Relativistic quantum reference frames: the operational meaning of spin
RRelativistic quantum reference frames: the operational meaning of spin
Flaminia Giacomini,
1, 2
Esteban Castro-Ruiz,
1, 2 and ˇCaslav Brukner
1, 2 Vienna Center for Quantum Science and Technology (VCQ), Faculty of Physics,University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria Institute of Quantum Optics and Quantum Information (IQOQI),Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria (Dated: September 4, 2019)The spin is the prime example of a qubit. Encoding and decoding information in the spin qubitis operationally well defined through the Stern-Gerlach set-up in the non-relativistic (i.e., low ve-locity) limit. However, an operational definition of the spin in the relativistic regime is missing.The origin of this difficulty lies in the fact that, on the one hand, the spin gets entangled with themomentum in Lorentz-boosted reference frames, and on the other hand, for a particle moving in asuperposition of velocities, it is impossible to “jump” to its rest frame, where spin is unambiguouslydefined. Here, we find a quantum reference frame transformation corresponding to a “superpositionof Lorentz boosts,” allowing us to transform to the rest frame of a particle that is in a superposition ofrelativistic momenta with respect to the laboratory frame. This enables us to first move to the parti-cle’s rest frame, define the spin measurements there (via the Stern-Gerlach experimental procedure),and then move back to the laboratory frame. In this way, we find a set of “relativistic Stern-Gerlachmeasurements” in the laboratory frame, and a set of observables satisfying the spin su (2) algebra.This operational procedure offers a concrete way of testing the relativistic features of the spin, andopens up the possibility of devising quantum information protocols for spin in the special-relativisticregime. I. INTRODUCTION
The description of physical systems is standardly given in terms of coordinates as defined byreference frames. Thanks to the principle of covariance, stating the equivalence of all descriptionsregardless of the choice of the reference frame, it is possible to choose the reference frame where therelevant dynamical quantities can be most conveniently described. For example, it is typically easierto describe the dynamics of a system from the point of view of its rest frame, because only internaldegrees of freedom contribute to the dynamics in the rest frame.When the external degrees of freedom (momentum) of the system are in a quantum superposi-tion from the perspective of the laboratory, no classical reference frame transformation can map thedescription of physics from the laboratory to the rest frame. However, this can be achieved via aquantum reference frame (QRF) transformation between two frames moving in a superposition ofvelocites relative to one another. In order to achieve such change of quantum reference frame in thenonrelativistic regime, a formalism was introduced in Ref. [1] to change the description to a refenceframe which is in a quantum relationship with the initial one. This QRF transformation only de-pends on relational quantities, and it has also been derived starting from a gravity inspired symmetryprinciple in a perspective neutral model [2, 3] . An immediate consequence of the formalism is thatentanglement and superposition are QRF-dependent features. This formalism naturally leads to thepossibility of identifying the rest frame of a quantum system in an operational way.Here, we further develop this approach in the case of a relativistic quantum particle with spin,with the goal of finding an operational description of the spin in a special-relativistic setting. Spin isoperationally defined in the rest frame of a particle (or, to a good approximation, for slow velocities)via the Stern-Gerlach experiment. When the particle has relativistic velocities, the spin degree offreedom transforms in a momentum-dependent way. If a standard Stern-Gerlach measurement isperformed on a particle in a pure quantum state moving in a superposition of relativistic velocities,the operational identification of the spin fails, because no orientation of the Stern-Gerlach apparatus a r X i v : . [ qu a n t - ph ] S e p returns an outcome with unit probability. This happens because, as shown in Ref. [4], the reduceddensity matrix of the spin degree of freedom is mixed when a Lorentz boost is performed and themomentum is traced out. The question arises whether it is possible to find ‘covariant measurements’of the spin and possibly momentum, which predict invariant probabilities in different Lorentzianreference frames also for the case of a quantum relativistic particle moving in a superposition ofvelocities. In this case, it would be possible to map the unambiguous description of spin in the restframe of the particle to the frame of the laboratory, and therefore derive the corresponding observablesto be measured in the laboratory frame to verify spin with probability one.The question of finding such covariant measurements is motivated by the ubiquitous applicationswhere the spin degree of freedom is used as a