = |ψ(e)iB (α ψ(g1 ) A ψ(g2g1 ) C † U 0→i
−1 −1 + β ψ(h1 ) A ψ(g2h1 ) C) . (82) Z 0 0 i † 0 ⊗n−2 † = dgi gi g0 i ⊗ 10 ⊗ UR(gi ) ◦ SWAP0,i Let us show that unitarity of the change of refer- g0∈G i ence frame implies that the states |ψ(g)i are mutually Z = SWAP ◦ dg0 g0 gi ⊗ 1 ⊗ U † (g0)⊗n−2 orthogonal. We compute the inner product of two ini- i,0 i i 0 0 i R i 1 3 0 tial states ψ and ψ : gi ∈G Z i 0 i i ⊗n−2 1 3 = SWAPi,0 ◦ dg0 gi g0 ⊗ 1i ⊗ UR(g0) ψ ψ = α + β hψ(g1)|ψ(h1)i , (83) i 0 B g0∈G 1 = U i→0 and the inner product of the two final states φ and 3 φ : From the second to the third line, we commuted the 1 3 1 2 SWAP operator through to the left by changing the φ φ = α + β φ φ , (84) labels of partitions 0 and i. From the third to the 0 fourth line, we used the fact that the integral over gi where i is the same as the integral over g0. Moreover, it holds 1 2 −1 −1 −1 −1 † 0 i i 0 φ φ = ψ(g1 ) ψ(h1 ) A ψ(g2g1 ) ψ(g2h1 ) C . that UR(gi ) = UR(g0) because UR(g0)UR(gi ) |gi = i 0 i † 0 (85) gg0gi = |gi, hence UR(g0) = UR(gi ). −1 −1 Since ψ(g2g1 ) = U(g2) ψ(g1 ) and B.3 Proof of Lemma3 −1 −1 ψ(g2h1 ) = U(g2) ψ(h1 ) we have that i→j k→i k→j −1 −1 −1 −1 We prove that U U = U . ψ(g2g1 ) ψ(g2h1 ) C = ψ(g1 ) ψ(h1 ) C. There- For this, we show how the operators act on an ar- fore: bitrary basis state: 1 3 −1 −1 2 φ φ = α + β( ψ(g1 ) ψ(h1 ) ). (86) i→j k→i k k k k U U |ei g ... g ... g ... g k 0 0 i i j j n−1 n−1 By assumption there are two commuting ac- =U i→jSWAP |ei gkgi ... gk ... gkgi ... gk gi tions: U (g) |ψ(h)i = |ψ(gh)i and U (g) |ψ(h)i = k,i k 0 k 0 i i j k j n−1 k n−1 L R ψ(hg−1) . = U i→j |ei gkgi ... gk ... gkgi ... gk gi i 0 k 0 i k j k j n−1 k n−1 1 3 1 3 Now ψ ψ = φ φ requires hψ(g1)|ψ(h1)i = E E E E i j k j j i j −1 −1 2 = SWAPi,j |eii g0gi ... gi gi ... gi ... gn−1gi ψ(g1 ) ψ(h1 ) . We assume this holds and show it 0 k j n−1 E E E E implies that hψ(g)|ψ(h)i = δ(h, g). j j j j = |eij g0 ... gk ... gi ... gn−1 0 k i n−1 = U k→j |ei gk ... gk ... gk ... gk . −1 −1 2 k 0 0 i i j j n−1 n−1 hψ(g1)|ψ(h1)i = ψ(g1 ) ψ(h1 ) 2 −1 −1 ⇔ hψ(g1)|ψ(h1)i = ψ(g1 ) ψ(h1 ) C On the unitarity of the change of −1 −1 ⇔ UL(g1 )ψ(g1) UL(g1 )ψ(h1) 2 reference frame operator −1 −1 −1 −1 = UR(h1 )ψ(h1 ) UR(h1 )ψ(g1 ) Let us consider |ψ(g)i, with U (k) |ψ(g)i = |ψ(kg)i. 2 L ⇔ ψ(e) ψ(g−1h ) = ψ(e) ψ(g−1h ) Take two initial states defined relative to A: 1 1 1 1 −1 1 Which only holds when ψ(g h1) is orthogonal to ψ = |ψ(e)i |ψ(g1)i |ψ(g2)i , (77) 1 A B C |ψ(e)i (when g 6= h ). Therefore for all g ∈ G it is 2 1 1 ψ = |ψ(e)iA |ψ(h1)iB |ψ(g2)iC . (78) the case that hψ(e)|ψ(g)i = δ(g, e).

