Relativistic causality in particle detector models: Faster-than-light signalling and "Impossible measurements"
RRelativistic causality in particle detector models:Faster-than-light signalling and “Impossible measurements”
Jos´e de Ram´on,
1, 2
Maria Papageorgiou,
1, 2, 3 and Eduardo Mart´ın-Mart´ınez
1, 2, 4 Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada Division of Theoretical and Mathematical Physics,Department of Physics, University of Patras, 26504, Patras, Greece Perimeter Institute for Theoretical Physics, 31 Caroline St N, Waterloo, Ontario, N2L 2Y5, Canada
We analyze potential violations of causality in Unruh DeWitt-type detector models in relativisticquantum information. We proceed by first studying the relation between faster-than-light signalingand the causal factorization of the dynamics for multiple detector-field interactions. We show inwhat way spatially extended non-relativistic detector models predict superluminal propagation ofthe field’s initial conditions. We draw parallels between this characteristic of detector models,stemming from their non-relativistic dynamics, and Sorkin’s “impossible measurements on quantumfields” [arXiv:gr-qc/9302018]. Based on these features, we discuss the validity of measurements inQFT when performed with non-relativistic particle detectors.
I. INTRODUCTION
Relativistic Quantum field theory (QFT) shares manyof its defining features with non-relativistic formulationsof quantum mechanics. However, the two crucially de-part in the sense that QFT is a theory explicitly formu-lated in a relativistic (possibly curved) spacetime man-ifold, which allows to incorporate notions such as Ein-stein causality or general covariance. In particular, theformal and conceptual differences between QFT and non-relativistic quantum mechanics become patent in the con-text of quantum measurement theory , since the standarddescription of measurements has been found to be at oddswith the notion of relativistic causality [1].In this context, particle detector models (such as theUnruh-DeWitt (UDW) model [2, 3]) can be thought ofas a framework to model the measurement, or ‘probing’,of quantum fields. As their name suggests, these mod-els were originally concocted to account for particle phe-nomenology in cases where a naive particle notion fails,e.g., for non-inertial motion or curved spacetimes. Gener-ically defined to be non-relativistic quantum-mechanicalsystems that couple locally to a fully relativistic QFT,particle detectors have been able to deliver physical in-sight in scenarios ranging from fundamental problems inQFT, such as horizon radiation and the entanglementstructure of quantum fields [4–7], to practical descrip-tions of the light-matter interaction in quantum opticsand quantum information experiments [7–9].The UDW model was first proposed by Unruh in hisseminal paper [2], with the intention of addressing thequestion of whether an accelerated observer will experi-ence the vacuum of a field theory as a thermal bath. Inthis first work, the detector was modeled in two differ-ent, yet thought of as complementary ways. First, thedetector was prescribed as a confined quantum mechani-cal particle in a box, and second as an auxiliary quantumfield. Later, DeWitt simplified the former by further con-sidering the detector as a point-like two-level system [10]. Yet elegant in its simplicity, the point-like model suffersfrom several technical complications such as spurious ul-traviolet (UV) divergences when the interaction betweendetector and field is not switched on and off adiabati-cally [11].Beyond the point-like model, the practice of smearingthe interaction of the detector with the field in spacetimehas become popular in the particle detector literature.Indeed, one can consider smeared detectors as straight-forward generalizations of the point-like model that donot suffer from UV divergences [11–13], i.e. as regular-ized detectors. In addition, from the practical point ofview, smeared detector-field interactions can better cap-ture the physics of experimental set-ups involving, e.g,the light-matter interaction [8, 14, 15], or the physics ofsuperconducting qubits [16]. Despite these advantages,smeared couplings are not devoid of their own fundamen-tal issues: coupling non-relativistic systems to smeared field operators causes problems with relativistic causal-ity [17] and the covariance of the model [9, 18].The main subject of this paper is to deepen the analysisof the friction between smeared detector models and rel-ativistic causality for general detector models in curvedspace-times, with an emphasis in the problem of the so-called “impossible measurements in QFT” [1, 19, 20].Crucially, the causality issues we will tackle are in-troduced by the very fundamental construction of themodel per-se, and not by extra approximations that in-troduce non-locality, such as the rotating-wave approxi-mation [21], or other a-posteriori non-relativistic approx-imations [22].A fully relativistic measurement scheme for QFT, whiletechnically involved, is of course devoid from causality is-sues (e.g., the FV-framework [23]). However it is perhapsstill reasonable to approach measurements in QFT frommuch simpler, effective, non-relativsitic detector modelssuch as the UDW. In this paper we will be concernedwith structural aspects of UDW-type models related tothe interplay between their non-relativistic nature and a r X i v : . [ qu a n t - ph ] F e b their spacetime localization through their smearing. Wewill pay especial attention to the possibility of superlu-minal signaling in smeared models, a phenomenon thatoughts to be unacceptable in relativistic physics.This work is divided in the following sections. In sec-tion II we discuss the challenges that arise in the for-mulation of measurement schemes in relativistic QFT,to highlight the differences with respect to the standard‘quantum-mechanical’ theory of measurement. This isnecessary for identifying which of these challenges arerelevant for detector models in QFT. In section III weestablish a unifying language in terms of which one cantackle the causality issues of general detector models ingeneral globally hyperbolic space-times.In section IV, we give general (non-peturbative) argu-ments about signaling between two detectors, and showthe absence of faster-than-light signalling, that is, theabsence of signalling when the detectors’ couplings areconstrained to spacelike separated regions.In section V we discuss superluminal propagation ofinitial data through the following set-up: The initial datais encoded in the field state through a local unitary overa region A. A detector C, partially in the causal future ofregion A, ‘mediates’ as a repeater information about thisinitial data to a detector B that is partially in the causalfuture of C, but spacelike separated from region A. Inthis section we will establish links and parallels betweenthis feature of detector models and Sorkin’s no-go resulton impossible measurements in QFT [1].In section VI we further analyze the interplay ofthe concepts of superluminal signaling and superluminalpropagation in three-detector scenarios. We show thatparticle detector models do not suffer superluminal sig-nalling at the first three orders in perturbation theory inthe coupling constants of the detectors. We conclude andsummarize in section VII. II. MEASUREMENT SCHEMES IN QFT