qubit, to encode and transmit quantum information.Such protocols are no longer valid in a relativistic context, thus limiting the range of applicability oftechniques involving spin as a quantum information carrier. It is then important to explore possiblealternative methods which could overcome this limitation. In the context of relativistic quantum in-formation, this question has been extensively discussed [4–17] in relation to Wigner rotations [18–20]and has been related to the problem of identifying a covariant spin operator. The problem of identify-ing such covariant spin operator has arisen long before the birth of relativistic quantum information,and dates back to the early times of quantum mechanics [21–23]. Since then, a multitude of rela-tivistic spin operators have been proposed [24], such as the Frenkel [22], the Pauli-Luba´nski [25–27],the Pryce [28], the Foldy-Wouthuysen [29, 30], the Czachor [31] the Fleming [32] the Chakrabarti[33], and the Fradkin-Good [34] spin operators. A comparative description of spin observables canbe found in Ref. [17].Here, we introduce ‘superposition of Lorentz boosts’ which allow us to “jump” into the rest frameof a relativistic quantum particle even if the particle is not in a momentum eigenstate. In the rest frame,the spin observables fulfill the spin su (2) algebra (the algebra of a qubit) and are operationally definedthrough the Stern-Gerlach experiment. We transform the set of spin observables in the rest frameback to an isomorphic set of observables in the laboratory frame. The transformed observables are ingeneral entangled in the spin and momentum degrees of freedom. The set fulfills the su (2) algebra andis operationally defined through a ‘relativistic Stern-Gerlach experiment’: we construct the interactionand the measurement between the spin-momentum degrees of freedom and the electromagnetic fieldin the laboratory frame which gives the same probabilities as the Stern-Gerlach experiment in therest frame. This set of observables in the laboratory frame allows us to partition the total Hilbertspace into two (highly degenerate) subspaces corresponding to the two outcomes “spin up” and “spindown”. Hence, with QRFs techniques the relativistic spin can effectively be described as a qubit inan operationally well-defined way. II. A RELATIVISTIC STERN-GERLACH EXPERIMENT
In the following, we build a QRF transformation between the reference frame of a laboratory C ofmass m C and the rest frame of the external degrees of freedom A of a relativistic quantum particle ofmass m A > with spin degrees of freedom ˜ A , as illustrated in Fig. 1. We allow the particle to haveany quantum state, and in particular to move in a superposition of momenta. This implies that there isa non-classical relationship between the initial and the final reference frame, i.e., that the rest frame Aand the laboratory frame C are not related by a standard boost transformation. We show in this sectionhow to generalise the boost transformation to this case. Formally, the situation we consider can bedescribed by taking the one-particle sector of the positive-energy solutions of the Dirac equation inthe Foldy-Wouthuysen representation [29]. For simplicity, we only consider spin- / particles, but the method can be straightfowardly applied to arbitrary spin. AC ˜ A − v − v ˜ A C C A ˜ A v v A ( a ) ( b ) FIG. 1: (a) The state of a Dirac particle A with spin ˜ A as seen from the laboratory perspective (C). When the state is in asuperposition of relativistic velocities − v and − v , the spin degree of freedom and the momentum degree of freedom areno longer separable. (b) The state of the spin ˜ A and of the laboratory C as seen in the rest frame of the quantum particle A.In this quantum reference frame, the spin is operationally defined by means of the Stern-Gerlach experiment. Following Ref. [1] (see Supplemental Information for a review of the original formalism), whenwe “stand” in the rest frame of a particle, we describe all the systems external to the particle, but notthe external degrees of freedom (i.e., the momentum degrees of freedom) of the particle itself. Hence,the quantum state describes the relational information in a given reference frame. In the referenceframe in which A is at rest, the quantum state is assigned to the internal degrees of freedom ˜ A andthe laboratory C. For simplicity, we consider that the particle and the laboratory are moving withconstant, yet not necessarily well-defined, relative velocity and define the x axis along the directionof the relative motion. The total state of the spin and the laboratory is assumed to be | Ψ (cid:105) ( A )˜ AC = | (cid:126)σ (cid:105) ˜ A | ψ (cid:105) C , (1)where | (cid:126)σ (cid:105) ˜ A is any vector representing the state of the spin in the rest frame A. In the rest frame,the spin state can in principle be tomographically verified by performing a series of standard Stern-Gerlach measurements. The state of the laboratory has a momentum-basis representation alongthe x direction (at this stage, we neglect the quantum state in the y and z direction) | ψ (cid:105) C = (cid:82) dµ C ( π C ) ψ ( π C ) | π C (cid:105) C , where