Accepted in Quantum 2020-11-16, click title to verify. Published under CC-BY 4.0. 19 Now consider hψ(g)|ψ(h)i for arbitrary g, h ∈ Then one can obtain T (s0)j = Λ0→jT (s0) = −1 −1 0 0 G. This is equal to UL(g )ψ(g) UL(g )ψ(h) = ΓR(mj )T (s ): −1 −1 ψ(e) ψ(g h) = δ(e, g h) = δ(g, h). This entails 0 j  j 0 j 0 j 0 j  that the states |ψ(g)i form an orthonormal basis for T (s ) = (m0, ..., mk−1m0), (mkm0, ..., ml−1m0) . H or a subspace of it if they do not span the full i (96) space. Hence, the operator which performs the required 0 j j 0 j Does mi m0 = ni , i.e. does mi m0nj = ni. There are change of reference frame and obeys the principle of two cases: i ∈ {0, . . . , k − 1} and i ∈ {k, . . . , l − 1}. coherent change of reference frame is unitary if and Let us consider the second: only if the states |ψ(g)i form an orthonormal basis for H or a subspace thereof. 0 0 i mi qi n0p0 = ni (97) 0 −1 −1 mi = ni(p0 n0p0) (98)

D Imperfect reference frames 0 −1 0 −1 −1 since qi = p0 . Similarly mj = nj(p0 n0p0) . Therefore D.1 Proof of Theorem2 0 j 0 j −1 −1 −1 −1 −1 mi m0 = mi (m0) = ni(p0 n0p0) (p0 n0p0)nj , −1 s = ((g0, ..., gk−1), (gk, ..., gl−1)) (87) = ninj . (99) = ((n p , ..., n p ), (n , ..., n )) (88) 0 0 k−1 k−1 k l−1 j This is indeed ni since it maps nj to ni. j 0→j 0
Now let us consider the case i ∈ {0, . . . , k − 1}: We first find the state s˜ = Λ RN (T (s)) . Let us truncate s: 0 0 mi qi n0p0 = nipi (100)

s˜ = T (s) = ((n0, ..., nk−1), (nk, ..., nl−1)) . (89) 0 −1 where qi is not necessarily equal to p0 . Then we obtain Then the state s˜0 is: m0mj = m0(mj )−1 = n p (q0n p )−1(p−1n p )n−1 s˜0 = R0 (n0, ..., n0 ), (n0 , ..., n0 )
. (90) i 0 i 0 i i i 0 0 0 0 0 j N 0 k−1 k l−1 (101) j 0→j 0 −1 −1 0 −1 −1 −1 Then the state s˜ = Λ s˜ is: = nipip0 n0 (qi ) p0 n0p0nj (102)

j  0 j 0 j 0 j 0 j  j T (s) = (n0n0, ..., nk−1n0), (nkn0, ..., nl−1n0) which is not equal to ni in general. For a specific 1 (91) example we can look at R = Z o S . Consider  j j j j  = (n0, ..., nk−1), (nk, ..., nl−1) (92) s = (n0L + x0, n1L + x1, n2L, n3L). (103)

