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https://doi.org/10.1038/s41467-018-08155-0 OPEN and the covariance of physical laws in quantum reference frames

Flaminia Giacomini1,2, Esteban Castro-Ruiz1,2 & Časlav Brukner1,2

In physics, every observation is made with respect to a . Although refer- ence frames are usually not considered as degrees of freedom, in all practical situations it is a physical system which constitutes a reference frame. Can a quantum system be considered

1234567890():,; as a reference frame and, if so, which description would it give of the world? Here, we introduce a general method to quantise reference frame transformations, which generalises the usual reference frame transformation to a “superposition of coordinate transformations”. We describe states, measurement, and dynamical evolution in different quantum reference frames, without appealing to an external, absolute reference frame, and find that entangle- ment and superposition are frame-dependent features. The transformation also leads to a generalisation of the notion of covariance of dynamical physical laws, to an extension of the weak equivalence principle, and to the possibility of defining the rest frame of a quantum system.

1 Vienna Center for Quantum Science and Technology (VCQ), Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria. 2 Institute of Quantum Optics and (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria. Correspondence and requests for materials should be addressed to F.G. (email: fl[email protected])

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he state of a physical system has no absolute meaning, but via a ‘generalised coordinate transformation’ from an initial QRF Tis only defined relative to the observer’s reference frame in to a final QRF, which does not only involve the observed system, the laboratory. The same system may be associated to but also the degrees of freedom of the quantum system con- different states in different reference frames, which are normally sidered as a reference frame. The resulting transformation takes related via some reference frame transformation. From a physical the states of all systems external to the initial QRF as input, and point of view, a frame of reference is an abstraction of an idealised outputs the states of all systems external to the final QRF. We physical system: for example, an ideal rigid body can serve as a find that a quantum state and its features—such as superposition reference frame to define relative spatial distances and orienta- and entanglement—are only defined relative to the chosen tions of other objects. In classical physics, a coordinate trans- reference frame, in the spirit of the relational description of formation is used to transform the description of the system physics16–19,23,24. For example, a quantum system which is in a under consideration between two different reference frames. well-localised state of an observable for a certain observer may, These transformations include, for example, spatial rotations and for another observer, be in a superposition of two or more states translations in space and or constant relative motion of the or even entangled with the first observer. As the transformations frames (e.g., Galilean tranformations). In general, the dynamical between different QRFs are unitary, the observed probabilities physical laws are invariant under some group of transformations. (i.e. relative number of counts) are invariant. However, the For instance, the laws of non-relativistic physics are invariant measured systems and observables are different in different QRFs under Galilean transformations. and we find the transformation that maps the measured obser- In every physical laboratory situation, the reference frame is vables and systems in one QRF with those in the other QRF. realised through a physical system. As any physical system, it Turning to the dynamics, we propose an extension of the notion ultimately behaves according to the laws of quantum mechanics. of covariance of the physical laws to include genuine quantum Therefore, one might see the standard treatment of reference- transformations, where one frame of reference is in a super- frame transformations as an approximation to a more funda- position of different relative positions, momenta or velocities with mental set of transformations. Specifically, one should take into respect to another frame of reference. We find Hamiltonians that account the possibility that one laboratory, from the perspective are symmetric under such “superpositions of Galilean transla- of another laboratory, might appear in a superposition or even tions” and “superpositions of Galilean boosts”. Finally, we find become entangled with the system. Hence, the relationship that the weak equivalence principle can be extended to QRFs: The between the two laboratories becomes more than a simple effects as observed in a “superposition of uniform gravitational coordinate transformation between classical reference frames; it fields” are indistinguishable from those in a frame in a “super- becomes a fundamentally quantum relationship. We may then position of accelerations” in flat space-time. In all these trans- speak about transformations between ‘quantum reference frames’ formations the quantum system considered as a reference frame (QRFs). For example, we can imagine that the laboratory and the acts as a control for the transformation on the observed system. instruments of one observer are fixed to a platform that is in a superposition of position states with respect to the laboratory of a Results second observer, as illustrated in Fig. 1. Can we meaningfully fi Transformations between quantum reference frames. When de ne transformations between such QRFs? Which transforma- reference frames are considered as abstract entities, the reference tions relate quantum states of systems defined with respect to one fi frame transformation consists in a coordinate transformation, frame of reference to those de ned with respect to a second frame where the new coordinates x′ in which the system under con- of reference? What are the dynamical physical laws that are sideration is expressed are functions of the old cordinates x and invariant under such ‘quantum transformations’? ′ = ′ 1–22 time t, i.e. x x (x, t). In the transformation, the relation QRFs have been extensively discussed in the literature . between the old and new reference frame, such as the relative Previous works on QRFs, and a comparison with our approach, is position or velocity, enters as a parameter. A special case of these discussed in detail in Methods (Comparison of previous transformations, discussed in detail in Supplementary Note 1, are approaches to quantum reference frames). 25 fi the extended Galilean transformations, introduced in ref. . In this work, we nd (unitary) transformations that relate These are transformations of the type x′ = x − X(t), and contain states, dynamical evolution and measurements from the point of as particular cases spatial translations, Galilean boosts and view of different QRFs. This is achieved by changing perspective transformations to a uniformly accelerated reference frame. In quantum theory, all these transformations can be represented via their unitary action on the quantum state of the system ψ′ ^ ψ j i¼Uij i, where the index i labels the transformation. In the case of spatial translations, the coordinate transformation x′ = x B i ^ − ^ X0p fi A C X0 induces the transformation UT ¼ eh , where X0 is a xed parameter, with the physical dimension of a length, describing the displacement of the new reference frame with respect to the old one. When the new reference frame moves with constant and ⎢〉  ⎢ 〉 A = dx (x) x A uniform velocity v from the point of view of the initial one (Galilean boost), we have X(t) = vt, and the transformation which Fig. 1 Illustration of the notion of quantum reference frames. Two quantum i ^ ^ hvG ^ reference frames A and C and quantum system B. The reference frames A changes the state to the new frame is Ub ¼ e . Here, G ¼ ^ À ^ and C are pictorially represented as two laboratories equipped with their pt mx is the generator of the Galilean boost, with m being the own instruments. In a realistic situation, however, the system A could be an mass of the boosted particle. Finally, if the new reference frame is atom, B a photon and C a laboratory (or another atom). The reference constantly and uniformly accelerated with acceleration a from the point of view of the initial one, XðtÞ¼1 at2 and theR operator to frame associated to A is in a superposition of positions as observed from 2 t i _ ^ i ^ im _ 2 ^ ÀmXðtÞx XðtÞp Àh 2 dsX ðsÞ the laboratory C (the superposition is illustrated by the fuzziness of be applied on the state is Ua ¼ e h eh e 0 . laboratory A). Given the quantum states of A and B relative to the reference The usual coordinate transformations which describe the frame of C, what are the states of B and C as defined with respect to the change between reference frames rely on the assumption that reference frame of A? reference frames are abstract, ideal entities, to which we do not