dµ C ( π C ) = dπ C (2 π ) / √ m C c + π C ) is the Lorentz-covariant inte-gration measure.We now construct the transformation corresponding to the “superposition of Lorentz boosts” tothe QRF of the laboratory. The unitary operator to boost to the QRF C is ˆ S L = P ( v ) CA U ˜ A (ˆ π C ) , (2)where U ˜ A (ˆ π C ) is a unitary transformation acting on the total Hilbert space H ˜ A ⊗ H C (notice that ˆ π C is an operator), and P ( v ) AC is the ‘generalised parity operator’ introduced in Ref. [1], whose explicit ex-pression is P ( v ) CA = P AC exp (cid:16) i (cid:126) log (cid:113) m A m C (ˆ q C ˆ π C + ˆ π C ˆ q C ) (cid:17) , where P AC is the parity-swap operatormapping ˆ x A → − ˆ q C and ˆ p A → − ˆ π C (and viceversa), where ˆ q C , ˆ π C are canonically-conjugated one-particle operators of C in the reference frame of A and ˆ x A , ˆ p A are canonically-conjugated one-particleoperators of A in the reference frame of C. Additionally to the action of P AC , the operator P ( v ) AC rescales the momentum of A by the ratio of the masses of A and C, i.e., P ( v ) AC ˆ p A P ( v ) † AC = − m A m C ˆ π C .This enforces the physical condition that the velocity of A is mapped to the opposite of the velocity ofC via the transformation . The operator ˆ S L can be defined via its action on a basis of the total Hilbertspace of the spin and the laboratory ˆ S L | (cid:126)σ (cid:105) ˜ A | π (cid:105) C = | − m A m C π ; Σ π (cid:105) A ˜ A , where the state | p ; Σ p (cid:105) A ˜ A isdefined via a standard Lorentz boost ˆ U ( L p ) from the rest frame as | p ; Σ p (cid:105) A ˜ A = ˆ U ( L p ) | k ; (cid:126)σ (cid:105) A ˜ A and For a relativistic particle the relation between the i -th velocity component and the momentum is v i = p i m i (cid:16) | (cid:126)p | m i c (cid:17) − / , where | (cid:126)p | is the norm of the spatial momentum. Therefore, only the ratio between momentumand mass determines the velocity. k = ( mc,(cid:126) is the momentum in the rest frame. In Supplemental Information we derive the transfor-mation ˆ S L in terms of standard Lorentz boosts connecting two relativistic reference frames where theparameter of the boost transformation is promoted to an operator.The state of A and ˜ A expressed in the laboratory frame is | Ψ (cid:105) ( C ) A ˜ A = ˆ S L | Ψ (cid:105) ( A )˜ AC , and is explicitlywritten as | Ψ (cid:105) ( C ) A ˜ A = (cid:90) dµ A ( p A ) ψ (cid:18) − m C m A p A (cid:19) | p A ; Σ p A (cid:105) A ˜ A , (3)where dµ A ( p A ) = dp A (2 π ) / √ m A c + p A ) and the spin degree of freedom cannot be separated anymorefrom the momentum degree of freedom, which means that the state is not a product state in the labo-ratory frame. Notice that the effect of the ˆ S L transformation is to apply the usual boost transformationconditional on C’s momentum degree of freedom. In the laboratory frame C, unless particle A is ina sharp momentum state, no spin measurement in a standard Stern-Gerlach experiment would give aresult with probability one, because of two reasons: the spin and momentum are no longer separable,and the relation between the laboratory and the rest frame is not a standard (classical) reference frametransformation. Our goal is to devise a different measurement in the laboratory reference frame, pos-sibly involving both the spin and momentum degrees of freedom, which gives the same probabilitydistribution as a standard Stern-Gerlach would give, if performed in the rest frame.In order to devise such measurement we note that, in the laboratory frame, it is possible to definethe observables corresponding to the spin operators in the rest frame by transforming the spin, asdefined in the rest frame, with a QRF transformation ˆΞ i = ˆ S L (ˆ σ i ⊗ C ) ˆ S † L , i = x, y, z. (4)In terms of the momenta and of the manifestly covariant Pauli-Luba´nski operator ˆΣ ˆ p A = ( ˆΣ p A , (cid:126) ˆΣ ˆ p A ) ,the operators ˆΞ i are expressed as (see Supplemental Information) (cid:126) ˆΞ = (cid:126) ˆΣ ˆ p A − ˆ γ A ˆ γ A +1 (cid:16) (cid:126) ˆΣ ˆ p A · (cid:126) ˆ β A (cid:17) (cid:126) ˆ β A ,where ˆ γ A = (cid:114) ˆ p A m A c and (cid:126) ˆ β A = (cid:16) ˆ β xA , ˆ β yA , ˆ β zA (cid:17) , where each component is ˆ β iA = ˆ p iA (cid:113) m A c + (cid:126) ˆ p A with i = x, y, z . The operators ˆΞ i are equivalent to the Foldy-Wouthuysen [29] or Pryce spin operator[28]. By definition, these operators satisfy the su (2) algebra (cid:104) ˆΞ i , ˆΞ j (cid:105) = i(cid:15) ijk ˆΞ k , and have the sameeigenvalues as the Pauli operators ˆ σ i , i = x, y, z . This last property can be easily checked by choosingan eigenvector | λ i (cid:105) of the operator ˆ σ i in the rest frame A, such that ˆ σ i | λ i (cid:105) = λ i | λ i (cid:105) and by noting that ˆΞ i ˆ S L | λ i (cid:105) ˜ A | ψ (cid:105) C = λ i ˆ S L | λ i (cid:105) ˜ A | ψ (cid:105) C . Hence, it is possible to partition the total Hilbert space H A ⊗ H ˜ A into two equivalence classes, defined as H = (cid:110) | Ψ (cid:105) A ˜ A ∈ H A ⊗ H ˜ A s.t. | Ψ (cid:105) A ˜ A ∼ ˆ S L | (cid:105) ˜ A | ψ (cid:105) C , ∀ | ψ (cid:105) C ∈ H C (cid:111) , (5a) H = (cid:110) | Φ (cid:105) A ˜ A ∈ H A ⊗ H ˜ A s.t. | Φ (cid:105) A ˜ A ∼ ˆ S L | (cid:105) ˜ A | φ (cid:105) C , ∀ | φ (cid:105) C ∈ H C (cid:111) , (5b)where | (cid:105) ˜ A and | (cid:105) ˜ A are the eigenvectors of ˆ σ z and two states are said to be equivalent, i.e., | Ψ (cid:105) A ˜ A ∼ ˆ S L | i (cid:105) ˜ A | ψ (cid:105) C , with i = 0 , , if they are both eigenvectors of the ˆΞ z operator with the same eigenvalue.We can then build a partition of the Hilbert space into two highly degenerate subspaces, one corre-sponding to the “spin up” and the other to the “spin down” eigenvalue, and on which it is possibleto define a set of operators satisfying the su (2) algebra, which can be used to encode or decodeinformation of a single qubit. Notice that we could have chosen any other Pauli operator to define this partition. spin p laboratory − v − v UP DOWN ⃗ n UP DOWN ⃗ n p particle v v ̂ H ( A ) in t = μ ⃗ σ ⋅ ⃗ B ̂ H ( C ) in t = μγ −1 A ⃗ Ξ ⋅ ⃗ 𝒮 Λ ( ⃗ B ( A ) ) FIG. 2:
The relativistic Stern-Gerlach experiment as seen from the QRF A (above) and from the QRF C (below). Inthe rest frame of particle A, the spin is operationally defined via the Stern-Gerlach experiment. To measure spin alongdirection (cid:126)n the spin (Pauli operator) (cid:126)σ is coupled to an inhomogeneous magnetic field oriented along (cid:126)n . The particle isthen deflected towards the direction (cid:126)n and − (cid:126)n corresponding to outcome “spin up” and “spin down” respectively. Whentransforming to the laboratory frame C, the magnetic field and the spin transform with a superposition of Lorentz boostsfor v and v . The interaction Hamiltonian is also transformed, giving rise to a coupling between the transformed vector (cid:126) S Λ ( (cid:126)B ( A ) ) = ˆ γ A (cid:104) (cid:126)B ( C ) − ˆ γ A ˆ γ A +1 (cid:16) (cid:126) ˆ β A · (cid:126)B ( C ) (cid:17) (cid:126) ˆ β A + (cid:16) (cid:126) ˆ β A × (cid:126)E ( C ) (cid:17)(cid:105) aligned in the same direction (cid:126)n as the magnetic fieldin the rest frame, and the transformed spin operator (cid:126) Ξ . The particle is again deflected either to (cid:126)n or − (cid:126)n corresponding tothe outcome “spin up” and “spin down” respectively. The probability of detecting the outcomes “spin up” and “spin down”is preserved under change of QRF. The operators (cid:126) ˆΞ in general act on both the external and the internal degrees of freedom of the par-ticle. Operationally, they can be defined via a “relativistic Stern-Gerlach experiment,” illustrated inFig. 2. Traditionally, in a Stern-Gerlach experiment, the spin measurement is performed by applyinga magnetic field, which interacts with the spin as (cid:126)B · (cid:126)σ and is inhomogeneous along the direction ofits orientation, i.e., (cid:126)B = B ( (cid:126)r · (cid:126)n ) (cid:126)n , where (cid:126)n gives the direction and (cid:126)r = ( x, y, z ) . If the magneticfield is aligned precisely in the direction in which the spin state is prepared, the outcome is obtainedwith certainty. However, if the particle carrying the spin is moving in a superposition of relativisticvelocities, no measurement of the spin alone in the laboratory frame will return the result with prob-ability one in general. To treat such a case we set up a hypothetical Stern-Gerlach experiment in therest frame of the particle, where the interaction Hamiltonian is H ( A ) int = µ (cid:126)B ( A ) · (cid:126)σ and µ is a couplingconstant. We assume that the direction in which the magnetic field is aligned (cid:126)n is orthogonal to thedirection of the boost x . Formally, this geometric configuration requires to enlarge the Hilbert spaceof the laboratory to the z direction, which we identify with the direction (cid:126)n of deflection, and modifyour previous definition of the state in Eq. (1) as | ψ (cid:105) C = | ψ x (cid:105) C | ψ z (cid:105) C , where | ψ x (cid:105) C transforms with ˆ S L and | ψ z (cid:105) C is left invariant by the transformation ˆ S L , except for the fact that the label is changedfrom C to A, i.e., ˆ S L | ψ z (cid:105) C = | ψ z (cid:105) A . Additionally, we assume that the motion in the z directionis non relativistic. We then transform the Hamiltonian to the laboratory frame via the QRF trans-formation ˆ S L . Knowing that the magnetic field transforms under superposition of Lorentz boostsas (cid:126) ˆ S Λ ( (cid:126)B ( A ) ) = ˆ γ A (cid:104) (cid:126)B ( C ) − ˆ γ A ˆ γ A +1 (cid:16) (cid:126) ˆ β A · (cid:126)B ( C ) (cid:17) (cid:126) ˆ β A + (cid:16) (cid:126) ˆ β A × (cid:126)E ( C ) (cid:17)(cid:105) , we find that the interactionHamiltonian H ( A ) int is transformed to H ( C ) int = µ ˆ γ − A (cid:126) ˆ S Λ ( (cid:126)B ( A ) ) · (cid:126) ˆΞ . (6)It is straightforward to check that the direction of (cid:126) ˆ S Λ ( (cid:126)B ( A ) ) is also (cid:126)n , therefore the deflection ofthe particle in the laboratory frame happens in the same direction as in the rest frame. Notice that,since both the quantum state and the observables transform unitarily, probabilities are automaticallyconserved after the change of QRF. In particular, if in the rest frame of the particle A the Stern Ger-lach measurement detects that the spin is “up” with probability one, the “relativistic Stern-Gerlach”experiment in the laboratory frame with the interaction Hamiltonian of Eq. (6) will also detect “spinup” with probability one. Note that the specific form of the electromagnetic field in Eq. (6) is notcrucial to our result, but we can design the coupling between the particle and the electromagneticfield according to our experimental capabilities in each reference frame. However, it is crucial thatthe electromagnetic field couples to the operator (cid:126) ˆΞ , unlike in the standard Stern-Gerlach experiment.In Supplemental Information, we set up a different experiment, where we couple an ihnomogenousmagnetic field in the laboratory frame to give an explicit analysis of a relativistic Stern-Gerlach ex-periment.It is worth noting that the interaction Hamiltonian of Eq. (6) is covariant, because the quantity H := ˆ γ A H ( C ) int transforms like the zero-component of a -vector. Therefore, the Schr¨odinger equa-tion in the reference frame of A, i (cid:126) ddt A | ψ (cid:105) ( A )˜ AC = H ( A ) int | ψ (cid:105) ( A )˜ AC , where t A is the proper time in the restframe of A, is mapped to i (cid:126) ddt C | ψ (cid:105) ( C )˜ AA = H ( C ) int | ψ (cid:105) ( C )˜ AA , where t C is the proper time in the rest frameof C and the relation t C = ˆ γ A t A holds. The general, manifestly covariant expression of H is H = 12 η ρ (cid:15) ρµνλ ˆΣ µp A F νλ , (7)where η µν = diag (1 , − , − , − is the Minkowski metric, F νλ is the electromagnetic tensor and (cid:15) ρµνλ is the totally antisymmetric tensor such that (cid:15) = 1 .In order to complete the measurement, we now have to project the position of the particle alongthe z direction. Formally, this is achieved by defining the two operators ˆΠ ( A )+ = (cid:82) + ∞ dz c | z C (cid:105) C (cid:104) z C | and ˆΠ ( A ) − = (cid:82) −∞ dz c | z C (cid:105) C (cid:104) z C | , distinguishing whether the particle is respectively deflected upwardsor downwards. For a thorough analysis of a concrete detection of spin via the “relativistic Stern-Gerlach” proposed here and more details on the measurement, see Supplemental Information.The QRF transformation provides the description of the same experiment from the point of view oftwo different QRFs, which move in a superposition of velocities relative to each other. This treatmentof the relativistic Stern-Gerlach experiment makes it possible to associate an operational meaning tothe spin of a relativistic quantum particle, thus solving the problem of encoding quantum informationin a particle with spin degrees of freedom as in a qubit. III. CONCLUSIONS
In this paper, we have provided an operational description of the spin of a special-relativisticquantum particle. Such operational description is hard to obtain with standard methods due to thecombined effect of special relativity, which makes the spin and momentum not separable, and quan-tum mechanics, which makes it impossible to jump to the rest frame with a standard reference frametransformation. We have introduced the ‘superposition of Lorentz boosts’ transformation to the restframe of a quantum particle, moving in a superposition of relativistic velocities from the point ofview of the laboratory. We have found how the state transforms under such quantum reference frametransformation and identified a set of observables in the laboratory frame which satisfies the su (2) al-gebra and has the same eigenvalues as the spin in its rest frame. In addition, this set complies with thedesiderata for a relativistic spin operator in Ref. [24]: it commutes with the free Dirac Hamiltonian,it satisfies the su (2) algebra, and it has the same eigenvalues as the spin in its rest frame. In addition,it has the correct nonrelativistic limit. It can be easily shown, in fact, that our operator (cid:126) ˆΞ coincideswith the Foldy-Wouthuysen spin operator [29, 30]. Thanks to the unitarity of the transformation,probabilities are the same in the rest frame and in the laboratory frame. Finally, we have generalisedthe Stern-Gerlach to the special-relativistic regime by means of a transformation of the interactionHamiltonian from the rest frame to the laboratory frame. Such generalisation opens up the possibilityof performing quantum information protocols with spin in the special-relativistic regime. ACKNOWLEDGMENTS
We would like to thank Carlos Pineda for helpful discussions. We acknowledge support from theresearch platform Testing Quantum and Gravity Interface with Single Photons (TURIS), the AustrianScience Fund (FWF) through the project I-2526-N27 and I-2906, the ¨OAW Innovationsfonds-Projekt“Quantum Regime of Gravitational Source Masses”, and the doctoral program Complex QuantumSystems (CoQuS) under Project W1210-N25. We also acknowledge financial support from the EUCollaborative Project TEQ (Grant Agreement 766900). This work was funded by a grant from theFoundational Questions Institute (FQXi) Fund. This publication was made possible through the sup-port of a grant from the John Templeton Foundation (Project 60609). The opinions expressed in thispublication are those of the authors and do not necessarily reflect the views of the John TempletonFoundation.