In this section we review some general considerationsregarding measurements that are particular to relativityand QFT. Aspects of quantum measurement theory, aswell as the measurement problem, are most commonlydiscussed in the context of non-relativistic quantum me-chanics. This can be helpful to isolate the aspects of theproblem that are present in any quantum theory, rela-tivistic or not. Nevertheless, when one tries to formulatethe problem in the context of relativistic QFT, there is avariety of technical and conceptual challenges that comefrom the explicitly relativistic nature of the theory [24].Despite the success of the quantum field theoretical de-scription in high energy physics and cosmology, not muchprogress has been made in the development of a mea-surement theory in QFT (only recently in the context ofthe algebraic approach [23]). In contrast, quantum me-chanics comes with (at least a standard) measurementtheory, i.e. a formalism for the mathematical description of measurements on which basis interpretational issuescan be addressed. Commonly, the physical predictionsare encoded in probability distributions derived from thetheory by means of projector valued measures (PVM),which assign probability distributions to sets of commut-ing observables following Born’s rule. More generally, onecan implement measurement schemes in which a systemis measured with the aid of an auxiliary system, or de-tector, which effectively generates probabilities describedby positive operator valued measures (POVM) [25]. Ineither case, it is further assumed that quantum mechani-cal systems undergo a “state update” once a property ofthe system is measured, and, in non-relativistic quantummechanics, this update is generally modelled by Luder’srule [25, 26].These concepts of standard quantum-mechanical mea-surement theory happen to be incompatible with rela-tivistic causality. Indeed, it can be proved that any lo-calized operator valued measure (in the sense of beingassociated to spatial sub-intervals of a hypersurface) isincompatible with the microcausality condition, i.e. thatobservables associated with spacelike separated regionscommute. This follows from a series of no-go theorems,the most relevant ones being the initially formulated byMalament [27] for PVM’s and the extension to POVM’sby Hegerfeld [28]. These theorems are most commonlydiscussed in the context of the localisation problem andthe non-existence of position operators in relativistic set-ups, and they seem to hint the necessity of a field theo-retical description of relativistic quantum physics.Perhaps surprisingly, formulating a measurement the-ory that is consistent with relativity has not been anyeasier in the context of an explicitly relativistic quan-tum field theory. The notion of instantaneous state up-date becomes problematic since joint probability distri-butions will depend on the order in which the measure-ments are performed, which generally depends on thereference frame. A modified prescription is required forthe extraction of frame-independent probabilities of suc-cessive measurements. A covariant version of the stateupdate rule that gives rise to frame independent proba-bility assignments was developed by Hellwig and Kraus[26].Moreover, in the algebraic approach to quantum fieldtheory (AQFT), local spacetime regions are associatedwith local Von Neumann algebras, which should be re-garded as the building blocks for a local measurementtheory. However, such algebras in AQFT, under somemild assumptions [29], are known to be of type III. Thistype of algebras do not contain finite-rank projectors,which poses several challenges to the usual constructionof measurements schemes, e.g. the Kraus representationof POVM’s [24, 30]. Furthermore, the type III characterof the local algebras has non-trivial consequences for thelocality and the entanglement properties of the theory[31, 32].Regarding the possibility of superluminal signaling,perhaps the most important challenge was posed bySorkin’s no-go result [1] on the impossibility of (ideal-ized) measurements in QFT. Sorkin’s work demonstratesthat signaling between two spacelike separated regions Aand B can be ‘mediated’ by an operation on a third regionC that is partially in the future of A and partially in thepast of B. Since superluminal signaling is not compatiblewith the axioms of relativity, Sorkin’s result proves thata naive ‘quantum-mechanical’ set of measurement ruleswould fail in relativistic QFT. Furthermore, the resultraises the issue of the consistent description of successive measurements when more than two measurements in dif-ferent spacetime regions are involved. The issue stemsfrom the fact that a partial causal order can be definedbetween pairs of extended (bounded) regions, but cannotbe naturally extended to multiple regions (unless they arepointlike). Sorkin suggests that a resolution can be givenin the more ‘space-time oriented’ formulation of sum-overhistories approach (See, e.g., [33]).Sorkin’s result was further analyzed recently in [19],where they studied (from a purely formal point of view)what conditions can be imposed as requirements for local,field-valued POVM measurements to avoid superluminalsignalling. Moreover, a formal resolution of the Sorkinproblem has been proposed recently in the context ofFewster and Verch framework for measurements in alge-braic QFT [20], although a connection from the formalresults to the description of experiments where quantumfields are measured remains elusive in this context.As we discussed in our introduction, particle detec-tor models do provide an alternative framework for thedescription of local measurements in QFT. Thus, wewould like to study whether particle detector modelspresent similar incompatibilities with relativity as thoseon Sorkin’s “impossible measurements”. Due to the non-relativistic nature of the detector system, the compatibil-ity with the premises of the underlying relativistic theory(QFT) is generally not guaranteed. To ground the for-malism of particle detectors on a relativistic QFT we needto consider the problem as made of two related (but sepa-rate) components: 1) the possible violation of relativisticcausality (related to the presence or not of FTL signallingin the framework) and 2) the possible breakdown of rela-tivistic covariance (related to coordinate independence ofpredictions). In fact, these two components are separatewithin QFT even before any notion of measurement isintroduced [34, 35]. Both requirements would be easierto fulfill in the case that the detector is also modeled asa relativistic system (probe quantum field) [23], but themain advantage of non-relativistic detectors is that theycome with a standard measurement theory. For exam-ple, one can measure them projectively (in contrast tothe quantum field) and infer the induced field observableassociated with each outcome.Even though the standard algebraic approach to QFTdoes not provide any measurement axioms, AQFT for-malizes a great variety of causality conditions [34, 36]that can be recruited to