2 NATURE COMMUNICATIONS | (2019) 10:494 | https://doi.org/10.1038/s41467-018-08155-0 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-018-08155-0 ARTICLE assign a physical state. In a real experimental situation this Before giving an example of a transformation to a quantum idealisation may not be accurate, since real physical systems reference frame, we should stress that the requirement of standardly serve as reference frames, and therefore the assump- canonicity leads to important consequences. If we were to tions usually made may be untenable. An instance of the approach the transformation naively, we would be tempted to differences occurring when a physical system is considered as a define a transformation to relative coordinates and momenta, 26 reference frame is presented in ref. , where a vibrating wire where the relative coordinates of a set of N particles of mass mi = r serves as a quantum non-inertial reference frame. The wire, (i 1,...,N) relative to particle 0 of mass m0 are xi ¼ xi À x0 and placed in an interferometer traversed by an atom, induces a phase r μ pi p0 μ mim0 the relative momenta are pi ¼ i0 À , i0 ¼ being the shift on the atom, leading to a loss of interference. mi m0 miþm0 fi We next give the basic elements of our formalism, where a reduced mass. If we now compute the Poisson bracket, we nd r; r ≠ ≠ description of a set of physical systems is given relative to another that fxi pj g 0 for i j, thus violating the canonicity require- set of physical systems (the latter set serving as a QRF), within the ment. This argument can also be found in ref. 20,21. This means framework of either classical or quantum theory. We find general that, whenever we perform a transformation to a quantum transformations between the descriptions that different QRFs reference frame, we have to choose the relative variables we are provide for their respective ‘rests of the world’. We will see that interested in, and then complete the transformation of the the notion of ‘jumping’ to a QRF becomes ill-defined: not all conjugated variable by canonicity. Note that this feature does not the variables can be cast in relational terms, and a choice has to only arise in quantum mechanics, but in classical physics too. be made as to which degrees of freedom are relevant to the Moreover, in the Lagrange formalism, a transformation to the situation studied. All calculations are done in one dimension to relative variables is a point transformation, and is therefore keep the notation simple. An extension to three dimensions is automatically canonical when mapped to the phase space. straightforward. We now move on to illustrate the idea of a transformation We consider three quantum systems, as illustrated in Fig. 1:C between QRFs through an example. We describe the situation in is the initial reference frame, A is the new reference frame to which the new reference frame A is simply translated with respect whose perspective we want to change, and B is, in general, a to the old one C, but where the quantum state of A, instead of composite system to which the transformation from C’stoA’s being sharp in position, is in a superposition of positions. In this reference frame will be applied. Our approach is operational in case it is clear that a mere coordinate transformation of the type that primitive laboratory operations—preparations, transforma- discussed previously in this section and in detail in Supplemen- tions and measurements—have fundamental status. This empha- tary Note 1 is no longer adequate, because the position of the new sis on the operational approach enables the theory to be specified reference frame is not localised, and therefore there is no unique purely in terms of notions that have immediate physical meaning. distance between the two reference frames. As a consequence, a Note, however, that the approach does not entail the necessity of transformation which captures the quantum features of A is having macroscopic superpositions. In a realistic situation, for necessary in order to describe a quantum system B in the new example, system A could be a particle with external degrees of reference frame. We suggest that a natural procedure is to make freedom in superposition with respect to laboratory C, which use of the linearity of quantum mechanics and ‘coherently serve to define the new set of relative coordinates, and with translate’ the state of B relative to the position of A, via the ix^ ^p internal degrees of freedom used as a ‘detector’ in reference frame operator eh A B , where the indices refer to the two quantum A. We assume that a dynamical description relative to a reference systems A and B. Note that here the position operator of the ^ frame does not involve the frame itself, but only the systems system A, xA, replaces the classical parameter of the usual external to it. An explanation of this is that, for instance, the translation operator. position and of the reference frame are not The full spatial translation to change from C’stoA’s reference dynamical variables when considered from the reference frame frame consists of a change of relative position coordinates as seen itself (this can also be related to the so-called self-reference from C (illustrated in Fig. 2a) to the relative position coordinates problem27,28). Therefore, the reference frame is not a degree of as seen from A (in Fig. 2b). This can be achieved via the canonical freedom in its own description, but external systems to it are. transformation ’ Hence, from the perspective of C s reference frame, A and B are ^x 7!^q À ^q ; ð1aÞ external systems, and from the perspective of A’s reference frame B B C so are B and C. ^ ^ ; In C’s reference frame the systems A and B are described by xA7!ÀqC ð1bÞ ðCÞ ðCÞ quantum states in the Hilbert space HA HB . To change the reference frame we apply a canonical transformation, the most ^p 7!π^ ; ð1cÞ general transformation which preserves the symplectic structure B B of the phase space. Quantum canonical transformations have 29,30 ^ π^ π^ ; been object of study in Refs. , and are defined as invertible pA7!Àð C þ BÞ ð1dÞ transformations C^ which map the initial operators ð^x; ^pÞ,to^q ¼ ^^^À1 π^ ^^^À1 ^; π^  CxC and ¼ CpC such that ½Š¼q ih. The general theory ab of quantum canonical transformations involves technical issues, B B for example that not all quantum canonical transformations are xB 30 isometries . For simplicity, in this work we restrict our q consideration only to unitary transformations, which by defini- C C B tion are isometries. Such unitary transformations take the form q ^ : fi xA C C H1 !H2, where H1, H2 are the initial and nal Hilbert ψ; ϕ A A spaces, such that, for all states 2H1, the scalar product is ϕ ψ ^ϕ ^ψ preserved, i.e. h j i1 ¼hC jC i2, where hÁjÁii is the scalar Fig. 2 Transformation to relative coordinates. a Relative position ; product on the Hilbert space Hi, i ¼ 1 2. Notice that the coordinates of A and B from the point of view of C. b Relative position functional form of the two scalar products might differ, because coordinates of B and C from the point of view of A. It is immediate to verify ^ ^ ^ ^ ^ the measure of the Hilbert space is allowed to change. that xB 7! qB À qC and that xA7!ÀqC

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^ ; ^ ; where xi pi, i ¼ A B, are the positions and momenta relative to ab C, and ^q ; π^ , j ¼ B; C, are the positions and momenta relative to A A j j B A. The transformation (1) can be represented via the canonical B C C ^ ^y transformation SxðÁÞ ¼ Sx Á Sx on the phase space observables of x x the systems A and B, where S ðÁÞ is a superoperator and ^S the x x C ^ : ðCÞ ðCÞ ðAÞ ðAÞ fi B C unitary transformation Sx HA HB !HB HC de ned B as A A q q ^ ^ i^x ^p S ¼ P eh A B : ð2Þ x AC cd ^ : ðCÞ ðAÞ A B A Here, PAC HA !HC is the parity-swap operator acting as B ^ ψ ψ C PAC AðxÞ¼ CðÀxÞ in the coordinate representation of the C ρðCÞ L L x x states. If C assigns the quantum state AB to the joint system of A and B, the transformed state from A’s perspective is C B C B ρðAÞ ^ ρðCÞ^y BC ¼ Sx AB Sx. Note that our transformation can be applied to A A both pure and mixed states. This transformation satisfies the q q transitive property, meaning that changing reference frame from C to B and subsequently from B to A with Eq. (2) has the same Fig. 3 Examples of relative states in different quantum reference frames. effect as changing it from C to A. In particular, this means that The relative states are described from the reference frame of C (above in fi fi changing from C to A and then back to C is equivalent to an each sub gure) and A (below in each sub gure) in the position basis. ÀÁ Product states are represented as curves whose area is shaded, while ^ðC!AÞ ^ðA!CÞ y identity operation, i.e. Sx ¼ Sx . This is shown in entangled states as curves whose area is not shaded. In a A’s state is well- Supplementary Note 2. The transformation in Eq. (2) can be seen localised from the point of view of C. In A’s reference frame, B has the same as a translation of system B controlled by the position of system A state as seen from C, but translated, and C is well-localised. This case followed by the parity-swap operator. corresponds to the translation of a classical reference frame. In b A and B ‘ ’ Note that no absolute reference frame ( external perspective) are in a product state, and A is in a superposition of two sharp-position was needed to establish Eq. (2) (throughout the paper, we states that do not overlap. From A’s point of view B and C are entangled, ‘ ’ ‘ ’ ‘ interchangeably use the terminology absolute , abstract , exter- but the relative distance between the states is unchanged. In c A and B are ’ ‘ ’ nal , and classical to refer to such reference frames). This entangled and perfectly correlated, i.e. the relative distance between them transformation can be obtained by performing a point transfor- is always L.InA’s reference frame B is in a well-defined position and C is in mation to relative coordinates in the Lagrangian formalism, from a superposition of positions. Finally, in dRA and B are entangled in an EPR which the relations between momenta can be derived. Alter- ’ ψ state from C s point of view, i.e. jiAB ¼ dxji x Ajix þ X B. Changing to A, B natively, in the Hamiltonian formalism, the transformation in Eq. appears in a fixed position, while C is spread over the whole space (2) can be fixed uniquely (up to a constant in Eq. (2)) by requiring that it is canonical, linear in phase-space observables, and does there is no unambiguous notion of ‘jumping’ to a reference frame. not mix coordinates and momenta. In this paper, we work with Note that this feature arises both in classical and quantum infinite-dimensional Hilbert spaces isomorphic to L2ðRÞ with mechanics from the requirement of canonicity of the reference- standard measure dμ(x) = dx. It may happen that the system frame transformation when the reference frames are considered constituting the new reference frame, say A, is a composite system as physical degrees of freedom, and therefore attributed a phase from the point of view of C. For instance, A could be composed of space. The expression “jumping” to a QRF is to be intended, in different particles (for example, imagine an atom made of protons the rest of the paper, in a loose sense, up to the choice of a specific and neutrons). Our formalism can be applied to this situation by transformation and basis. defining the transformation to jump, for instance, to the degrees Some examples of transformed states according to the map in of freedom of the centre of mass. In this case, the internal degrees Eq. (2) are given in Fig. 3. In particular, we see in Fig. 3a that of freedom can be transformed like any external system B. In when the new reference frame A is very sharp in position basis Methods (Application: notion of rest frame of a quantum system) and the initial state in C’s reference frame is |ψ〉 = |x 〉 |ψ〉 , we provide another example of how our formalism can be applied AB 0 A B from A’s point of view the state of B is translated by x , and the to a composite system with discrete internal degrees of freedom. 0 state of C is also sharp.R The state in the new reference frame The general procedure that we follow to perform the canonical would then be jψi ¼ dq ψðq þ x Þjq i jÀx i . This cor- transformation is to choose a basis in which we want to express BC B B 0 B B 0 C responds to the translation of a classical reference frame by an the relative quantities, and then complete it canonically. Note that amount x , since transformation ^S applied to the well-localised any quadrature in the phase space could be considered as the 0 x state of A takes the form of the standard translation operator relative variable. Different choices of relative coordinates would ^ ^ ix ^p S jx i ¼ P eh 0 B jx i . (Up to the parity-swap operator that induce different transformations between QRFs. In Eq. (1)we x 0 A AC 0 A specifies the relative position of the two reference frames, which is have chosen position basis to define the relative coordinates in C usually ignored in the standard framework.) and A, but we could have chosen the eigenbasis of, for instance, In Fig. 3b we illustrate the case in which theÀÁ state of A is a relative momenta. In this case the transformation is 1 i^ ^ ϕ pffiffi ^ ^ ÀpAxB superposition of two sharp states, i.e. jiA ¼ jix1 Aþ jix2 A . Sp ¼ PACe h , and it gives rise to the following canonical 2 ^ π^ π^ ^ π^ ^ ^ In general, if C describes the joint state of A and B as a transformation: pB 7! B À C, pA 7!À C, xB 7! qB, and ^ ^ ^ product state |ϕ〉 |ψ〉 , the state in the reference frame of A xA7!ÀðqC þ qBÞ. The possibility of choosing different relative A B coordinates shows that, when we promote a physical system to a is entangled and is obtainedR as the convolution product of ^ ϕ ψ ϕ ψ reference frame, the question what the description of the rest of the two, SxjiAjiB ¼ dqBdqC ðÀqCÞ ðqB À qCÞjiqB BjiqC C. the world is relative to the reference frame is ill-posed unless a Analogously, if the states of A and B are entangled in the initial choice of relative coordinates is met. An equivalent statement is reference frame, this property might not hold after changing to that, when the reference frame is considered as a physical system, the reference frame of A. Examples of this situation are given