Appendix A: Review of the formalism for quantum reference frames in the Galilean case
In Ref. [1], a formalism to describe quantum states, dynamics, and measurements from the pointof view of a quantum reference frame (QRF) was introduced. This formalism is operational, inthat primitive laboratory operations —preparation, transformations, and measurements of quantumstates— have fundamental status, and relational, because everything is formulated in terms of rela-tional quantities and the formalism does not require the presence of any external or absolute referenceframe.The simplest situation is composed of three systems C (the initial QRF), A (the final QRF), and B(a quantum system). Since only the relational degrees of freedom play a role, from the point of viewof C the relational degrees of freedom of A and B relative to C are described, and from the point ofview of A the relational degrees of freedom of B and C relative to A are described. For instance, ifwe want to consider relative coordinates, C associates the position operator ˆ x A to A, and the positionoperator ˆ x B to B. Operationally, these operators indicate the relative distance between A or B andthe origin of the QRF C. From the point of view of A, B has the position operator ˆ q B associatedto it, while C has the position operator ˆ q C . The operators in the two QRFs are related via a QRFtransformation ˆ S x , acting as ˆ S x ˆ x B ˆ S † x = ˆ q B − ˆ q C and ˆ S x ˆ x A ˆ S † x = − ˆ q C . The explicit expression ofthe transformation to change QRF from C to A is ˆ S x = P AC e i (cid:126) ˆ x A ˆ p B , (A1)where P AC is the “parity-swap” operator acting as P AC ˆ x A P † AC = − ˆ q C and P AC ˆ p A P † AC = − ˆ π C and ˆ π C is the canonically conjugated operator to ˆ q C . Intuitively, this transformation acts as a “controlledtranslation” on the state of the QRF A. One of the main consequences of this transformation is that thenotion of entanglement and superposition is frame-dependent. To illustrate this point, let us assumethat C assigns to A and B the quantum state | Ψ (cid:105) ( C ) AB = √ ( | x (cid:105) A + | x (cid:105) A ) | x (cid:105) B , which is separable.After the QRF transformation, the state in A’s QRF becomes entangled, i.e. | Φ (cid:105) ( A ) BC = ˆ S x | Ψ (cid:105) ( C ) AB = √ ( | x − x (cid:105) B |− x (cid:105) C + | x − x (cid:105) B |− x (cid:105) C ) .Another relevant property of the formalism for QRFs introduced in Ref. [1] is that the probabilityto observe an outcome of a measurement is conserved under change of QRF thanks to the unitarity ofthe QRF transformation. Specifically, if the probabilities in the QRF of C are calculated as p ( b ∗ ) = Tr (cid:104) ˆ ρ ( C ) AB ˆ O ( C ) AB ( b ∗ ) (cid:105) , (A2)where ˆ ρ ( C ) AB is the quantum state, ˆ O ( C ) AB the operator, and ˆ O ( C ) AB ( b ∗ ) the projector on a specific outcomein C’s perspective, in A’s QRF, with analogous notation, we find that p ( b ∗ ) = Tr (cid:104) ˆ ρ ( A ) BC ˆ O ( A ) BC ( b ∗ ) (cid:105) , (A3)where ˆ ρ ( A ) BC = ˆ S x ˆ ρ ( C ) AB ˆ S † x and ˆ O ( A ) BC = ˆ S x ˆ O ( C ) AB ˆ S † x .Notice that in this work we use the terminology “superposition of Lorentz boosts” because wedescribe physics from the point of view of two QRFs moving in a superposition of relativistic veloci-ties relative to each other. This is different to the definition of such transformations given in Ref. [1],where the transformation is written as the classical reference frame transformation with the param-eter of the transformation promoted to an operator, because the transformation of the spin degree offreedom alone does not correspond to a Lorentz boost. Notice, however, that the QRF transformationwould be completely analogous to those in Ref. [1] if we considered a Klein-Gordon field instead ofthe spin ˜ A . In this case, the QRF transformation would be given by a Lorentz boost, with the param-eter v replaced by a function of the operator ˆ π C , followed by a generalised parity-swap operator. Appendix B: Derivation of the quantum reference frame transformation