analyze any particular measure-ment model. Typically, the microcausality axiom (i.e. commutativity of field observables in spacelike separa-tion) is associated with non-signaling, at least when non-selective measurements are introduced [34]. The co-variance axiom guarantees the coordinate independenceof the dynamics of the theory and, crucially, it is inde-pendent from the microcausality axiom. For example, wecould design a theory in which superluminal signaling isnot possible but the laws of physics are dependent on thereference frame in which we describe them.These two topics were addressed separately by one ofthe authors in the context of detector models in QFT.Covariance requires examining the transformation prop-erties of the interaction Hamiltonian density betweenthe field and the detector under changes of referenceframes [18]. Causality issues in Unruh-DeWitt detec-tor models were analyzed in [17]. Also, a treatment ofthe causality porblem that shares some similarities withour approach has been explored for the particular caseof accelerated point-like detectors in a recent paper byScully et al. [37]. In what follows we will present a gen-eral analysis that will allow us to comment on the inter-play between these two notions (covariance and causal-ity of general detector models) as well as relate it withSorkin’s identification of the problem with standard mea-surements in QFT. III. THE DETECTOR MODEL
In this work we will consider a version of the Unruh-DeWitt detector model that is suitable for curved back-grounds. In QFT in curved spacetimes, one is oftenforced to restrict the analysis to globally hyperbolic man-ifolds, which are equipped with a Lorentzian metric g µν [38]. It can be shown, at least in some contexts (e.g.locally covariant AQFT [39]) that one can formulate aquantum theory of fields propagating in this type of man-ifolds and the fields will generally obey some hyperbolicequation of motion [40]. In the case of a scalar field onepossible such equation is the minimally coupled Klein-Gordon equation ∇ µ ∇ µ ˆ φ ( x ) + m ˆ φ ( x ) = 0 . (1)To define a detector model in general relativistic set-ups, it is convenient to introduce the interaction Hamil-tonian defining the coupling between detector and fieldin terms of Hamiltonian densities, which are covariantobjects [9].Now, consider a quantum system coupling to the fieldthrough the following Hamiltonian density in the inter- A non-selective measurement is a linear CPTP map that sends astate to a convex combination of states associated with possiblesoutcomes of a measurement, whereas a selective measurementsis a map, generally non-linear, that sends a state to one of thosestates associated to a particular outcome. action picture: ˆ h ( x ) = ˆ h ( x ) (cid:112) | g | , (2)where the Hamiltonian weight ˆ h ( x ) is defined asˆ h ( x ) := λ Λ( x ) ˆ J ( x ) ⊗ ˆ φ ( x ) . (3)Λ is a space-time function of compact support, ˆ J is acurrent operator associated with the detector and λ is acoupling constant. This Hamiltonian density generalizesmany particle detector models in the literature.From this Hamiltonian density one can define a timedependent Hamiltonian for the joint system asˆ H ( t ) = (cid:90) E ( t ) d E ˆ h ( x ) , (4)where E ( t ) is a one-parameter family of spacelike sur-faces. The parameter t is given by the values of a globalfunction, T ( x ), whose level curves can be taken to be theplanes of simultaneity of some observer. Under some as-sumptions [9, 18], t can be chosen to be the actual propertime of the detector. Finally, d E is a shorthand for thefamily of induced measures such that (cid:90) E ( t ) d E := (cid:90) d V δ ( T ( x ) − t ) (5)in the surfaces E ( t ), where d V = (cid:112) | g | d x n , n being thedimension of the manifold.Note that the operator ˆ J can typically depend on aprivileged foliation associated with the proper time ofthe center of mass of the detector, which we will denote τ ( x ). In fact, in the usual UDW prescription,ˆ J ( x ) = ˆ µ ( τ ( x )) , (6)where ˆ µ is the monopole operator given by ˆ µ ( τ ) = e iΩ τ ˆ σ + + e − iΩ τ ˆ σ − . The family of models (4) with thestandard UDW prescription for the operators acting overthe detector reduces to the standard UDW model whenthe time evolution is prescribed by an observer fiduciaryto the detector, that is T ( x ) = τ ( x ). In that caseˆ H ( t ) = λ (cid:90) E ( t ) d E Λ( x )ˆ µ ( τ ( x )) ⊗ ˆ φ ( x )= λ ˆ µ ( t ) ⊗ (cid:90) E ( t ) d E Λ( x ) ˆ φ ( x ) . (7)One can regard the current ˆ J as a property of the detec-tor and the function Λ as a property of the detector-fieldinteraction, introduced phenomenologically. This typeof detector models in curved space-times have been in-troduced very recently with finite dimensional detectors[9]. Hamiltonian (4) accounts for general detectors and general scalar fields in curved spacetimes, and so we willuse it as a starting point to analyse general features thatconcern all types of detector models. IV. FASTER-THAN-LIGHT SIGNALING INDETECTOR MODELS
This section is devoted to provide a general argumentconcerning the existence of faster-than-light signallingin measurement schemes modeled with particle detectormodels.In the underlying quantum field theory superluminalsignalling is prevented through the microcausality axiom,which states that the field operators commute in space-like separation, i.e. [ ˆ φ ( x ) , ˆ φ ( y )] = 0 , (8)if x and y are spacelike separated. The link between themicrocausality axiom and signalling stems from the no-tion of statistical independence of measurements. Con-sider for example two commuting observables [ ˆ A, ˆ B ] = 0of a closed quantum system. A non-selective measure-ment of observable ˆ A does not affect the expectationvalue of ˆ B . Indeed, let ˆ ρ be the initial state. Con-sider the spectral decomposition of ˆ A , i.e., ˆ A = (cid:80) a a ˆ P a .After a non-selective measurement of ˆ A the state of thesystem, denoted as ˆ ρ | a , is given byˆ ρ | a = (cid:88) a ˆ P a ˆ ρ ˆ P a (9)Since [ ˆ A, ˆ B ] = 0, it holds that [ ˆ P a , ˆ B ] = 0 ∀ a . Makinguse of this, we can easily see thattr( ˆ B ˆ ρ | a ) = tr( ˆ B ˆ ρ ) (10)which means that the expectation value of ˆ B does notdepend on a non-selective measurement (i.e., a measure-ment of which the outcome is not known) having hap-pened. In the context of quantum field theory, where thecommuting ˆ A and ˆ B are explicitly associated with space-like separated regions, microcausality guarantees condi-tion (10), which implies no-signaling in spacelike separa-tion.It is important to clarify that in the case of selectivemeasurements microcausality does not guarantee statis-tical independence, i.e., the statistics of ˆ B will generallydepend on the outcome A . It is well known that quan-tum field theory permits outcome-outcome correlations Above we denoted as ˆ ρ | a the state of the system given that theobservable A has been measured but the value of the outcomeA is not known, i.e., the mixture (9), which corresponds to anon-selective measurement. On the other hand, in a selectivemeasurement a particular outcome A is ‘selected’ and the stateof the system given this outcome is simply ˆ ρ | a = | a (cid:105) (cid:104) a | . In this in spacelike