4 NATURE COMMUNICATIONS | (2019) 10:494 | https://doi.org/10.1038/s41467-018-08155-0 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-018-08155-0 ARTICLE in Fig. 3c, d. In particular, in Fig. 3c we consider an quantum system has been confirmed as unitary only with respect entangled state of A and B in position basis, where there is a to an ‘abstract’ reference frame (in the sense explained in the perfect correlation between A and B, i.e. Introduction). Such an abstract reference frame can be approxi- jψi ¼ p1ffiffi ðjx i jx þ Li þjx i jx þ Li Þ. From the point of mated, in our description, as a very massive and classical-like AB 2 1 A 1 B 2 A 2 B – view of A, the state of B and C is in a product state. In particular, reference frame. Then the mathematical steps from Eqs. (3) (5) = show that, starting from a classical-like reference frame, the B appears localised at the positionÀÁqB L, while the state of C is in the superposition state p1ffiffi jiÀx þÀjix . Similarly, if A evolution can be described unitarily also from any QRF, including 2 1 C 2RC those which are not massive, provided that there can exist a ψ ^ and B are entangled in the EPR state jiAB ¼ dxji x Ajix þ X B as transformation like the S operator to change the reference frame. = in Fig. 3d, A sees B localised at position qB X, while C is spread Therefore, C can be taken as a QRF also in Eq. (3). over the whole space. A transformation that leaves the Hamiltonian invariant is From the examples considered it is clear that the notions of called a symmetry transformation. The symmetry of a transfor- superposition and entanglement are reference-frame dependent. mation implies the existence of a conservation of a dynamical fi Speci cally, this fact implies that a quantum particle in a spatial observable of the system. In the next sections we will identify superposition from the point of view of a laboratory would in symmetry transformations between QRFs. An important differ- turn attribute to the laboratory a state which is in a spatial ence between classical and quantum RF transformations is that, superposition. This notion of frame-dependence of the features of in the latter case, the Hamitonian as seen from one QRF not only a quantum state is different to the one typically found in the includes the observed system B but also the other QRF. We then literature, where the frame-dependence always appears due to the define a symmetry transformation as a map that leaves the fi decoherence of the state of the system as speci ed in the external functional form of the Hamiltonian invariant, i.e. the Hamilto- reference frame and after tracing out this frame (see, e.g., nian of A and B is the same function of operators as the 4,8,20,21 refs. .). (More details on the differences between our Hamiltonian of C and B, approach and the existing literature on QRFs can be found in ^  Methods (Comparison of previous approaches to quantum ^^ ^ ^y dS ^y ^ SH fm ; ^x ; p g ; S þ ih S ¼ H fm ; ^q ; π^ g ; ; ð6Þ reference frames)) i i i i¼A B dt i i i i¼B C The dependence of entanglement and superposition on the reference frame is known to appear in relativistic quantum theory where the operators and the mass of A are simply replaced by the 31 ones of C. It can be shown that if condition (6) is satisfied, then due to relativity of simultaneity in different reference frames . ^ Frame-dependent entanglement between momentum and the transformation S allows to define a map between the degrees of freedom also appears in relativistic quantum informa- dynamicalno conserved quantities in the reference frame C, tion when a state is Lorentz-boosted to a reference frame moving ^ðCÞ Ci , with N 2 N, to the dynamical conserved quan- with constant and uniform velocity with respect to the initial i¼1;:::;N no 32 ^ðAÞ one . The boost on the state is represented through a Wigner tities in the reference frame A, Ci . In particular, these rotation, which couples the spin and the momentum. Here we i¼1;:::;N show that the effect can arise due to genuine quantum quantities have the same functional form, but with all the labels A relationships between reference frames even in non-relativistic and C swapped. We show this in Supplementary Note 3. quantum mechanics. As discussed at the beginning of Results (Transformations between quantum reference frames) and in detail in Supplemen- tary Note 1, when reference frames are treated as abstract entities, Dynamics and symmetries from a quantum reference frame. the relationship between the old and the new reference frame i n ^ n We will now derive the Schrödinger equation as seen from a QRF. ^ Π  f ðtÞOB fi enters as a function in the transformation Ui ¼ neh , where More speci cally, starting from the Hamiltonian in the initial fn(t) depends on the specific transformation between two reference frame C, we will derive the dynamical law relative to A. ^ reference frames, and specifically on the displacement of the Here, the transformation S from C to A is a completely general two reference frames X(t) and its time-derivatives. The operator unitary operator, and may depend explicitly on time. In addition, ^ ’ OB acts on B s Hilbert space. The product over the index n is due we assume that the states of A and B with respect to the reference to the fact that, in a general transformation, we might not be able frame C satisfy the Schrödinger equation: to decompose the transformation into a single product of a hi d^ρðCÞ function of A and an operator of B. This condition translates to  AB ^ ðCÞ; ^ρðCÞ ; ð3Þ n ih ¼ HAB AB our formalism by promoting the functions f (t) to time- dt ^n fi dependent operators fA ðtÞ. This transformation is speci ed in ^ρðCÞ the time-dependent operators of A, i.e., in the Heisenberg picture. where AB is the state of A and B (which can be either pure or ^ ^ ðCÞ We want to apply the transformation S to the states of A and B at mixed) and HAB their Hamiltonian relative to C. time t, i.e. to their states in the Schrödinger picture, and to obtain ^ρðAÞ ^^ρðCÞ^y The state of B and C relative to A is given by BC ¼ S AB S . the transformed state of B and C at time t. This implies the Differentiating this expression with respect to time and applying following structure of general ^S operator: the Leibniz rule we obtain ^ ^ À iH^ t ^ðiÞ if nðtÞO^ n i H^ t hi S ¼ e h C P Π eh A B eh A ; ð7Þ d^ρðAÞ AC n ih BC ¼ H^ ðAÞ; ^ρðAÞ ; ð4Þ dt BC BC where the prescription to change QRF can be described through the following steps: (a) we first map the state of A to the where Heisenberg picture by evolving it back in time with the ^ ^ ðAÞ ðAÞ dS Hamiltonian HA, (b) we then apply the generalisation of ^ ^^ ^y  ^y: ð5Þ ^n HBC ¼ SHBC S þ ih S the classical transformation using the operators f ðtÞ and (c) dt ‘ ’ ^ðiÞ A we apply the generalised parity operator PAC to exchange the At first sight, the Schrödinger equation in Eq. (3) for a general equations of motion of C and A (e.g. depending on the specific quantum reference frame C might look unjustified. According to transformation i chosen, the parity operator ensures that the our current status of experimental tests, the evolution of a position, velocity or acceleration of A from the point of view of C

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Action on system B Classical RF (action on A is omitted) transformation

i Relative xˆ qˆ – qˆ X (t)pˆ ˆ B B C ˆ = eh Sx  Ux coordinates pˆB ˆB

 ˆC xˆ qˆ − qˆ + (t − ) i X pˆ Superposition B B C 0 ˆ mC ˆ = eh ST  UT of translations pˆB ˆB  ˆC ˆ qˆ i xB B − m t (t p ˆ – m xˆ ) Superposition ˆ C ˆ h S Ub = e of boosts b  mB  pˆB ˆB – ˆC mC  ˆC 1 dV (q ˆ C ) xˆ qˆ − tt+ 2 2 2 3 Superposition B B m 2m – i mat xˆ i at pˆ – i ma t ˆ C A dqˆC ˆ h h 2 h 6 SEP Ua = eee of accelerations  mB  pˆB ˆB – ˆC mC

Fig. 4 Table summarising different quantum reference frame (QRF) transformations on system B. (The action on A is omitted) The time-independent transformation ^S is the QRF generalisation of a standard reference frame (RF) transformation where the reference frame is moving along X(t), and the x ^ ^ ^ relative coordinates describe the distance between systems A and B at time t. The three QRF transformations ST, Sb, and SEP generalise the extended Galilean transformations to a reference frame which is respectively translated, moving with constant and uniform velocity, and moving with constant and uniform acceleration. In particular, ^S transforms the old coordinate x into the relative position between system B at time t and system A at time τ, and ^ T ^ B reduces to S for t ¼ τ. The transformation S performs a Lorentz boost on system B, where the velocity is written in terms of dynamical variables of x ^ b system A. Finally, the transformation SEP, generalises the transformation to an accelerated reference frame to when system A moves in a superposition of accelerations. Importantly, the extension of the RF transformation to dynamical quantities of quantum systems makes it possible to introduce a generalised fi ^ ^ notion of symmetric transformation, which is exempli ed, in this work, in the case of the ST and Sb transformations are opposite to the same quantities of C from the point of view of position xB of particle B at time t in C into the relative position of A, see below for more details); finally (d) we map the state of C B at time t and the position of C at time τ (see Fig. 4 and ^ back to the Schrödinger picture via the Hamiltonian HC. Methods). In addition, this transformation is a symmetry, in the Equation (7) is in fact the most general QRF transformation extended sense of QRFs, for the free particle Hamiltonian, ^2 ^2 π^2 π^2 achievable in one dimension when A and B do not interact, and ^ ðCÞ pA pB ^ ðAÞ B C because it maps HAB ¼ þ into HBC ¼ þ . contains as particular cases the (extended) Galilean transforma- 2mA 2mB 2mB 2mC tions in one dimension. Via this transformation, it is possible to The second case we consider is the generalisation of the “ ” identify a general method to quantise a reference frame Galilean boosts to the superposition of boosts, corresponding to transformation. Firstly, one should identify the type of transfor- when particle A moves in a superposition of momenta (velocities) mation (e.g., translation, boost, accelerated reference frame, etc.). from the point of view of C. This case is discussed in detail in Secondly, by solving the equations of motion of the system Methods (Boosts between quantum reference frames). In this constituting the new reference frame, one should quantise the case, the QRF transformation is dynamical variables of the classical reference frame appearing in  π^2 ^ ^2 i ^ðvÞ i pA i p the transformation by using the phase space operators of the new ^S ¼ exp À C t P exp G^ exp A t ; ð9Þ QRF. This constitutes the central part of Eq. (7). Finally, one b h 2m AC h m B h 2m ^ ^ C A A should add the two propagators with Hamiltonians HA and HC and choose a generalised parity-swap operator in such a way that and has the physical meaning of “jumping” to the rest frame of a the solutions of the equations of motion of system C from the quantum system described by a free-particle Hamiltonian point of view of A are of opposite sign to those of the equations of ^ ðCÞ ^p2 ^p2 H ¼ A þ B . This transformation is also a symmetry for motion of A from the point of view of C, i.e., they are equal up to AB 2mA 2mB a minus sign. the free-particle Hamiltonian, which is transformed into π^2 π^2 We next exemplify this procedure through the generalisation of ^ ðAÞ B C HBC ¼ 2m þ 2m . the extended Galilean transformations to QRFs, which is Finally,B we generaliseC the transformation to a constantly discussed in detail in Methods and summarised in Fig. 4.In ^ accelerated reference frame to the QRF transformation SEP (see particular, the Galilean translations can be generalised by Methods (The weak equivalence principle in quantum reference considering the relative coordinates as defined relative to the τ frames)), describing the change to a QRF moving in a quantum state of a system A at some time (see Methods superposition of accelerations. This superposition is achieved by (Translations between quantum reference frames) for more choosing as initial Hamiltonian details). Following the general scheme introduced in Eq. (7), this ^p2 ^p2 ^ ðCÞ ^ ^ A B ^ ^ transformation can be written as HAB ¼ HA þ HB ¼ 2m þ 2m þ VðxAÞ, where VðxAÞ is piecewise  A B π^2 ^2 linear, and the system A evolves in a superposition of two ^ i C τ ^ðxÞ i ^ ^ i pA τ : amplitudes, each of which is localised in a region corresponding ST ¼ exp À  ðt À Þ PACexp  xApB exp  ðt À Þ h 2mC h h 2mA to a single gradient of the potential. The QRF transformation is ð8Þ expressed as  π^2 ^p2 i C mC ^ i A ^ In this case, the QRF is provided by the quantum state of À þ VðÀq Þ t  þVðx Þ t ^ h 2mC mA C ^ðvÞ ^ h 2mA A ; ð10Þ fi τ ^ SEP ¼ e P Qte system A at some xed time . The transformation ST maps the AC

6 NATURE COMMUNICATIONS | (2019) 10:494 | https://doi.org/10.1038/s41467-018-08155-0 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-018-08155-0 ARTICLE where our formalism to the situations in which the reference frames are   R 2 related via ‘superposition of translations’ and ‘superposition of ^ ^ ^ t ^ i mB ^ dVðxAÞ ^ i ^ 1dVðxAÞ pB i mB ^ dVðxAÞ À p À ^ t x  p À ^ t t À ds pAÀ ^ s ’ ^ hmA A dxA B h A 2 dxA mA h2m2 0 dxA Galilean boosts , and formulate an extension of the weak Qt ¼ e e e A equivalence principle for such quantum reference frames. ð11Þ This work has been carried out within Galilean relativity, however the framework is general and can be applied in a special- is the straightforward extension of the usual transformation to an relativistic or in a general-relativistic context. This would lead to accelerated reference frame. The new Hamiltonian from the point interesting insights as to, for instance, the flow of proper time of view of A is when there is no classical worldline describing the motion of the system serving as reference frame. More specifically, our form- π^2 π^2 m m dV H^ ðAÞ ¼ B þ C þ C VðÀ^q ÞÀ B ^q ; ð12Þ alism could be able to describe situations, such as those studied in BC C ^ ^ B 33–35 2mB 2mC mA mA dxA ÀqC refs. , in which clocks—quantum systems with internal degrees of freedom—move in superpositions of classical wor- where a non-inertial term appears, and particle B moves in a dlines in the gravitational field. As a result, the clock’s internal gravitational potential with gravitational acceleration depending and external degrees of freedom get entangled, because the clock’s on the acceleration of particle C. Once the transformation of the proper time depends on the worldline taken in the superposition. quantum state is taken into account, it is possible to see (see In these situations, proper time is measured by the clock in its rest Methods for details) that the system B evolves as if it were in a frame, but currently no complete formalism is known which superposition of uniform gravitational fields. The weak equiva- would allow to transform to the rest frame of a clock that is in lence principle states that physical laws as seen from a reference superposition of positions or momenta with repect to the frame moving with a constant and uniform acceleration are laboratory frame. Already in the present work we provide a indistinguishable from those as seen in an uniform gravitational solution to this problem in the low-velocity limit to explain the field. Our result extends the principle to the equivalence between Doppler-shift induced transitions for atoms in superpositions of the physical laws as seen from a QRF in a superposition of momenta (see Methods (Application: notion of rest frame of a constant and uniform accelerations and those as seen in a quantum system). We move to the rest frame of the atom, superposition of uniform gravitational fields. compute the transition probabilities for the incoming light fre- This completes our discussion of how the extended Galilean quencies in this frame, and then move back to the laboratory transformations can be generalised to QRFs. In all the examples frame. provided, we considered situations in which A and B initially do An alternative future direction of our work concerns the not interact. The most general case, in which systems A and B application to future experiments, in particular those able to test evolve in a general, interacting potential, will be object of future relative variables, such as the techniques in refs. 36–39, and those investigation. involving ‘macroscopic’ systems (e.g. nanomechanical oscillators), This method to quantise a reference frame transformation which could play the role of ‘large’ quantum reference frames, allows us to define the transformation to the rest frame of similarly to the situation considered in ref. 26. Experiments with a quantum system, e.g., a system moving in a superposition these systems could shed light on some conceptual issues of of velocities from the point of view of the laboratory. This quantum gravity at low energies, such as those related to quan- has important applications in the study of the internal degrees tum fluctuations of the spacetime or superposition of large of freedom of quantum systems. In Methods (Application: masses. notion of rest frame of a quantum system) we provide an It would also be interesting to investigate whether allowing example of such a situation. Finally, in Methods (Measurements observers to be in a superposition or entangled with other systems as seen from a quantum reference frame) we show how the could lead to scenarios with indefinite causal structures, such as description of a measurement procedure changes with the those in ref. 40, where a global time-order cannot, at least in change of a QRF. general, be imposed, but the observers are in well-defined posi- tions. Our formalism for quantum reference frames can be seen Discussion as a dual picture to this work: while a global time order can still be In this work we introduced an operational formalism to apply found, at least in the Galilean-relativistic case, the observers are quantum mechanics from the point of view of a reference frame not localised. attached to a quantum particle, which we call quantum reference frame. This reference frame has its own degrees of freedom, Methods which can be in quantum superposition or entangled and evolve Translation between quantum reference frames.Wefirst consider the case in in time according to their own Hamiltonian with respect to the which we change to the reference frame described by the position of the quantum system A at a particular instant of time τ, when the initial Hamiltonian for A and B ^2 ^2 laboratory frame of reference. We adopt a relational view, ^ ðCÞ pA pB ^ is HAB ¼ 2m þ 2m . In this case, the operator XAðtÞ generalises the function A B ^p2 p^2 according to which any reference frame is described as a quantum i A τ i A τ ^  À XðtÞ¼X , with X being a constant, and takes the form X ðtÞ¼eh2mA ^x e h2mA , degree of freedom relatively to another reference frame: hence, 0 0 ^ A A the laboratory frame of reference is a quantum system relative to and the full operator S is the quantum reference frame of a particle, much like the particle is a quantum system relative to the laboratory frame. This allows  π^2 ^2 ‘ ’ ^ i C τ ^ðxÞ i ^ ^ i pA τ ; us to avoid assuming the existence of an external perspective of ST ¼ exp À  ðt À Þ PACexp  xApB exp  ðt À Þ ð13Þ an absolute reference frame. h 2mC h h 2mA We find transformations between quantum reference frames, and show how the state, the dynamics, and the measurement ð Þ π^2 ^ x ^ i C τ change under these transformations. We show that the notion of where PAC ¼ PAC in Eq. (2) and we have introduced the term expðÀ h 2m ðt À ÞÞ to ensure that the position of the system A at time τ tranforms into the symmetricC entanglement and superposition are observer-dependent features, ^ ^p ^y π^ position of the system C, i.e. S ð^x À A ðt À τÞÞS ¼Àð^q À C ðt À τÞÞ. T A mA T C mC and we write the Schrödinger equation in quantum reference Notice that for t = τ the operator ^S in Eq. (13) is precisely the operator ^S in Eq. (2). frames. Furthermore, we introduce a generalised notion of cov- ^ T x Therefore, we can interpret Sx as the operator which performs the translation to a ariance of physical laws for quantum reference frames. We apply quantum reference frame when the dynamics is “frozen” at time τ. The