The QRF transformation ˆ S L can be derived by considering a tripartite Hilbert space of the externaldegrees of freedom of the particle (A), the spin of the particle ( ˜ A ), and the laboratory (C). A generalbasis element, in the rest frame of the particle A is | k A ; (cid:126)σ (cid:105) A ˜ A | π (cid:105) C , where k µA = ( m A c,(cid:126) . If we wishto change to the QRF of the laboratory C, we need to apply a standard Lorentz boost to the state ofthe particle A ˜ A , where the boost parameter is controlled by the momentum of the laboratory ˆ π C , andthen boost the laboratory state by the momentum of A. The full transformation reads ˆ S ext = ˆ U † C ( L mCmA ˆ p A ) ˆ U A ˜ A ( L − mAmC ˆ π C ) , (B1)where ˆ U X ( L ˆ p Y ) (with X (cid:54) = Y ) is the standard Lorentz boost by the momentum operator ˆ p Y (i.e.,the standard Lorentz boost where the parameter p has been promoted to the operator ˆ p Y ), actingon the Hilbert spaces X and Y as ˆ U A ˜ A ( L − mAmC ˆ π C ) | k A ; (cid:126)σ (cid:105) A ˜ A | π (cid:105) C = | − m A m C π ; Σ π (cid:105) A ˜ A | π (cid:105) C and ˆ U † C ( L mCmA ˆ p A ) | − m A m C π ; Σ π (cid:105) A ˜ A | π (cid:105) C = | − m A m C π ; Σ π (cid:105) A ˜ A | k C (cid:105) C . The action on a basis of the totalHilbert space is ˆ S ext | k A ; (cid:126)σ (cid:105) A ˜ A | π (cid:105) C = | − m A m C π ; Σ π (cid:105) A ˜ A | k C (cid:105) C , (B2)with k µC = ( m C c,(cid:126) . We notice that the Hilbert spaces of the QRF, A and C respectively beforeand after the transformation, are irrelevant and can be omitted, as they are only labelled by the zeromomentum -vector k A and k C and represent no dynamical degrees of freedom (their quantum statesdo not change in time). Therefore, in accordance with the QRF formalism introduced in Ref. [1], wecan drop them and write the QRF transformation in the more compact form ˆ S L in the main text ˆ S L | (cid:126)σ (cid:105) ˜ A | π (cid:105) C = | − m A m C π ; Σ π (cid:105) A ˜ A . (B3)This transformation can be explicitly written as ˆ S L = P ( v ) AC ˆ U ˜ A (ˆ π C ) , (B4)where ˆ U ˜ A (ˆ π C ) is a unitary operator acting on the joint Hilbert space H ˜ A ⊗ H C , and P ( v ) CA = P AC exp (cid:16) i (cid:126) log (cid:113) m A m C (ˆ q C ˆ π C + ˆ π C ˆ q C ) (cid:17) is the ‘generalised parity-swap operator’ introduced inRef. [1]. Here, P AC is the parity-swap operator mapping ˆ x A → − ˆ q C and ˆ p A → − ˆ π C (and viceversa).Additionally to the action of P AC , the operator P ( v ) AC rescales the momentum of A by the ratio of themasses of A and C, i.e., P ( v ) AC ˆ p A P ( v ) † AC = − m A m C ˆ π C , such that the velocity operators of A and C aremapped as ˆ v A (cid:55)→ − ˆ v C . Appendix C: Action of the spin operators
The spin of a particle has a natural definition in the rest frame. In a different reference frame,quantum field theory predicts that a more general quantity, the total angular momentum, is conserved,and spin alone is no longer operationally well defined, because the splitting of the angular momentumand spin momentum is not unique. However, it is possible to associate a -vector to the spin operator,the Pauli-Luba´nski spin operator, which can then be transformed in a covariant way. The treatmentcan be found, e.g., in Refs. [20, 35]. In the rest frame, the components of the covariant spin are ˆ σ ν =(ˆ0 , ˆ σ x , ˆ σ y , ˆ σ z ) , where ˆ σ i , i = x, y, z , are the Pauli operators, which generate the group SU (2) . Wenow want to boost the state of the particle to a reference frame that moves in general in a superpositionof velocities. To this end we introduce the momentum operator (cid:126) ˆ p = mγ(cid:126) ˆ v , where m is the mass ofthe particle, ˆ γ = (cid:113) (cid:126) ˆ p m c = (cid:16) − (cid:126) ˆ v c (cid:17) − / and (cid:126) ˆ v is the velocity operator of the particle in thenew reference frame. We define by U ( L − ˆ p ) the operator representing a pure Lorentz boost to the newreference frame, where the boost operator L ˆ p is explicitly written as the matrix L ˆ p = ˆ p mc − ˆ p i mc − ˆ p i mc δ ij + ˆ p i ˆ p j mc (ˆ p + mc ) , (C1)where i, j = x, y, z . The Pauli-Luba´nski spin operator is defined (up to a factor m A ) as ˆΣ µ ˆ p =ˆ U † ( L ˆ p )ˆ σ µ ˆ U † ( L ˆ p ) = ( L − ˆ p ) µν ˆ σ ν [20]. Formally, this relation can be derived via the action on a genericbasis element ˆ U ( L p ) | k, (cid:126)σ (cid:105) = | p, Σ p (cid:105) , where | k, (cid:126)σ (cid:105) = (cid:80) λ c λ | k, λ (cid:105) is represented in some spin basis,and ˆ U ( L p ) is the standard Lorentz boost from the rest frame to the frame where the particle hasmomentum p . We can now write the