2 i r j j Then T (s) = (n0L, n1L, n2L, n3L) and T (s) = (n0 − where njni = nj and ni nr = ni . n2, n1 − n2, 0, n3 − n2)L. For the other order we get j 0→j 0
0 0 Now we compute s˜ = Λ T (R (s)) . s = R s 0 G G s = (0, n1L+x1 −(n0L+x0), n2L−(n0L+x0), n3L− is: (n0L + x0)). Let us assume that x1 − x0 < 0, then 0 0 2 T (s ) = (0, n1−n0−1, n2−n0, n3−n0)L and T (s ) = s0 = (g0, ..., g0 ), (g0, ..., g0 )
(93) 0 k−1 k l−1 (−(n2 − n0), n1 − n0 − 1 − (n2 − n0), n2 − n0 − (n2 − 0 0 0 0 0 0 0 0
0 2 = (m0q0, ..., mk−1qk−1), (mkqk, ..., ml−1ql−1) , n0), n3 − n0 − (n2 − n0))L which gives T (s ) = (n0 − 2 (94) n2, n1 − n2 − 1, 0, n3 − n2)L and does not equal T (s) .

j j j where gi = mi qi is the decomposition into NP . Let D.2 Averaging and invariants for the example 0 0 us consider the elements mj qj for j ∈ {k, . . . , l − 1}. of SO(3) T (R3) 0 0 0 o We have that gj g0 = nj → mj qj n0p0 = nj. Now observe that since m0 and n in N this implies that Now let us return to the case where the configu- j j 3 0 0 0 ration space is R but the transformation group is q n0p0 = n ∈ N. Now there is a unique q such that j j E+(3) =∼ T (R3) SO(3) and the case of three particles. this holds, and observe that p−1n p = m0. There o 0 0 0 Here E+(3) is the special Euclidean group consisting is a unique m ∈ N such that mm0 = n0. Therefore of translations followed by a rotation. Let us say that q0 = mp−1. However since q0 ∈ P this implies m = e j 0 j particle 0 uses a coordinate system (x0, y0, z0) such 0 −1 and therefore qj = p0 . that it assigns itself the state (0, 0, 0) = ~0. There Let us truncate: are infinitely many such coordinate systems corre- sponding to all reference frames centred on parti- 0 0 0 0 0
T (s ) = (m0, ..., mk−1), (mk, ..., ml−1) . (95) cle 0, related by rotations O ∈ SO(3). We write