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momenta transform as a C ^ ^ π^ π^ π^ A A ^ ^y mCqC þ mBqB C þ B C ^ ^y mA B S ^x S ¼À þ t À t; S ^p S ¼À π^ ; ð17Þ b A b m m m b A b m C   A A C C 1 2 pB

π^ m b ^S ^x ^Sy ¼ ^q À C t; ^S ^p ^Sy ¼ π^ À B π^ : C b B b B b B b B C ð18Þ C mC mC A B B   – 1 pB – 2 pB ^ This transformation, similarly to the transformation ST discussed previously, is also a symmetry of the free-particle Hamiltonian, because it maps the initial ^2 ^2 Fig. 5 Schematic illustration of the descriptions in two quantum reference ^ ðCÞ pA pB Hamiltonian HAB ¼ þ to the Hamiltonian in the new reference frame frames that are boosted with respect to each other: a the state of A and B 2mA 2mB ðAÞ π^2 π^2 H^ ¼ B þ C through Eq. (5). Hence, this constitutes a Galilean boost as described from C, and b the state of B and C as described from A. In a, BC 2mB 2mC fi the state of A and B is a product state, and the state of A is in a transformation for quantum reference frames, which allows the system de ning the reference frame to be in a superposition of velocities. superposition of the two velocities v1 and v2. By applying a ‘superposition of To illustrate this point, we consider the situation depicted in Fig.R 5.Weconsidera boosts’ by the velocity of A, we find that, as seen from A, the state of B and i ^ ðCÞ Ψ ÀhHAB t ϕ ψ ϕ ϕ state jit AB¼ e 0 A 0 B, where the initial state j 0iA ¼ dpA 0ðpAÞjpAiA C is entangled. In particular, the entanglement is such that if C moves with of system A is in a superposition of momenta with respect to the initial reference velocity −vi, i = 1, 2, from A’s point of view, B is boosted by −vi frame C. We now change perspective to the reference frame A. No simple coordinate transformation of reference frames could capture this change. Our method gives, as a ^ result of the transformationSb, the entangled state of B and C π2 π2  R i C B π ^ À þ t i C ^ transformation implemented by S is ^ h 2mC 2mB mA À GB T S jiΨ ¼ dπ dπ e ϕ À π ψ ðπ Þe hmC jiπ jiπ . The state b t AB C B 0 mC C 0 B C C B B π^ π^ π^ of B is boosted by the velocity of A (which corresponds to the opposite of the velocity ^ ^ ^y ^ C τ B þ C τ ; ^ ^ ^y π^ π^ ; STxAST ¼ÀqC þ ðt À ÞÀ ðt À Þ STpAST ¼Àð B þ CÞ of C, given Eq. (18)) for each momentum in the superposition state of A, while the mC mA system C evolves as a free particle with opposite velocity to A. ψ 〉 ð14Þ In the special case of a free particle B in the general state | 0 B and the reference fi ϕ 〉 = 〉 frame A having a state with a well-de ned momentum (velocity) | 0 A |pA A, the π2 π i C i C ^ À t À G ^ Ψ h2mC π hmC B ψ ψ 〉 transformed state Sbjit AB¼ e jiC Ce j t iB, where | t B is the time π^ m y y π C ^ ^ ^ ^ ^ ^ C τ ; ^ ^ ^ π^ : evolved state and C ¼Àm pA, reduces to the standard boost transformation Ub STxBST ¼ qB À qC þ ðt À Þ STpBST ¼ B ð15Þ A mC in the usual description of reference frames, with the difference that here C is a degree of freedom and hence evolved in time. With a similar reasoning to the one presented in the previous section, it is ^ ; ^ ; ^ ; ^ Equation (15) implies that the position at time t of system B from the point of possible to show that the set of the conserved quantities pA pB GA GB in the view of C is mapped into the relative position between system B and the position of reference frame C is mapped to the set of the conserved quantities in the reference π^ ; π^ ; ^ ; ^ A at time τ, while the momentum of B remains unchanged. In addition, this frame A B C GB GC. Analogously to the generalised translations in the previous fi ^ fi transformation is a symmetry of the free particle according to the de nition given Subsection, this choice of the operator Sb also satis es the transitive property, i.e. ^ ðCÞ ^ðC!AÞ ^ðB!AÞ^ðC!BÞ in Eq. (6), because the Hamiltonian HAB is mapped through Eq. (5)to Sb ¼ Sb Sb . ðAÞ π^2 π^2 ^ ^ B C ^ Notice that a time-independent version of the transformation Sb, mapping to HBC ¼ þ . Therefore, the transformation ST in Eq. (13) constitutes a 2mB 2mC instantaneous relative velocities, would not preserve the invariance of the generalisation of the Galilean translations to quantum reference frames. The Hamiltonian. This example is discussed in Supplementary Note 4. simplest example of dynamical conserved quantities, in this case, are the two momenta C^ðCÞ ¼ ^p and C^ðCÞ ¼ ^p . It is immediate from Eqs. (14) and (15)tosee 1 A 2 B The weak equivalence principle in quantum reference frames ^ðAÞ ^ ^ðCÞ^y π^ ^ðAÞ ^ ^ðCÞ^y ^ ^ðCÞ^y π^ . In this section that the choice C1 ¼ STC2 ST ¼ B and C2 ¼ÀSTC1 ST À STC2 ST ¼ C we generalise the weak equivalence principle to quantum reference frames. By this, leads to the corresponding conserved quantities in the reference frame A. A similar we mean that the physical effects as seen from a reference frame moving in a procedure holds when we consider the extended set of conserved quantities superposition of uniform gravitational fields are indistinguishable from those as ^ ^ ^ ^ ; composed of translations pi and Galilean boosts Gi ¼ pit À mixi, i ¼ A B. Notice seen from a system in superposition of accelerations. To achieve a superposition of ^ fi that this construction of the ST operator satis es the transitive property, meaning accelerations, let us consider the situation depicted in Fig. 6, in which two particles that changing the reference frame from C to A directly has the same effect as A and B evolve in time according to the Hamiltonian changing the reference frame first from C to B and then from B to A, i.e. ^ðC!AÞ ^ðB!AÞ^ðC!BÞ ^2 ^2 S ¼ S S . ^ ðCÞ ^ ^ pA pB ^ T T T HAB ¼ HA þ HB ¼ þ þ VðxAÞ ð19Þ 2mA 2mB Boosts between quantum reference frames. The second example we consider in the reference frame of an observer C. is the change to a reference frame moving with the velocity of a quantum ^ system A, which is described as a free-particle from the point of view of the initial For the purpose of further analysis we will now consider the potential VðxAÞ to observer C. The total Hamiltonian for both systems A and B from C’s point of be piecewise linear and particle A to evolve in time t as a superposition of wave ^2 ^2 ^ ðCÞ pA pB amplitudes, each localised in an interval that corresponds to a constant yet different view is HAB ¼ þ . This section generalises the usual Galilean boost i ^ 2mA 2mB  ^ vG ^ ^ potential gradient. For concreteness consider the superposition of two such U ¼ eh B , with G ¼ p t À m ^x being the generator of the boost on system B, b B B B B amplitudes, ψ ðtÞ ¼ p1ffiffi ψ ðtÞ þ ψ ðtÞ (see Fig. 7). The state then is in a introduced in Results (Transformations between quantum reference frames), to 0 A 2 1 A 2 A superposition of accelerations, i.e. the ‘acceleration operator’ applied on the state situations in which the velocity of the reference frame is distributed according to ^ ^ dXA ðtÞ pA gives: its quantum state. With reference to Eq. (7), in this case we have dt ¼ m ^ A  generalising the parameter v in Ub. The complete transformation, which we call 1 dVð^x Þ 1 ^ À A ψ ðtÞ  pffiffiffi a ψ ðtÞ þ a ψ ðtÞ ; ð20Þ Sb,is ^ 0 A 1 1 A 2 2 A mA dxA 2  π^2 ^ ^2 ^ i C ^ðvÞ i pA ^ i pA ^ ^ S ¼ exp À t P exp G exp t ; ð16Þ 1 dVðxA Þ 1 dVðxA Þ b  AC  B  where a1 ¼À ^ j and a2 ¼À ^ j , where x1(t) and x2(t) h 2mC h mA h 2mA mA dxA x1ðtÞ mA dxA x2ðtÞ are the mean values of position operator for the individual localised amplitudes. qffiffiffiffiffi Notice that the scalar accelerations a1 and a2 as well as the scalar positions x1(t) ^ðvÞ ^ i mC ^ ^ ^ ^ where the ‘generalised parity operator’ P ¼ P exp  log ðx p þ p x Þ AC AC h mA A A A A and x2(t) should be understood as multiplied by the identity operator. maps the velocity of A to the opposite of the velocity of C via the standard fi ^ ^ In order to nd the generalised version of the operator Ua, discussed in Results parity-swap operatorPAC and an operator scaling coordinates and momenta. (Transformations between quantum reference frames) and in detail in ð Þ ð Þ y ð Þ ð Þ y fi ^ v ^ ^ v mC ^ ^ v ^ ^ v mA π^ Speci cally, PACxA PAC ¼Àm qC, PACpA PAC ¼Àm C. This choice of Supplementary Note 1, and get an expression analogous to the one in Eq. (7), we ^ A C ^ Sb ensures that the velocity of A in the reference frame of C is opposite to the need the time derivative of the position operator xA at time t. To calculate the ^ ^p ^y π^ velocity of C in the reference frame of A, i.e. S A S ¼À C . The coordinates and evolved position operator, we write an explicit expression for b mA b mC