action of the Pauli-Luba´nski operator as ˆΣ µ ˆ p | p, Σ p (cid:105) = ˆΣ µp ˆ U ( L p ) | k, (cid:126)σ (cid:105) = ˆ U ( L p ) ˆ U † ( L p ) ˆΣ µp ˆ U ( L p ) | k, (cid:126)σ (cid:105) == ˆ U ( L p )( L − p ) µν ˆΣ νp | k, (cid:126)σ (cid:105) = (cid:88) λ c λ ˆ U ( L p )( L − p ) µν ˆ σ ν | k, λ (cid:105) == (cid:88) λ,λ (cid:48) c λ ˆ U ( L p )( L − p ) µν [ σ ν ] λ (cid:48) λ | k, λ (cid:48) (cid:105) = (cid:88) λ,λ (cid:48) c λ ( L − p ) µν [ σ ν ] λ (cid:48) λ | p, Σ p ( λ (cid:48) ) (cid:105) == ( L − p ) µν ˆ σ ν | p, Σ p (cid:105) . (C2)Thus, by making use of Eq. (C1), it is immediate to verify that ˆΣ p = ˆ γ (cid:126) ˆ β · (cid:126) ˆ σ ; (cid:126) ˆΣ ˆ p = (cid:126) ˆ σ + ˆ γ ˆ γ + 1 ( (cid:126) ˆ β · (cid:126) ˆ σ ) (cid:126) ˆ β, (C3)where (cid:126) ˆ β = (cid:126) ˆ vc = (cid:126) ˆ p √ m c + | ˆ p | .0Notice that the introduction of the fourth spin component does not add degrees of freedom, be-cause the -vector has to satisfy the covariant constraint η µν ˆ p µ ˆΣ ν ˆ p = 0 , where η µν = diag (1 , − , − , − is the Minkowski metric. With an analogous method it is possible to show that ˆΞ i | p, Σ p (cid:105) =ˆ U ( L p )( ⊗ (cid:126)σ ) ˆ U † ( L p ) | p, Σ p (cid:105) = ( L p ) iµ ˆΣ µp | p, Σ p (cid:105) . Thus, the action of the two operators ˆΣ p and ˆΞ is obtained by Lorentz-transforming the other.Formally, the total Hilbert space can be described as a fiber bundle, the base manifold beingthe space of square-integrable functions and the fibers being the two-dimensional Hilbert space H p ,representing the spin degree of freedom. Appendix D: Concrete analysis of the relativistic Stern-Gerlach
Let us consider an experiment performed in the laboratory frame C. We describe the x and z external degrees of freedom of particle A and the spin degrees of freedom ˜ A , and take the state attime t = 0 to be | Ψ (cid:105) ( C ) A ˜ A = cos θ (cid:12)(cid:12) Ψ +0 (cid:11) A ˜ A + sin θ (cid:12)(cid:12) Ψ − (cid:11) A ˜ A , (D1)where (cid:12)(cid:12) Ψ ± (cid:11) A ˜ A = | ψ z (cid:105) A (cid:12)(cid:12) φ ± x (cid:11) A ˜ A (D2)and | ψ z (cid:105) A = (cid:82) dp z ψ z ( p z ) | p z (cid:105) A with ψ z ( p z ) = πs z ) / e − p z s z being a gaussian wavepacket centeredin zero with standard deviation s z , and | φ ± x (cid:105) A ˜ A = (cid:82) dµ ( p x ) φ x ( p x ) (cid:12)(cid:12) p x ; Σ ± p x (cid:11) A ˜ A . Here, φ x ( p x ) is ageneral wavepacket and (cid:12)(cid:12) p x ; Σ ± p x (cid:11) A ˜ A is the eigenvector with eigenvalue λ ± = ± of the operator ˆΞ z .As in the main text, we assume that the motion along z is nonrelativistic. In the laboratory frame,we engineer an interaction between the system A and the magnetic field such that the interactionHamiltonian is ˆ H ( C ) int = µB ( C ) z ( z )ˆΞ z , with B ( C ) z ( z ) = B z − αz , α > . Evolving the state in theinteraction picture, we get | Ψ t (cid:105) ( C ) A ˜ A = e − i (cid:126) ˆ H ( C ) int t | Ψ (cid:105) ( C ) A ˜ A = cos θ (cid:12)(cid:12) Ψ + t (cid:11) A ˜ A + sin θ (cid:12)(cid:12) Ψ − t (cid:11) A ˜ A , (D3)where (cid:12)(cid:12) Ψ ± t (cid:11) A ˜ A = (cid:82) dµ ( p x ) dp z φ x ( p x ) ψ z ( p z ) e ∓ i (cid:126) B z t e ± i (cid:126) αµt ˆ z A | p z (cid:105) A (cid:12)(cid:12) p x ; Σ ± p x (cid:11) A ˜ A . This state can berewritten as (cid:12)(cid:12) Ψ ± t (cid:11) A ˜ A = e ∓ i (cid:126) B z t (cid:82) dµ ( p x ) dp z φ x ( p x ) ψ z ( p z ∓ p ∗ ( t )) | p z (cid:105) A (cid:12)(cid:12) p x ; Σ ± p x (cid:11) A ˜ A , where p ∗ ( t ) = αµt (cid:126) . Hence, under the effect of the interaction with the magnetic field, the gaussian wavepacket along z gets split into two wavepackets, moving in opposite directions according to the state of the spin.The two gaussians become distinguishable under the condition | A (cid:104) ψ + z | ψ − z (cid:105) A | (cid:28) , where we havedefined | ψ ± z (cid:105) A = (cid:82) dp z ψ z ( p z ∓ p ∗ ( t )) | p z (cid:105) A . This condition is satisfied if t > (cid:126) s z αµ (neglecting factorsof order one). If we now define the projectors ˆΠ ± z = | ψ ± z (cid:105) A (cid:104) ψ ± z | , we can calculate the probabilitiesto find spin “up” or spin “down” as p ± = Tr (cid:104) | Ψ t (cid:105) ( C ) A ˜ A (cid:104) Ψ t | ˆΠ ± z (cid:105) , (D4)where | Ψ t (cid:105) ( C ) A ˜ A denotes the time-evolved state of Eq. (D1). 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