separation [31]. It is also well-understoodthat outcome-outcome correlations do not lead to su-perluminal signaling. Rather, they are a consequence ofthe fact that the state of the field (for example, a ther-mal state, or the vacuum) can display entanglement andclassical correlations even between spacelike separated re-gions [41].In the context of detector models, measurements areto be implemented through the dynamical coupling of adetector to the quantum field. The dynamical coupling isimplemented through a unitary operator that acts overjoint detector-field system, which is meant to describethe evolution of the joint system from an “in” spacetimeregion to an “out” spacetime region. The detector systemis supposed to be initially uncorrelated with the field inthe “in region”. After the interaction has finished, thestatistics of the detector system are analyzed. Effectively,this implies to analyze some subset of the statistics of thefield since the detector and the field are now correlateddue to the interaction.In this section we will make use of the most generalpossible (linear) detector model of the family of modelsdescribed by the Hamiltonian density (3). Further, if weare to restrict our set of measurements of the field tothis kind of detector-based protocols, it is necessary todefine signalling in terms of interactions between severaldetectors.Let us study then the dynamics of a set of independentdetectors interacting with the same quantum field. First,consider two detectors that couple to the same quantumfield undergoing an interaction generated by the Hamil-tonian densityˆ h ( x ) = ˆ h a ( x ) + ˆ h b ( x )= λ a Λ a ( x ) ˆ J a ( x ) ⊗ b ⊗ ˆ φ ( x )+ λ b Λ b ( x ) a ⊗ ˆ J b ( x ) ⊗ ˆ φ ( x ) . (11)This Hamiltonian density generates a joint Hamilto-nian for the joint system of the formˆ H ( t ) = ˆ H a ( t ) + ˆ H b ( t ) (12)where ˆ H a , b ( t ) = (cid:90) E ( t ) d E ˆ h a , b ( x ) . (13)Note that this Hamiltonian generates evolution with re-spect to the same parameter t for both detectors. Al-though we will not concern ourselves with this in thepresent work, since it has already been studied in [18],it is clear that one needs to properly reparametrize thelocal Hamiltonians to generate time translations with re- case, one can see that the analogue of (10) does not hold, i.e.,tr( ˆ B ˆ ρ | a ) (cid:54) = tr( ˆ B ˆ ρ ). FIG. 1. Causal relations between simply connected, non in-tersecting sets (grey and white) in two spacetime dimensions.Black lines represent the future or past lightcone of the sets orpoints between them. a ) , b ): Examples of non causally order-able sets. c ) , d ): Examples of sets that are causally orderable,but that do not causally precede each other according to ourdefinition. e ): Spacelike separated sets. f ): Example of a setcausally preceding another set. spect to the same parameter, which in general cannotcorrespond to the proper time of both detectors.In order to analyze causal relations between detectors,we need first to define causal relations between subsetsof spacetime. Given a globally hyperbolic spacetime, thefuture lightcone of a region O , J + ( O ), is the set of allpoints that lay in the causal future of some point of O .Similarly, J − ( O ), the causal past of a region O , is theset of all points that lay in the causal past of O . • We say that A and B are causally orderable if J − ( O a ) ∩ O b or J − ( O b ) ∩ O a are empty. • We say that A and B are spacelike separated if( J + ( O a ) ∪ J − ( O a )) ∩ O b or ( J + ( O b ) ∪ J − ( O b )) ∩O a are empty. Notice that this is a particular caseof causally orderable. • Finally, we have that if O b ⊂ J + ( O a ) /O a , and O a ⊂ J − ( O b ) /O b , we say that A causally precedesB. Notice that this is a particular case of causallyorderable since although J − ( O b ) ∩ O a = O a (cid:54) = ∅ ,it holds that J − ( O a ) ∩ O b = ∅ .Specifically, we have defined O a to precede O b if forevery observer all the events in O a precede any event in O b , that is, O a “comes first” for all observers.These are covariant statements that are independentof the observer, but one can also define causal relationswith respect to a particular foliation T ( x ). We say thatA precedes B with respect to T ( x ) if T ( x ) < T ( y ) forall x ∈ O a and for all y ∈ O b . The two notions are linkedby the following facts: • If A and B are causally orderable, one precedes theother with respect to some foliation. • If A and B are spacelike separated, then there areat least two foliations such that A precedes B withrespect to one and such that B precedes A withrespect to the other. • If A causally precedes B, then A precedes B withrespect to all foliations.We will say that two detectors obey any of the causalorder relations above if the regions O a = supp(Λ a ), O b = supp(Λ b ) obey the respective causal relations de-scribed above. See figure 1 for examples with simplyconnected sets.Given these definitions of causal relations, we can ana-lyze further the implications of the microcausality axiomin detector physics. The Hamiltonians defined by (13)are defined respect to some time function T ( x ), so thetwo detectors will naturally have causal relations withrespect to the foliation defined by its level curves. If theunderlying field theory were not relativistic, we wouldexpect that different foliations give rise to different dy-namics for spacelike separated detectors, because in thatcase the order in which the measurements are done wouldtypically matter. This is exactly what is to be avoided ina relativistic theory, and in the following we will exam-ine this condition in detector models departing from themicrocausality condition of the underlying QFT.Now, recall that the microcausality axiom in curvedspacetimes implies that, for two compactly supportedspacetime functions m ( x ) and l ( x ), (cid:90) d V (cid:90) d V (cid:48) l ( x ) m ( y )[ ˆ φ ( x ) , ˆ φ ( y )] = 0 (14)where d V = d x n (cid:112) | g | and d V (cid:48) = d y n (cid:112) | g (cid:48) | , if the sup-ports of l and m are spacelike separated. Therefore, themicrocausality axiom implies that (cid:2) ˆ h a ( x ) , ˆ h b ( y ) (cid:3) = 0 (15)if Λ a and Λ b have spacelike separated supports. This inturn implies that (cid:2) ˆ H a ( t ) , ˆ H b ( t (cid:48) ) (cid:3) = 0 (16)for all t, t (cid:48) .The joint evolution in the of the detectors and the fieldcan be described as a unitary operator acting over thejoint state of the system. That is, if ˆ ρ initial is the den-sity operator describing the state of the field-detectorssystem before the interactions are switched-on (respectto the parameter t ). The notation A+B indicates thatthe operator accounts for the interaction of the two de-tectors, whereas ˆ S a , b will denote the scattering matricesassociated with the individual interactions generated bythe individual interaction Hamiltonians. The total statein the asymptotic future will be given by the transforma-tion ˆ ρ final = ˆ S a + b ˆ ρ initial ˆ S † a + b (17) where ˆ S a + b is the so-called scattering operator . Thescattering operator is unitary and can be formally writtenas the Dyson seriesˆ S a + b = (cid:88) n (cid:18) − i (cid:126) (cid:19) n n ! (cid:90) ∞−∞ · · · (cid:90) ∞−∞ d t n × T ( ˆ H a ( t ) + ˆ H b ( t ) . . . ˆ H a ( t n ) + ˆ H b ( t n )) . (18)Intuitively, we would like to ensure that if two detectorsA and B are coupled to the field in spacelike separation,one cannot conclude whether the other one is coupledto the field or not. Therefore, a minimum non-signalingrequirement would be that if A interacts first with thequantum field in any foliation, i.e. if B is not in the causalfuture of A, then all expectation values of observables ofdetector B should not depend on magnitudes of detectorA, e.g. the coupling constant λ a . If the expectationvalues of observables of B depend on λ a , then its valuecould be used to encode, and then signal information.It is not a priori obvious why the causal behaviourof the underlying QFT, e.g. the microcausality axiom,would guarantee the causal behaviour of detectors. How-ever, we will see that this is guaranteed under some con-ditions [17]. As we will show below, in the context of par-ticle detector models, faster-than-light signalling is pre-vented if the joint scattering matrix factorizes when thedetectors are causally orderable. In particular, if A doesnot intersect with the past of B, we would haveˆ S a + b = ˆ S b ˆ S a . (19)We will refer to this property as causal factorization.To see that causal factorization prevents acausal sig-nalling, consider the local statistics of the detector A,given by the partial traceˆ ρ a = tr b ,φ ( ˆ S a + b ˆ ρ initial ˆ S † a + b ) . (20)Now, if causal factorization holds, thenˆ ρ a = tr b ,φ ( ˆ S b ˆ S a ˆ ρ initial ˆ S † a ˆ S † b ) . (21)But ˆ S b depends only on operators acting over the sub-spaces associated with the field and the detector B, there-fore it can be permuted within the partial trace:ˆ ρ a = tr b ,φ ( ˆ S a ˆ ρ initial ˆ S † a ˆ S † b ˆ S b )= tr b ,φ ( ˆ S a ˆ ρ initial ˆ S † a ) . (22)Therefore, we have shown that if causal factorizationholds, there is no local (space-time compact) measure-ment carried though a detector interaction that can be It is common in the UDW literature to denote the evolutionoperator by ˆ U . In this work, however, we prefer to denote itwith ˆ S to emphasize the fact that these maps represent scatteringoperators and we adopt a notation analog to, e.g., [20] . used to receive signals from another detector outside thecausal past of such interaction.Note that in the particular case where A and B arespacelike separated, then causal factorization impliesˆ S a + b = ˆ S b ˆ S a = ˆ S a ˆ S b , (23)which implies that neither detector A can signal to de-tector B nor detector B can signal to detector A, that is,it prevents faster than light signaling.We provide a proof of causal factorization in appendixA, which relies heavily on the microcausality conditionfulfilled by the field operators. It is rather intuitive whycondition (19) should hold if, e.g. A precedes B respectto the concrete foliation in which the interaction has beendefined, as the unitary evolution factorizes by construc-tion. This can be used to argue that the factorization willbe independent of the foliation if A causally precedes Bin the sense given at the beginning of this section. Theproof is also simple if A and B are spacelike separated,in which case the factorization also holds independentlyof the foliation. What is less trivial, however, is that thefactorization holds if the detectors are causally orderable,which is a covariant statement that does not depend onthe foliation either.In conclusion, causal factorization prevents faster-than-light signalling, as far as only two detectors areinvolved. The result can be extended to some limitedscenarios with many detectors. For example, if one hasmore than two detectors, say A , B , . . . , B N , one can al-ways define the collection of all the detectors that are notA as a single detector A c . If all the detectors in A c andA are causally orderable, with A preceding the rest, thenagain causal factorization will hold andˆ S Σ b i + a = ˆ S Σ b i ˆ S a = ˆ S a c ˆ S a (24)and the measurements on A will not be affected by theother detectors.It could be tempting to claim that this implies thatthe signals sent by a detector can only reach other detec-tors in the causal future of its interaction region. Indeed,causal factorization ensures this as long as we considerschemes involving two detectors. Obviously, if a detectorB can only receive signals from its causal past, then an-other single detector A can only send signals to B if B isin the causal future of A.However, if more than two detectors are involved, thencausal factorization does not solve all the possible fric-tions that the detector models can have with relativisticcausality. We deal with this in the next section. V. “IMPOSSIBLE MEASUREMENTS” ANDSUPERLUMINAL PROPAGATION OF INITIALDATA
We have defined signalling so far as the transmissionof information between detectors through their interac- O C O A O B FIG. 2. Two detectors A and B are coupled to the field overregions O a , b . The initial data are encoded in the field statethrough a unitary intervention over region O c . Notice thatregion O a is partially invading the past and future lightconesof regions O b and O c respectively. tion with the field. We have seen that, in a two-detectorscenario, a detector localized in some region is irrelevantfor another detector localized in its causal complement,which means that the detector only influences, in somesense, its own causal future.However, as pictured in Sorkin’s impossible measure-ments paper [1], there are subtleties associated with thedetectors not being in a definite causal ordering whenconsidering more than two measurements. Namely, evenif the response of the detector A cannot be influencedby the detector B, the influence of detector A over Bcan still carry information about events that happenedoutside the causal past of B, which is obviously not ac-ceptable.In order to understand how Sorkin’s problem can man-ifest in measurement’s models with particle detectors, weshall first analyze a different kind of signalling in whichthe information is not encoded in the interaction, but inthe initial state of the system.Indeed, a detector can also be thought of as a repeater,that is, given some initial state of the field (possibly com-ing from another interaction), the detector can registerthe initial data and propagate it back to the field. Inthis case, one may fear that a detector can re-emit in-formation in a non-causal manner. In this subsection wewill prove that this is indeed a reasonable concern, sincesuperluminal propagation of initial data is a widespreadphenomenon when considering non-relativistic systems.For instance, one could imagine a scenario in which adetector A partially precedes and is partially spacelikeseparated from a second detector B (see figure 2). Con-sider that the state of the system is initially given byˆ ρ initial = ˆ ρ a ⊗ ˆ ρ b ⊗ e i λ f ˆ φ ( f ) ˆ ρ φ e − i λ f ˆ φ ( f ) (25)where ˆ ρ a , b ,φ are arbitrary states of the detector A, B andthe quantum field respectively, and ˆ φ ( f ) is a smearedfield operator which is compactly supported in region O c , spacelike separated from B, but not from A. If O c One can think of this as a third party Charles encoding informa-tion in the field in region through a spacetime localized unitaryaction in region O c . is spaceilike separated from B, the local statistics of Bshould not be affected by the value of the constant λ f ,otherwise detector A would be acting as an agent forsuperluminal signalling.More generally, one can consider the case in which theinitial state of the detectors plus field has the formˆ ρ initial = ˆ U ˆ ρ ˆ U † , (26)where ˆ ρ is an arbitrary reference state of the joint sys-tem, and ˆ U = a ⊗ b ⊗ ˆ U φ is an arbitrary unitary actingon the field’s Hilbert space, so that ˆ U φ is localized in O c (contained in the causal complement of the interactionregion O b ). It is clear that[ ˆ U , ˆ S b ] = 0 . (27)ˆ U can be thought of as encoding a set of initial data .The statistics of detector B can only depend on ˆ U if de-tector A’s interaction region overlaps with the causal pastof B and ˆ S a does not commute with ˆ U (e.g., as shown inFig. 