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abC C B V (xA) B g1

Δx C A B g2   2 1 2 2 0 a 1 x1 x2 x Δ A x 2 (t)  A 1 2 2(t) t   x1 x2 a2 Fig. 7 Representation of the conditions under which particle A effectively Fig. 6 Generalisation of the weak equivalence principle for quantum moves in a superposition of accelerations. The state of particle A and the reference frames. The quantum system A is initially in a superposition of piecewise linear potential Vð^x Þ are represented for the initial time 0 and ^ A two localised wave amplitudes in a piecewise linear potential VðxAÞ from for the time t in two different yellow planes. At the initial time, in the the point of view of another system C. The individual wave amplitudes are background, the state is chosen to be a superposition of two coherent localised in spatial intervals corresponding to two different potential σ (Gaussian) states localised around x1, with width 2 1, and x2, with width gradients. The system B evolves instead as a free particle. If we consider σ 2 2. The individual states are localised within the spatial intervals that the motion for sufficiently short , such that the two amplitudes remain correspond to constant but different potential gradients. Under time localised within the corresponding intervals, the system A evolves as if it evolution, the point where each of the two localised states is centred moves were in a superposition of the accelerations a and a . We can then change δ ; 1 2 by xi, i ¼ 1 2, and the wave-packet spreads. For each of them, it is possible perspective to the accelerated reference frame of A, by applying a to identify a maximal time such that the individual localised states in the transformation corresponding to a ‘superposition of accelerations’, and superposition still remain in the region where the gradient of the potential is describe how the quantum system A sees the quantum systems B and C constant. Up to this time A evolves in a superposition of accelerations and their evolution. After the transformation, the system B evolves as if it was moving in a superposition of linear gravitational potentials, where the fi well de ned position xiðtÞ and momenta piðtÞ, where the momentum at time t is ^ gravitational accelerations are such that ~g ~a , where i 1; 2. This ^ ^ dVðxA Þ i ¼À i ¼ calculated analogously to ^x ðtÞ, i.e. p ðtÞ¼p À ^ t. We denote the state of A A A dxA means that the effects of a superposition of accelerations are α ; each amplitude as j iðtÞi, with i ¼ 1 2. Hence we obtain indistinguishable from the effects of a superposition of gravitational fields 1 1 ^S pffiffiffi ðjα ðtÞi þjα ðtÞi Þjϕi ¼ pffiffiffi ðjα′ðtÞi Q1jϕðtÞi þjα′ðtÞi Q2jϕðtÞi Þ; EP 2 1 A 2 A B 2 1 C t B 2 C t B i ^ i ^ ^ HA t ^ ÀHA t xAðtÞ¼eh xAe h : ð25Þ ^p 1 1 dVð^x Þ  ^x ðtÞ¼^x þ A t À A t2: ð21Þ A A ^ α′ 〉 mA ; mC mA 2 mA dxA where the transformed coherent state | i(t) C is centred in À xiðtÞ À piðtÞ  R mC mA p t a t2 m t m i i i ^ i B 2 i B ^  À pB À 2 dsðÞ pi ÀmA ai s i À ðÞpi ÀmA ait xB h mA 2 h2m 0 and Q ¼ e hmA e e A . From A’s point of view, B ^ ^ ^ t From this expression we come to the generalised XAðtÞ¼xAðtÞÀxA, evolves in a superposition of gravitational accelerations, which is controlled by the which describes the change in time of the position of A as compared to the state of C. initial position and replaces the function X(t) in the extended Galilean We conclude that we have a generalised form of the weak equivalence principle transformation U^ . a ^ which holds when the reference frame is a quantum particle in superposition of Following Eq. (7), the overall transformation S reads ^ EP accelerations. This analysis can be extended to a general potential VðxAÞ acting for  fi π^2 ^p2 in nitesimal times, as we show in Supplementary Note 5. i C mC ^ i A ^ À þ VðÀq Þ t  þVðx Þ t ^ h 2mC mA C ^ðvÞ ^ h 2mA A ; ð22Þ The extension of the weak equivalence principle to quantum reference frames SEP ¼ e PACQt e provides a good opportunity to test this framework in an experiment. In general where the operator P^ðvÞ was defined below Eq. (16) and the operator Q^ is defined terms, a suitable technique to verify the predictions of this section would be AC t measuring relative degrees of freedom, for instance as done in refs. 36–39. However, as fi   a speci c proposal on how to do so goes beyond the scope of this work. R 2 ^ ^ ^ t ^ i mB ^ dVðxA Þ ^ i ^ 1dVðxA Þ pB i mB ^ dVðxA Þ It is possible to recover the usual notion of the weak equivalence principle if the À p À ^ t x  p À ^ t t À ds pA À ^ s ^ hmA A dxA B h A 2 dxA mA h2m2 0 dxA ð23Þ ^ ^ Qt ¼ e e e A potential is linear in the entire space, i.e. VðxAÞ¼mAaxA. The generalised form of ^ 2 the displacement of the reference frame reads X^ ðtÞ¼ pB t À at and the operator B mA 2 ^ _ R and represents the straightforward extension of the operator U . Note that ^ ^ À im X^ ðtÞ^x iX^ðtÞ^p À1 mB t ^_ 2 DE DE a S is analogous to Eq. (22) with Q ¼ e h B A B e h B e h 2 dsX ðsÞ, where 2^ 2^ EP t 0 A d xA d qC 2 ¼À 2 . Using the transformation in Eq. (22), the new Hamiltonian ^ fi dt dt XAðtÞ has been previously de ned and the dot indicates the time derivative. The from the point of view of A is ^ ðCÞ initial Hamiltonian HAB in Eq. (19) is transformed to

π^2 π^2 m m dV π^2 π^2 ^ ðAÞ B C C ^ B ^ : ^ ðAÞ B C HBC ¼ þ þ VðÀqCÞÀ qB ð24Þ H ¼ þ À m a^q À m a^q : ð26Þ 2m 2m m m d^x À^q BC C C B B B C A A A C 2mB 2mC

From Eq. (24), we can see that B evolves in a potential which is determined by This result shows that the weak equivalence principle holds also if the reference fi ^ the rst derivative of the potential at the position ÀqC, while C moves in a potential frame is treated as a quantum system (and can therefore be delocalised) with its given by the sum of mC VðÀ^q Þ and the interaction term involving its derivative. mA C own dynamics. Hence, the quantum system B moves, in the reference frame of A, as if it were in a linear gravitational potential with a gravitational acceleration being an operator Application: notion of rest frame of a quantum system. In this section, we show ^g ¼ 1=m dVð^x Þ=d^x j ^ in the Hilbert space of C. This is a formulation of the A A A ÀqC how our formalism enables us to define the notion of rest frame when the system is weak equivalence principle in QRF. in a superposition of momenta from the point of view of the initial laboratory ^ ϕ As an example, we will now apply SEP on an arbitrary state j ðtÞiB of B and on a frame. The rest frame of a system is the frame of reference in which the system is at ψ state j ðtÞiA of A, for which we assume that the two localised wave amplitudes rest. Physical laws standardly take a simple form in the rest frame; for example, the were initially prepared as non-overlapping coherent states (i.e. minimum rest frame Hamiltonian gives the dynamics of the internal degrees of freedom fi uncertainty wave-packets) with well de ned position xið0Þ and momenta pið0Þ, (e.g. spin). It is therefore useful to know how to map the descriptions in the rest i ¼ 1; 2. We evolve the state of A for time t such that the two amplitudes still have and the laboratory frames of reference. As long as the system moves along a