Accepted in Quantum 2020-11-16, click title to verify. Published under CC-BY 4.0. 20 0 0 0 0 1 ~ ~
~vi = (xi , yi , zi ) for the state of particle i in a refer- s0,1 = O−a, 0 for all O ∈ SO(3) from the point of ence frame centred at 0. The coordinates correspond- view of particle 1. In other words: what are the in- ing to Cartesian reference frames are in one to one variants? Here it is straightforward to see that any correspondence with group elements in T (R3)oSO(3). f(|~a|) is an invariant quantity. 0 0 0 0 Each state ~vi = (xi , yi , zi ) is stabilized by a SO(3) 0 subgroup: ~vi o O ∀O ∈ SO(3) . We can write R3 =∼ E+(3)/SO(3). E Wigner’s friend with additional ref- The description of the state of the three particles erence system 0 ~ 0 0
is s0,1,2 = 0,~v1,~v2 . What can particle 0 infer about the description used by particle 1, knowing only that One could argue that introducing an additional refer- the convention is such that particle 1 uses Cartesian ence system R into the box containing F and S might coordinates placed at its position? There are infinitely help to resolve the paradox of Wigner’s friend. In many such choices of coordinates, related to the co- this case, we can describe the measurement interac- 0 0 0 ordinates (xi , yi , zi ) by t−v0 o O for all O ∈ SO(3). tion from the point of view of W as follows:  1   1,O ~0 ~ 0 ~0 W W W W W W W W Hence the states s = −Ov , 0,O ~v2 − v for 0,1,2 1 1 |↑iR |↑iF |ψiS → |↑iR (α |↑iF |↑iS + β |↓iF |↓iS ). any O ∈ SO(3) correspond to coordinate choices cen- (108) tred on particle 1. In order to describe the ignorance The reference system is not affected by the measure- about O one needs a probabilistic representation of ment. Now, we want to apply the change of reference states: a state is now a measure on R3. The state frame to switch to the perspective of F. We apply the W→F −~a is the Dirac measure δ−~a and the description that operator UAS to the initial state: particle 0 assigns to particle 1 is: U W→F |↑iW |↑iW |ψiW = |↑iF |↑iF |ψiF , (109) Z   RS R F S W R S s1,av = Oδ , δ , δ (104) 0,1,2 −v~0 ~0 ~v0−v~0 O∈SO(3) 1 2 1 and the final state:   W W W W W U W→F |↑i (α |↑i |↑i + β |↓i |↓i ) = µ 2 ~0 , δ~0, µ 2 0 ~0 , (105) RS R F S F S S ,|v1 | S ,|~v2 −v1 | F F F F F = (α |↑iW |↑iR + β |↓iW |↓iR) |↑iS . (110) where µS2,|~a| is the normalised Haar measure on the sphere S2 of radius |~a| centred at the origin (in par- We see that, again, the state of the system is always ticle 1’s coordinates). We observe that there is no correlated with the state of the friend. Wigner and 1,average reversible transformation from s0,1 representing the additional reference system end up being entan- particle 0’s knowledge of particle 1’s description back gled. We conclude that introducing an additional ref- to the initial description of particle 0. Starting from erence system into the laboratory does not provide a s0 and mapping to s2 one obtains: resolution to the paradox. Z 2,av   s = Oδ 0 , δ 0 0 , δ0 (106) 0,1,2 −~v2 ~v1 −~v2 O∈SO(3) F Comparison to RF change operator   = µ 2 ~0 , µ 2 0 ~0 , δ~0 . (107) of [6] S ,|v2 | S ,|~v1 −v2 | We leave to future work the development of a full This is the formal proof that the operator given in [6] account of changes of reference frame in this proba- for the change between two reference frames A and B bilistic case (for instance it is not clear that there is on the real line (one-dimensional translation group) 1,av 2,av a well defined change s0,1,2 7→ s0,1,2). Here we are is equivalent to the operator given in Equation (15), interested only in the single change from s0 7→ s1. namely We also note that when describing ignorance one Z cannot integrate over the pure states, or configura- A→B U = SWAPA,B ◦ dxdy |−xihx|B ⊗ 1A ⊗ |y − xihy|C . tions. In the above example this would result in 1,average ~ ~
(111) s0,1 being 0, 0 which is nonsensical. This is similar to the case in quantum theory where inte- Let us take the most general state of three particles grating over a group action on pure states does not on the real line R relative to the first particle: represent ignorance (rather it is a projection) but in- Z Z Z tegrating over mixed states corresponds to ignorance. |ΨiABC = |0iA ⊗ dx dy ψ(x, y) |xiB ⊗ dy |yiC . One must move to a probabilistic description of states in terms of measures on X in order to infer states (112) of other particles for these types of cases. Alterna- In the formalism of [6], the reference system is not tively one could stay at the pure state description, explicitly included in the state of the joint system. and instead of finding the point of view that particle The equivalent state would be 0 infers particle 1 holds, we can search for what par- Z Z Z |Ψi = dx dy ψ(x, y) |xi ⊗ dy |yi . (113) ticle 0 knows holds true for all possible descriptions BC B C

Accepted in Quantum 2020-11-16, click title to verify. Published under CC-BY 4.0. 21 When changing from reference system A to reference system B, we apply the operator of [6]:

A→B i/ xˆBpˆC Sˆ = PˆABe ~ . (114)

The final state is then Z Z ˆA→B S |ΨiBC = dx dy ψ(x, y) |−xiA ⊗ |y − xiC . (115)

When applying the operator in Equation (111) to the initial state in Equation (112), we get the final state

A→B U |ΨiABC = SWAPA,B ◦  Z  0 0 0 0 0 0 00 |0iA dx dy dxdy ψ(x, y) |−x ihx | |xi |y − x ihy | |yi Z Z = |0iB dx dy ψ(x, y) |−xiA ⊗ |y − xiC . (116)

Hence, we see that both operators act in the same way on the most general states of particles on the real line.

Accepted in Quantum 2020-11-16, click title to verify. Published under CC-BY 4.0. 22