2). To avoid superluminal signalling, it should holdthat the local statistics of B do not depend on the choiceof ˆ U , i.e.ˆ ρ b = tr a ,φ ( ˆ S a + b ˆ U ˆ ρ ˆ U † ˆ S † a + b ) = tr a ,φ ( ˆ S a + b ˆ ρ ˆ S † a + b ) . (28)Further, since B is localized (at least partially) in thefuture of A, and it is spacelike separated from the set ofinitial data implemented by ˆ U , B cannot be fully con-tained in the causal past of A. We conclude that A doesnot causally precede B, in the terminology of the lastsection.Imposing condition (28) for all initial density operatorsis equivalent to tr a ,φ ( ˆ V ˆ σ ˆ V † ) = tr a ,φ (ˆ σ ) (29)for any arbitrary density operator ˆ σ , where ˆ V is a unitarygiven by ˆ V = ˆ S a + b ˆ U ˆ S † a + b . (30)This implies that if ˆ D b is an operator acting on detectorB (i.e. it commutes with the field operators and with theoperators acting on detector A) thentr( ˆ V † ˆ D b ˆ V ˆ σ ) = tr( ˆ D b ˆ σ ) (31)for all ˆ σ . For our purposes, this implies thatˆ V † ˆ D b ˆ V = ˆ D b , (32) We can always thing without loss of generality that the action ofˆ U is localized in a subset of a Cauchy surface in the causal pastof O c or equivalently [ ˆ D b , ˆ V ] = 0 (33)for all operators acting over detector B. Assuming that Aprecedes B, the connection with the propagation of initialdata is more clear when one uses causal factorization.Then, ˆ S a + b = ˆ S b ˆ S a and condition (33) can be written as[ ˆ S † b ˆ D b ˆ S b , ˆ S a ˆ U ˆ S † a ] = 0 , (34)for all unitaries in the causal complement of B. If wethink of ˆ S † b ˆ D b ˆ S b as an induced operator acting on thefield localized in region B and of ˆ S a ˆ U ˆ S † a as the evolutionof the initial data given by interaction A, we can interpretcondition (33) as that the interaction A does not prop-agate initial data superluminally, since the propagateddata still lays within the causal complement of region B.This condition is related to the unitary restriction of thecondition discussed in [19], but more general in the sensethat allows for auxiliary degrees of freedom representingthe devices used to implement the measurement.The relevant question now is whether condition (33)holds for general detector models. Unfortunately the an-swer is generally negative. It is easy to corroborate usingperturbation theory that the localization region of ˆ S a ˆ U ˆ S † a is not the causal future of ˆ U , but the causal future of A.Indeed, using Dyson’s expansionˆ S a ˆ U ˆ S † a = ˆ U − i (cid:126) (cid:90) d V [ˆ h a ( x ) , ˆ U ] − (cid:126) (cid:90) d V (cid:90) d V (cid:48) T (cid:104) ˆ h a ( x ) , [ˆ h a ( y ) , ˆ U ] (cid:105) + O ( λ a ) . (35)If we pay attention to the first term, which is given bythe density[ˆ h a ( x ) , ˆ U ] = λ a Λ a ( x ) ˆ J a ( x ) ⊗ b ⊗ [ ˆ φ ( x ) , ˆ U ] , (36)we realize that microcausality ensures that no x outsidethe lightcone of ˆ U can contribute to the integral. Thismeans that regardless of the localization of region B, theleading order propagation of initial data is still localizedin the lightcone of ˆ U and the propagation is causal.Now, at second order, the contribution will be givenby the kernel (cid:104) ˆ h a ( x ) , [ˆ h a ( y ) , ˆ U ] (cid:105) (37)where the time-ordering is implemented considering that y precedes x respect to the foliation T ( x ). Because ofmicrocausality, y will also be constrained to lie withinthe lightcone of O c , but x can be anywhere. One can useJacobi’s identity to expand this kernel as follows (cid:104) ˆ h a ( x ) , [ˆ h a ( y ) , ˆ U ] (cid:105) = (cid:104) [ˆ h a ( x ) , ˆ h a ( y )] , ˆ U (cid:105) + (cid:104) ˆ h a ( y ) , [ˆ h a ( x ) , ˆ U ] (cid:105) , (38)such that x has to lie in the lightcone of the initial datafor the second term not to vanish, but the first one willnot generally vanish when x is outside the lightcone of O c .One can see that in general, unless [ˆ h a ( x ) , ˆ h a ( y )] = 0when x and y are spacelike separated, the propagationwill not be causal anymore. Similar results were found in[18] when addressing violations of relativistic covariance.Indeed, one can further expand the commutator of theHamiltonian densities as[ˆ h a ( x ) , ˆ h a ( y )]= λ a Λ a ( x )Λ a ( y )[ ˆ J a ( x ) , ˆ J a ( y )] ⊗ b ⊗ ˆ φ ( x ) ˆ φ ( y )+ λ a Λ a ( x )Λ a ( y ) ˆ J a ( x ) ˆ J a ( y ) ⊗ b ⊗ [ ˆ φ ( x ) , ˆ φ ( y )] . (39)Again, microcausality ensures that the second term in(39) vanishes in spacelike separation, but the first onewill not vanish, nor will commute with ˆ U in general , un-less [ ˆ J a ( x ) , ˆ J a ( y )] = 0 in spacelike separation. In generalit is not difficult to argue (following a similar combina-toric procedure as in [18], together with a recursive useof Jacobi’s identity) that if the interaction Hamiltoinandensity of A is microcausal (for example for a pointlikedetector), the propagation of initial data is causal in allorders in perturbation theory.If this condition holds, it means that either all points insupp Λ a are causally connected (which is only possible fora pointlike detector) or that the detector is a relativisticfield. Since by assumption the system is non-relativisticand generally smeared, we conclude that the detector’sdynamics carry superluminal propagation of initial dataat second order in perturbation theory.Note that since for point-like detectors there is notsuperluminal propagation, one can disregard this kind offaster-than-light signalling for “small enough” detectors.Whether a detector is small or not will depend, of course,of the parameters of the problem.The preceding discussion provides a dynamical inter-pretation of the impossible measurements problem, in thesense that it links superluminal signalling with superlu-minal propagation within the device that is implementingthe measurement. It is clear then, that if the detector is arelativistic quantum field then there is not superluminalpropagation of initial data under some assumptions inthe dynamics of the coupling, as it is shown in full rigorin [20]. In our case, however, we have to understandthis kind of faster-than-light signalling as a fundamentalfeature of non-relativistic particle detector models thatrestricts their usage to regimes where these superlumi-nal features are negligible or irrelevant for the results athand. VI. IMPOSSIBLE MEASUREMENTS WITHWEAKLY COUPLED DETECTORS