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ω π ^ a  frequency holds, i.e. h B ¼ cj Bj. Finally, HA~ ¼ Eg jgiA~ hgjþEejeiA~ hej is the B Δ Hamiltonian of the internal degrees of freedom, with Ee À Eg ¼ E. C To change the reference frame, we apply a boost transformation between A and A˜ ΔE C, and the transformation which gives the Doppler shift on the photon. Overall, B   we obtain – 1 – 2 ^p2 π^2 À i A t ðvÞ i C t ^ h2m ^ ^B h2m ; ð27Þ SD ¼ e A P R π^ e C b   CA fC ð C Þ B1 B2 A B A˜ ΔE ^ðvÞ ^ðvÞ fi where the operator PCA is the adjoint of PAC de ned after Eq. (16), i.e. y ÀÁpffiffiffiffiffiffiffiffiffiffiffiffiffi   ^ðvÞ ^ðvÞ ^B i π^ ^ π^ π^ ^ 1 P ¼ P , and R π^ ¼ exp  log f ð Þðq þ q Þ , with 2 CA AC fC ð C Þ h C C B B B B π^ π^ C fi ^B fCð CÞ¼1 þ . Speci cally, the operator R ðπ^ Þ represents the Doppler shift of Fig. 8 Transformation to the rest frame of a quantum system. The cmC fC C the photon. Finally, the transformation between the spatial degrees of freedom of A interaction between a photon B and the internal degrees of freedom à of an and C is the boost transformation in Eq. (16). We obtain atom as described from the point of view of a the rest frame of the atom A  ^ itself and b the laboratory C. We consider the situation when the atom does ^ π^ ^y mA ^ ; ^ π^ ^y pA ^ : SD CSD ¼À pA SD BSD ¼ 1 þ pB ð28Þ not have a sharp momentum in the laboratory reference frame and mC mAc calculate which state of the photon and the atom we have to prepare to ^ ðAÞ maximise the probability of absorbing the photon. The description of the Applying this transformation to the Hamiltonian H ~ yields ABC situation is simplest in the rest reference frame (a). If the photon has ^2 ^ ΔE ^ ðCÞ pA ^ pA spectral frequency ω ¼ corresponding to the atom’s energy gap, the H ~ ¼ þ H~′ þ hω^ 1 þ ; ð29Þ B h AAB 2m A B m c probability is maximised. For simplicity of illustration, the state of the A A laboratory is described as a superposition of two amplitudes sharp around ω^ c ^ where B ¼ h jpBj. From the perspective of the laboratory C, the Hamiltonian the velocities −v1 and −v2. From the point of view of C, this situation is entangles the momentum of the atom A with the frequency of the photon, while described as in b, in which the state of the photon and the external degrees the internal degrees of freedom are unchanged. The state of the joint system of the of freedom of the atom are entangled. For each velocity v , i 1; 2 of atom atom, with its internal and external degrees of freedom, and the photon is i ¼ ÀÁ Z  vi p2 A, the frequency of the photon is Doppler-shifted as: ω ω 1 for i A Bi ¼ B À c ðCÞ À t pA jΨi / dp e h2mA ψðp Þjp i j1; ω 1 À i jgi~ : ð30Þ ; ω^ ’ AA~B A A A A B B A i ¼ 1 2. The measurement of the frequency B in A s reference frame, mAc ^p corresponds to the measurement of the observable ω^ 1 þ A in the B cmA laboratory reference frame (see the next section). This ensures that if the The state (30) is the one which has to be prepared in the laboratory reference photon is found absorbed (‘detected’) by the atom in its RF, so it will in the frame to maximise the absorption probability. We see that the frequency of the laboratory RF photon B is Doppler shifted by an amount that depends on the velocity of the atom A. In the next section we show that, by mapping the observables in reference frame A to those in reference frame C, the absorption of the photon is predicted consistently in both reference frames, i.e. if the photon is detected in A’s reference classical trajectory and is not treated as a dynamical degree of freedom the map can frame, so it will in C’s reference frame. be achieved through a coordinate transformation between the two reference frames. However, in quantum mechanics, a system can evolve in a superposition of Measurements as seen from a quantum reference frame. In this section we classical trajectories. How can we ‘move’ to the rest frame of a particle that is in analyse how a measurement procedure performed in one QRF looks like as seen superposition of momenta with respect to the laboratory reference frame? Here, by from another QRF. We assume that an observer in reference frame C performs a working out an explicit example, we show how our formalism can be used to measurement on the quantum systems A and B. How does an observer in the recover the notion of the rest frame of a quantum system, when the semiclassical reference frame A describe this procedure? Note that the procedure in general approximation fails. includes also a measurement on the reference frame of A itself. This situation We consider the situation illustrated in Fig. 8, in which an atom with its differs from the Wigner-friend scenario, in which one observer (friend) performs a external (A) and internal (Ã) degrees of freedom interacts with a photon (B), as measurement, while the other (Wigner) considers the process to be unitary24,41.In seen from the laboratory reference frame (C). We assume the internal degrees of the present case both observers agree that a measurement is performed, though, as freedom to be internal energy states of a two-level system. We want to find we will see, they might have a different view on which systems and which mea- conditions under which the photon is in resonance with the internal energy surement is performed. levels of the atom from different frames of reference. More precisely, we want to ’ ^ ðCÞ find the state of the atom and photon such that the probability for the transition Consider that in C s reference frame observable OAB is measured. The fi ’ ^ ðAÞ ^^ ðCÞ^y ^ ismaximised,inthecasewhentheatomdoesnothaveawell-de ned transformed observable in A s reference frame is OBC ¼ SOAB S , where S is a momentum in the laboratory reference frame. We know that when the source of general operator which implements the transformation from the reference frame of a photon (where the source is at rest from the point of view of the laboratory C to the reference frame of A. Using the cyclicity of the trace, it is immediate to reference frame) and the receiver (i.e. the atom) are in relative motion towards verify that each other, and in the limit of small relative velocity between emitterÀÁ and ω′ ω v ^ ðCÞ ^ρðCÞ ^ ðCÞ ^ρðAÞ ^ ðAÞ ^ ðAÞ ; receiver, the frequency is Doppler-shifted according to B ¼ B 1 þ ,where hOAB i¼TrABð AB OAB Þ¼TrBCð BC OBC Þ¼hOBC i ð31Þ ω ω′ c B, B are respectively the emitted and received frequency of the photon, v is the relative velocity between emitter and receiver, and c is the speed of light in the ^ρðAÞ where BC is the quantum state of B and C relative to A. An explicit example, using medium. ^ ^ The condition for the absorption of the photon is simplest in the rest frame of operator Sx in Eq. (2), is a measurement of the position operator qB of the quantum ψ ; ω system B in the reference frame of A, which is equivalent to the measurement of A. Suppose that in this frame the entire state is given as j t iCj1 BiBjgiA~ . Here, ^ ^ R π2  xB À xA in the reference frame of C. À i C t ψ π h2m ψ mA π π the state of the laboratory at time t is j t iC ¼ d Ce C À C j CiC (this is To make these statements more concrete, we adopt a measurement scheme (see, mC 42 related to the momentum distribution of atom A in laboratory reference frame C), for instance ) and check how the measurement procedure transforms when we ; ω change reference frame. The measurement procedure in C’s and A’s reference jgiA~ is the ground state of the internal degrees of freedom of the atom and j1 BiB Δ frames is depicted in Fig. 9. The measurement scheme consists in adding an is the 1-photon state of B with frequency ω ¼ E, where ΔE is the energy gap B h ξ ðCÞ between the ground and the excited state of the internal energy. The Hamiltonian ancillary system consisting of a pointer in the state E 2HE and of external π^2 (position) degrees of freedom of the measurement apparatus in the state ^ ðAÞ C ^ ^ ^ in the rest frame is taken to be H~ ¼ þ hω þ H~ , where hω is a simplified ðCÞ ðCÞ ðCÞ ABC 2mC B A B σ ^ ^ρ ω M 2HM . The measurement of the observable OAB on the quantum system AB photon Hamiltonian in the one-particle sector. Here, we promote the frequency B can be then described as an interaction between the pointer and the quantum to an operator because the frequency shift due to the Doppler effect changes the system, followed by a projection in the Hilbert space of the pointer. The probability mode of the photon state, but leaves the number of particles invariant. A more of measuring the outcome b* is complete description of the Hamiltonian would involve the creation and hihi fl annihilation operators, but we omit it here because it does not in uence our results. pðbÃÞ¼Tr ^ρðCÞO^ ðCÞðbÃÞ ¼ Tr Cð^ρðCÞ ξ Þ σ ð1 F^ ðbÃÞÞ ; ω^ AB AB AB ABEM AB E M ABM E The frequency operator B acts on the single-photon Hilbert space and is such ω^ ω ω ω that h Bj BiB ¼ h Bj BiB, where the usual relation between momentum and ð32Þ