We have seen that faster-than-light signalling is presentin smeared non-relativistic particle detector models.However, calculations involving particle detectors aremost commonly carried out in perturbation theory. In-deed, not only the justification of the model is jeopar-dized for strong couplings, but also some of the mostinteresting phenomenology, such emission and absorp-tion of particles, can be described at quadratic order inthe coupling strengths. Not only that, this is also theleading order for most phenomena in relativistic quan-tum information (e.g., detector’s responses [42], commu-nication [43], entanglement harvesting [7] and the FermiProblem [44–47], etc..). This section is devoted to ana-lyze the order in perturbation theory at which superlumi-nal propagation of initial data, described in last section,plays a role in measurement schemes involving more thantwo detectors.Let us slightly extend the set-up described in section Vby assuming that the unitary ˆ U in (26) is implemented bya weakly coupled detector C, in such a way that we canwrite ˆ U = ˆ S c . We can now determine at which order inperturbation theory the dynamics exhibits superluminalsignalling, that is, at which order in perturbation theorycondition (34) fails to hold.In order to do so, we first define the operatorˆ K := [ ˆ S † b ˆ D b ˆ S b , ˆ S a ˆ S c ˆ S † a ] . (40)If this operator vanished there would be no superlumi-nal propagation of initial data. We can determine thefirst order in the coupling strengths at which ˆ K does nottrivially vanish.We can expand ˆ K in the coupling strengths by writingˆ K = ˆ K (0) + ˆ K (1) + ... , where each ˆ K ( j ) contains integralsinvolving j Hamiltonians. Each term ˆ K ( j ) will containcontributions from orders in the coupling constants ofdetectors C+A+B in such a way that all the powers addup to j . It is easy to see thatˆ K (0) = [ ˆ D b ,
1] = 0 , (41)and that the linear term will also vanishˆ K (1) = (cid:20) i (cid:126) (cid:90) ∞−∞ d t [ ˆ H b ( t ) , ˆ D b ] , (cid:21) + [ ˆ D b , − i (cid:126) (cid:90) ∞−∞ d t [ ˆ H a ( t ) , D b , − i (cid:126) (cid:90) ∞−∞ d t ˆ H c ( t )] = 0 . (42)The fact that the first two terms in (42) vanish is obvious,while the third vanishes because B and C are spacelikeseparated.The higher order terms can be calculated similarly, butgiven the increasing complexity of the calculations it is0more practical to reason which terms will vanish basedon the following observations:1. The zeroth order in the coupling constant of detec-tor C cannot contribute to any order in ˆ K , becauseat that order condition (34) is satisfied trivially.2. The zeroth order in A cannot contribute at any or-der either, because B and C are spacelike separated.3. Finally, the zeroth order in B cannot contribute atany order because in that case the induced observ-able ˆ S † b ˆ D b ˆ S b acts trivially over the field.Therefore, ˆ K cannot have any contributions atquadratic order, because any quadratic contribution willinvolve the zeroth order of at least one of the detectors.Hence, ˆ K (2) = 0 . (43)This is not surprising if we take into account the re-sult of section IV, because at quadratic order the detec-tors interact only binary. Since the detectors only in-fluence each other pair-wise, the measurements cannotexhibit this type of superluminal signalling that involvesnecessarily three detectors. It is expected that this ar-gument carries through in general calculations involvingquadratic orders in perturbation theory.Interestingly, and perhaps less intuitively, the third or-der will also vanish. Indeed, the only term that can con-tribute at third order in perturbation theory, given theobservations made above, is the one that involves thelinear order of each in the three detectors.ˆ K (3) = − i (cid:126) (cid:20)(cid:90) ∞−∞ d t [ ˆ H b ( t ) , ˆ D b ] , [ (cid:90) ∞−∞ d t ˆ H a ( t ) , (cid:90) ∞−∞ d t ˆ H c ( t )] (cid:21) . (44)The operator in the first entry of the nested commuta-tor acts over the space of detector B and over the quan-tum field, whereas the operator on the second entry isgiven by (cid:90) d V (cid:90) d V (cid:48) [ˆ h a ( x ) , ˆ h c ( y )] . (45)Following the reasoning of section V, the microcausalitycondition forces this operator to be localized in the causalfuture of C, and therefore commutes with field operatorslocalized in region B. We then conclude thatˆ K (3) = 0 . (46)The interpretation in this case is that the superluminalpropagation of initial data happens only at quadratic or-der in the detector that may act as a repeater. Sincehaving a quadratic contribution in one of the detectors implies that at least one of the others contributes at ze-roth order, the arguments given above force ˆ K (3) = 0,and why no superluminal propagation can happen.The moral is that, assuming that all the measurementsare weakly performed with detectors, impossible mea-surements are not present in most calculations done inthe literature. One should be careful, however, whenhandling non-pertubative methods for smeared detectors. VII. CONCLUSIONS
In this work we have presented a detailed accountof the problem of formulating local measurements inQFT. We have explained how this problem motivates,both from the theoretical and practical points of view,the introduction of non-relativistic detector models. Wehave analyzed whether generalized Unruh-DeWitt-typedetector models fulfill minimum requirements regard-ing relativistic causality. In other words, we have dis-cussed whether non-relativistic systems coupled to quan-tum fields can be used to model repeatable measurementson quantum fields without incurring in incompatibilitieswith relativistic causality.In particular, we have investigated compatibility withrelativistic causality in detector-based measurements bydemanding that the signals emitted by each of the de-tectors should be constrained to lie within their associ-ated future light-cones. Furthermore, we have formulatedSorkin’s “impossible measurements” problem in termsof particle detector-based measurements, linking in thiscontext the “impossible measurements” issues to the non-relativistic dynamics of the detector. The physical intu-ition is that, when a detector is spatially extended, theinformation propagating inside the detector is not con-strained to travel subluminally since the detector is anon-relativistic system. However, we have shown thatwithin the usual assumptions—that is, weak couplingsor point-like or nearly point-like detectors— detector-based measurements are safe from the “impossible mea-surement” problem.
ACKNOWLEDGMENTS
The authors would like to thank Jason Pye for discus-sions that were as long as they were helpful and insight-ful. EMM acknowledges support through the DiscoveryGrant Program of the Natural Sciences and Engineer-ing Research Council of Canada (NSERC). EMM alsoacknowledges support of his Ontario Early Researcheraward. MP acknowledges support of the 2020 Constan-tine and Patricia Mavroyannis Scholarship Award by theAHEPA Foundation.1
Appendix A: Causal factorization in detector models
In this appendix we provide a proof of the causal fac-torization of the scattering operator for compactly sup-ported detectors. The proof relies in elementary proper-ties of unitary propagators. In this appendix we consider (cid:126) = 1 in order to ease the notation.Consider a general time-dependent interaction Hamil-tonian of the form,ˆ H ( t ) = ˆ H a ( t ) + ˆ H b ( t ) , (A1)and its associated Schrodinger equation ∂ t | ψ ( t ) (cid:105) a + b = − i( ˆ H a ( t ) + ˆ H b ( t )) | ψ ( t ) (cid:105) , (A2)or, more conveniently, in its integral form | ψ ( t ) (cid:105) a + b = | ψ ( t (cid:48) ) (cid:105) a + b − i (cid:90) tt (cid:48) d t ( ˆ H a ( t (cid:48)(cid:48) ) + ˆ H b ( t (cid:48)(cid:48) )) | ψ ( t (cid:48)(cid:48) ) (cid:105) a + b . (A3)By recursively applying this integral equation, disregard-ing domain issues, we can formally write the evolution ofthe state as the action of a two-parametric group of uni-tary operators, also known as the unitary propagator,ˆ U a + b ( t, t (cid:48) ): | ψ ( t ) (cid:105) = ˆ U a + b ( t, t (cid:48) ) | ψ ( t (cid:48) ) (cid:105) = (cid:88) n ( − i) n n ! (cid:90) tt (cid:48) · · · (cid:90) tt (cid:48) d t n × T ( ˆ H a ( t ) + ˆ H b ( t ) . . . ˆ H a ( t n ) + ˆ H b ( t n )) | ψ ( t (cid:48) ) (cid:105) , (A4)where the second line is the so-called Dyson expansion ofthe operator ˆ U a + b ( t, t (cid:48) ). Here we have defined the timeordering of two time-dependent operators ˆ A ( t ) and ˆ B ( t )as T ˆ A ( t ) ˆ B ( t (cid:48) ):= θ ( t − t (cid:48) ) ˆ A ( t ) ˆ B ( t (cid:48) ) + θ ( t (cid:48) − t ) ˆ B ( t (cid:48) ) ˆ A ( t ) , (A5)where the definition is similar for higher orders. It willbe useful in the following to define unitary propagatorsthat woudl be associated to local evolution, that isˆ U ν ( t, t (cid:48) ) | ψ ( t (cid:48) ) (cid:105) = (cid:88) n ( − i) n n ! (cid:90) tt (cid:48) · · · (cid:90) tt (cid:48) d t n × T ˆ H ν ( t ) . . . ˆ H ν ( t n ) | ψ ( t (cid:48) ) (cid:105) , (A6)where ν ∈ { A , B } . In order to describe the dynamics ofthe detection process, we are particularly interested inthe scattering operator, that is, the limitˆ S a + b = lim t (cid:48) →−∞ lim t →∞ ˆ U a + b ( t, t (cid:48) ) (A7) when the Hamiltonians are given by the expressionsˆ H a , b ( t ) = (cid:90) E ( t ) d E ˆ h a , b ( x ) . (A8)Consider now that the supports of Λ a and Λ b arecausally orderable, with A preceding B respect to somefoliation (possibly different from T ( x )). Then the scat-tering matrix factorizes, i.e.ˆ S a + b = ˆ S b ˆ S a . (A9)To show this we find the Schrodinger equation of thefactorized dynamics and prove that it coincides with thefull dynamics. Then we will use a uniqueness argumentto prove that therefore the dynamics coincide.Consider the family of states | ψ ( t ) (cid:105) ab = ˆ U b ( t, −∞ ) ˆ U a ( t, −∞ ) | ψ (cid:105) (A10)where | ψ (cid:105) is a fixed vector. It holds that ∂ t | ψ ( t ) (cid:105) ab = − i (cid:16) ˆ H b ( t ) + ˆ U b ( t, −∞ ) ˆ H a ( t ) ˆ U † b ( t, −∞ ) (cid:17) | ψ ( t ) (cid:105) ab . (A11)Let us first distinguish two trivial cases. First, considerthat A precedes B, respect to the foliation T ( x ). Then,there exists a number t c such thatˆ H b ( t c ) = ˆ H a ( t c ) = 0 (A12)and ˆ H b ( t ) = 0 t < t c (A13)ˆ H a ( t ) = 0 t > t c . (A14)This implies thatˆ U b ( t, −∞ ) ˆ H a ( t ) ˆ U † b ( t, −∞ ) = ˆ H a ( t ) (A15)for all t , since ˆ U a ( t, −∞ ) is only different from the iden-tity operator, ˆ
1, when ˆ H b ( t ) = 0.Second, if the supports are spacelike separated, then[ ˆ H b ( t ) , ˆ H a ( t (cid:48) )] = 0 (A16)for all t, t (cid:48) ∈ R , and thereforeˆ U b ( t, −∞ ) ˆ H a ( t ) ˆ U † b ( t, −∞ )= (cid:88) n ( − i) n n ! (cid:90) t −∞ · · · (cid:90) t −∞ d t n × T (cid:104) . . . [ ˆ H a ( t ) , ˆ H b ( t )] . . . , ˆ H b ( t n ) (cid:105) = ˆ H a ( t ) . (A17)More generally, assume that the detectors are causally2orderable. Then, essentially , thatˆ U b ( t, −∞ ) ˆ H a ( t ) ˆ U † b ( t, −∞ )= (cid:90) E ( t ) d E ˆ U b ( t, −∞ )ˆ h a ( x ) ˆ U † b ( t, −∞ ) . (A18)For each x ∈ supp(Λ a ), ˆ h a ( x ) is either causally con-nected or spacelike separated to the support of B, so wecan choose the corresponding proof from the two onesgiven above to show that it remains unchanged underthe adjoint action of ˆ U b ( t, −∞ ). Thereforeˆ U b ( t, −∞ ) ˆ H a ( t ) ˆ U † b ( t, −∞ )= (cid:90) E ( t ) d E ˆ h a ( x ) = ˆ H a ( t ) . (A19)Altogether, the conclusion is that if A precedes B for some observer then | ψ ( t ) (cid:105) ab fulfils (A2), and since (A2)is a linear differential equation, the vector | ϕ ( t ) (cid:105) = | ψ ( t ) (cid:105) ab − | ψ ( t ) (cid:105) a + b (A20)also fulfils (A2). Now, setting | ϕ ( −∞ ) (cid:105) = 0 implies | ϕ ( t ) (cid:105) = 0 for all t , since the solution is unique and | ϕ ( t ) (cid:105) = 0 is a solution with initial condition | ϕ ( −∞ ) (cid:105) =0. Therefore, we have shown that | ψ ( t ) (cid:105) a + b = | ψ ( t ) (cid:105) ab = ˆ U a ( t, −∞ ) ˆ U b ( t, −∞ ) | ψ (cid:105) , (A21)for all t , and more concretely,ˆ S a + b | ψ (cid:105) = | ψ ( ∞ ) (cid:105) a + b (cid:12)(cid:12) | ψ ( −∞ ) (cid:105) a + b = | ψ (cid:105) = ˆ S a ˆ S b | ψ (cid:105) , (A22)for all states in the Hilbert space. [1] R. D. Sorkin, in Directions in general relativity: Pro-ceedings of the 1993 International Symposium, Maryland ,Vol. 2 (1993) pp. 293–305.[2] W. G. Unruh, Phys. Rev. D , 870 (1976).[3] W. G. Unruh and R. M. Wald, Phys. Rev. D , 1047(1984).[4] G. W. Gibbons and S. W. Hawking, Phys. Rev. D ,2738 (1977).[5] A. Valentini, Phys Lett A , 321 (1991).[6] B. Reznik, Found. Phys. , 167 (2003).[7] A. Pozas-Kerstjens and E. Mart´ın-Mart´ınez, Phys. Rev.D , 064074 (2016).[8] E. Mart´ın-Mart´ınez and P. Rodriguez-Lopez, Phys. Rev.D , 105026 (2018).[9] E. Mart´ın-Mart´ınez, T. R. Perche, and B. de S. L. Torres,Phys. Rev. D , 045017 (2020).[10] B. DeWitt, in General Relativity: An Einstein CentenarySurvey , edited by S. W. Hawking and W. Israel (Cam-bridge University Press, Cambridge, 1979).[11] J. Louko and A. Satz, Class. Quant. Grav. , 055012(2008).[12] S. Schlicht, Class. Quant. Grav. , 4647 (2004).[13] A. Satz, Class. Quant. Grav. , 1719 (2007), arXiv:gr-qc/0611067.[14] N. Stritzelberger and A. Kempf, Phys. Rev. D ,036007 (2020).[15] R. Lopp and E. Mart´ın-Mart´ınez, Phys. Rev. A ,013703 (2021).[16] E. McKay, A. Lupascu, and E. Mart´ın-Mart´ınez, Phys.Rev. A , 052325 (2017).[17] E. Mart´ın-Mart´ınez, Phys. Rev. D , 104019 (2015).[18] E. Mart´ın-Mart´ınez, T. R. Perche, and B. d. S. L. Torres,Phys. Rev. D , 025007 (2021). We are assuming, without proof, that the adjoint action of theunitary evolution can be carried inside the integral (A18). [19] L. Borsten, I. Jubb, and G. Kells, Impossible measure-ments revisited (2019), arXiv:1912.06141 [quant-ph].[20] H. Bostelmann, C. J. Fewster, and M. H. Ruep, Impos-sible measurements require impossible apparatus (2020),arXiv:2003.04660 [quant-ph].[21] N. Funai and E. Mart´ın-Mart´ınez, Phys. Rev. D ,065021 (2019).[22] M. Papageorgiou and J. Pye, J.Phys. A Math. and Theor. , 375304 (2019).[23] C. J. Fewster and R. Verch, Commun. Math. Phys. ,851 (2020).[24] L. Ruetsche and J. Earman, in Probabilities in Physics (Oxford University Press, 2011) pp. 263 – 290.[25] M. A. Nielsen and I. L. Chuang,
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