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ab by construction, that the probabilities of the photon being absorbed are the same in Reference frame: C Reference frame: A the reference frame A and C. What changes is the measured observable in the two Before reference frames: in the rest frame A one measures the frequency ω^ , while in the C M A B A C M B B measurement laboratory frame the measurement involves both the external degres of freedom of interaction E E ^ the atom and the photon. In this case, the observable is 1 þ pA ω^ . cmA B After C click! A B A C click! B measurement M M E interaction E Comparison of previous approaches to quantum reference frames. In this Section, we compare different approaches to the topic of quantum reference Fig. 9 Measurement in quantum reference frames. The measurement frames, emphasising differences and similarities to our approach. procedure as seen from the point of view of C, in a, and of A, in b.Ina,C Much work has been done on the subject of QRFs starting from the seminal papers by Aharonov and Susskind1,2 and Aharonov and Kaufherr3. In refs. 1,2 the prepares an ancillary system, constituted by the position degrees of authors established a relation between superselection rules and the lack of a frame freedom of the apparatus M and a pointer E. The ancilla interacts with the of reference. The authors then challenged the existence of superselection rules via quantum systems A and B. Subsequently, a projective measurement of the some examples, where the superselection rule could be overcome by introducing a state of the pointer gives the outcome. In b, A describes the initial state as reference frame that was correlated with the system. The simplest example of this is described in ref. 4. There it is shown that, if two observers do not share information an entangled state of B, C, and M. The pointer E then interacts with both about their relative phase, this implies a superselection rule for photon number. systems B and C. Finally, system E is measured. The measurement This superselection rule can be overcome by introducing an appropriate quantum probability is the same as in the reference frame of C reference frame which is entangled with the system in such a way that the total photon number is conserved. In ref. 3 it was shown that it is possible to consistently formulate quantum theory without appealing to classical reference frames as well- localised laboratories of infinite mass. where C is a unitary channel entangling the states from the Hilbert spaces of A and QRFs have been considered as resources in quantum information protocols and ^ ðCÞ 4–12 B with those of the pointer E, and FE is a projector on HE . quantum communication in refs. . These works mainly focus on (a) the We measure the outcome b* as ObðCÞðÞ¼bà jibà hjbà . Then, if we choose consequences of the lack of a shared reference frame for quantum information AB AB tasks, on (b) the generalisation of the fact that superselection rules can be overcome F^ ðÞ¼bà jibà hjbà , ξ such that jjξ ðx Þ 2¼ δðx Þ and E E E E E E by choosing an appropriate quantum system as a reference frame, and on (c) “ ” Cð^ρðCÞ jξ i hξ jÞ ¼ C ð^ρðCÞ jξ i hξ jÞCy ; ð33Þ bounded reference frames. This means that, in a quantum communication AB E E E ABE AB E E E ABE protocol where a system is sent from A to B, the final reference frame only  possesses limited or no information on the initial reference frame. In order to i ^ ðCÞ^ fi ^ρðCÞ where CABE ¼ exp À h OAB pE , the condition (32) is satis ed for all AB . Notice obviate this problem, most approaches resort to an encoding of quantum ^ ^ information in the relational degrees of freedom (see, e.g. ref. 5). The tool used to that by xE and pE we mean the position of the pointer in some abstract space of the internal degrees of freedom of the apparatus, not the real position (relative to C) of achieve this encoding is the G-twirl operation, which consists in an average over the the measurement apparatus in space. group of symmetries of the external reference frame G. This operation consists in ρ We now turn to the description from the point of view of A. For concreteness, expressing the quantum state of the system under study, , in a way that does not we consider the transformation between two frames of reference C and A to be the contain any information about the externalR reference frame. The G-twirl operation map in Eq. (2); the formalism can be straightforwardly generalised to other maps. is mathematically expressed as GðρÞ¼ dμðgÞUðgÞρUyðgÞ, where UðgÞ is the Considering the degrees of freedom of the ancilla the map is modified to include unitary representation of the group element g 2 G, and μðgÞ is the group-invariant the external degrees of freedom of the measurement apparatus, i.e., measure. This approach shares some methodological similarities with our work, i^ ^ ^ ^ ^ xA ðpB þpM Þ such as the relevance of the relational degrees of freedom; however, there are some Sx;M ¼ PACeh , while the degrees of freedom of the pointer are considered as translational invariant, and therefore not transformed. From A’s reference frame important differences. Firstly, we do not assume the existence of an external the measurement process is reference frame, whose degrees of freedom need to be averaged out. This means hithat, differently to the other approaches mentioned, we do not need to apply fi pðbÃÞ¼Tr CðAÞ ð^ρðAÞ ξ ÞðCðAÞ Þyð1 F^ ðbÃÞÞ ; ð34Þ techniques, such as the G-twirling, in order to nd the relational quantities, because BCEM BCEM BCM E BCEM BCM E our formalism is genuinely relational from the very start. Secondly, we do not address the problem of communication in absence of a shared reference frame. ^ρðAÞ ^ ^ρðCÞ σ ^y fi where the state BCM ¼ Sx;Mð AB M ÞSx;M becomes entangled with the external Finally, we assume that the relation between the initial and nal reference frame is ðAÞ ^ ^y known and hence always given by a unitary transformation. Moreover, reference degrees of freedom of the measurement apparatus, and C ¼ S ; C S ¼  BCEM x M ABE x;M frames in our approach are not bounded; they allow to assign quantum states to all i ^ ðAÞ^ exp À h OBC pE . We see that the two observers in the reference frames C and A external systems to arbitrary precision. 13 disagree on which systems undergo the measurement and which observables are Other authors focus on the possible role of QRFs in quantum gravity , or point measured. For observer C, systems A and B are measured with the help of an out how QRFs, together with a relational approach, can lead to intrinsic fi 14,15 ancilla E whose internal and external degrees of freedom are initially in a state that decoherence due to the nite size of the systems considered . Considering factorizes out. For observer A, systems B and C are measured via the ancilla whose reference frames quantum mechanically is a fundamental ingredient in formulating external (but not internal) degrees of freedom are initially entangled with C. relational quantum theory, which makes no use of an external reference frame to 23 Therefore, a measurement model for C is transformed into a measurement model specify its elements . A relational approach to QRFs has been also considered in 16–18 for A. refs. , where the limit to an absolute reference frame was formalised, and in 19 Notice that the measurement procedure just considered is different from A ref. , where a symmetry group for transformations of a spin system was performing a measurement in her reference frame. In the previous paragraphs, reconstructed. 8 when we changed the reference frame from C to A, we still described the Among the works listed, special mention should be paid to ref. , which is the measurement performed by C from the point of view of A. Clearly, A can apply the closest to our work. There, the G-twirl operation is introduced to average over all same measurement procedure as C, with the observables defined in her reference the external information to the joint system of two particles, A and S, one of which frame O^ ðAÞ. (A) serves as a reference frame. After this operation, the only quantities remaining As a concrete example of this situation we consider the atom-photon are the relative variables of the system S with the quantum reference frame. This interaction from the previous section. We make the following identification: the description that the QRF A gives of the system presents some similarities with ours. atom’s external degree of freedom is system A, the photon is system B, the atom’s The main differences, at this level, are that in our method, after the transformation, internal degrees of freedom à are ancilla E, and finally the laboratory is system C. we describe the initial reference frame in addition to the second particle, and we do We consider that A ‘measures’ the frequency of the photon by transition of its not rely on any external frame. The rest of the paper, addressing the change to a internal level from the ground to the excited state. This transition can happen only different QRF, asks a different question to us, and thus arrives to a different result. 8 if the frequency of the photon matches the energy gap of the atom. How is this In ref. , the authors switch to the description of a third quantum system B, whose condition written in the laboratory reference frame, in which the photon frequency relationship with the initial QRF is not known and will be acquired through a is Doppler-shifted? quantum measurement. To this end, they propose a protocol, consisting of multiple ðAÞ ðAÞ ðAÞy steps: (1) the quantum state of the QRF B, initially factored out from the system In the rest frame of the atom A, the channel C ðÁÞ ¼ C ðÁÞC BA~ BA~ BA~ and the QRF A, undergoes a G-twirl operation, which removes the information entangling the states of the Hilbert spaces of B with those of à is such that about the external frame; (2) a quantum measurement on the joint state of the ðAÞ ^ðAÞ C ¼jei~ hgj j0; ωi h1; ωjþh:c:, and the project or F ¼jei~ hej.Changingto BA~ A B A~ A QRFs A and B is performed to establish the relation between the two QRFs; 3) The ^ initial QRF A is discarded. The result of this series of operation is the relational the laboratory frame C via the application of the SD operator in Eq. (27), the entangling ðCÞ ^B ^By description of the state of the system S with respect to B. As a result of the channel becomes C ~ ¼ 1 jei~ hgj R ^ j0; ωi h1; ωjR ^ þ h:c:,with ABA A A fA ðÀpA Þ B fA ðÀpA Þ ^ measurement, the group-averaging operation, and the discarding of the initial QRF ^ pA ^ðAÞ ^ðCÞ f ðÀp Þ¼1 À , while the projector is unchanged, i.e. F~ ¼ F~ . This ensures, the tranformation results in a decohered state of the system in general. The A A cmA A A

NATURE COMMUNICATIONS | (2019) 10:494 | https://doi.org/10.1038/s41467-018-08155-0 | www.nature.com/naturecommunications 11 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-018-08155-0 appearance of decoherence is a fundamental difference with our approach, where 8. Palmer, M. C., Girelli, F. & Bartlett, S. D. Changing quantum reference frames. we assume that the relative description of the two QRFs (in our language, the state Phys. Rev. A. 89.5, 052121 (2014). of the new QRF from the point of view of the old one) is known and hence the 9. Bartlett, S. D., Rudolph, T., Spekkens, R. W. & Turner, P. S. Degradation of a 8 change of reference frame is unitary. In addition, in ref. the system and the new quantum reference frame. N. J. Phys. 8.4, 58 (2006). QRF are never entangled at the beginning of the protocol, while our approach does 10. Smith, A. R. H., Piani, M. & Mann, R. B. 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Acknowledgements Journal peer review information: Nature Communications thanks the anonymous We thank Alessio Belenchia, Fabio Costa, and Philipp Höhn for useful discussions. We reviewers for their contribution to the peer review of this work. Peer reviewer reports are acknowledge the support of the Austrian Science Fund (FWF) through the Doctoral available. Programme CoQuS (Project no. W1210-N25) and the projects I-2526-N27 and I-2906, as well as the research platform TURIS. This project has received funding from the Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in ’ European Union s Horizon 2020 research and innovation programme under the TEQ published maps and institutional affiliations. collaborative project (Grant agreement no. 766900). This work was funded by a grant from the Foundational Questions Institute (FQXi) Fund. This publication was made possible through the support of a grant from the John Templeton Foundation. The Open Access This article is licensed under a Creative Commons Attri- opinions expressed in this publication are those of the authors and do not necessarily bution 4.0 International License, which permits use, sharing, adaptation, fl re ect the views of the John Templeton Foundation. distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons Author contributions license, and indicate if changes were made. The images or other third party material in F.G., E.C.R. and Č.B. contributed to all aspects of the research with the leading input this article are included in the article’s Creative Commons license, unless indicated from F.G. otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or Additional information exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/. Supplementary Information accompanies this paper at https://doi.org/10.1038/s41467- 018-08155-0. © The Author(s) 2019 Competing interests: The authors declare no competing interests.

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