aa r X i v : . [ qu a n t - ph ] F e b Entangling Power of an Expanding Universe
Greg Ver Steeg ∗ and Nicolas C. Menicucci
2, 3, † California Institute of Technology, Pasadena, California 91125, USA Department of Physics, Princeton University, Princeton, New Jersey 08544, USA Department of Physics, The University of Queensland, Brisbane, Queensland 4072, Australia (Dated: October 29, 2018)We show that entanglement can be used to detect spacetime curvature. Quantum fields in theMinkowski vacuum are entangled with respect to local field modes. This entanglement can beswapped to spatially separated quantum systems using standard local couplings. A single, inertialfield detector in the exponentially expanding (de Sitter) vacuum responds as if it were bathed inthermal radiation in a Minkowski universe. We show that using two inertial detectors, interactionswith the field in the thermal case will entangle certain detector pairs that would not become en-tangled in the corresponding de Sitter case. The two universes can thus be distinguished by theirentangling power.
PACS numbers: 04.62.+v, 03.65.Ud
Information in curved spacetime has played a promi-nent role in the attempt to understand the interface be-tween quantum physics and gravity [1, 2, 3, 4]. Whileabstract properties of curved-space quantum fields (in-cluding their entanglement) can be studied directly [5, 6,7, 8, 9], an operational approach involving observers withdetectors historically has been a critical component oftheoretical progress in this area [3, 10]. With the birth ofquantum information theory [11], quantum systems couldnow be analyzed in terms of their use for information-theoretic tasks like quantum computation [11], quantumteleportation [12], and quantum cryptography [13]. En-tanglement is a phenomenon that is uniquely quantummechanical in nature [14] and can be considered bothan information-theoretic and a physical resource [15].It is known that the Minkowski vacuum possesses long-range entanglement [9] that can be swapped to local in-ertial systems using standard quantum coupling mecha-nisms [16]. Variations on this theme can be considered,including accelerating detectors [17], thermal states [18],and curved spacetime. Our focus will be on curvature.For this, we choose an exponentially expanding (de Sit-ter) universe [7, 19] for its simplicity and because of itsimportance to cosmology [20].We wish to demonstrate a connection between aphysical property of spacetime (curvature) and aninformation-theoretic resource (entanglement). While itis possible to directly study the entanglement present ina quantum field in de Sitter spacetime, this sometimesleads to difficulties [6] that are not present in a more op-erational approach. Still, it is known that entanglementbetween field modes can directly encode a spacetime’scurvature parameters [8]. Motivated by a desire to beas operational as possible, we examine how curvature af-fects a field’s usefulness as an entangling resource —i.e.,its ability to entangle distant quantum systems (“detec-tors”) using purely local interactions. We begin by re-viewing the response of a single, inertial detector inter- acting with a massless, conformally coupled scalar field.The result in the vacuum de Sitter case is identical to thatin the case of a thermal ensemble of field particles in flatspacetime [3, 10]. Next, we ask the question, can entan-glement be used to distinguish de Sitter vacuum expansionfrom Minkowski-space heating?
We show that with twodetectors on comoving trajectories, there exists a param-eter regime in which the local systems that couple to thefield will become entangled despite the presence of extrathermal noise in each individual detector. Interestingly,this region of parameter-space in the expanding case is a proper subset of the same region in the locally equivalentthermal case. Thus, while both universes affect a localinertial detector in exactly the same way, entanglementbetween two detectors can be used to distinguish them.We start with the following experimental setup, whichis nearly identical to that used by Reznik et al. [16], us-ing units where ~ = c = k B = 1. We pose our prob-lem completely in operational terms, but our goal is toshow proof of principle—not necessarily practicality ofthe method. We suppose that the inhabitants of a par-ticular planet launch a satellite into space to measure thetemperature of the universe they inhabit. On board thissatellite is a qubit (a two-level quantum system), initiallyin the ground state | i , that gets coupled locally and fora limited time to a scalar field using a simple De Wittmonopole coupling [21]. The time-dependent interactionHamiltonian for this detector is, in the interaction pic-ture, H I ( τ ) = η ( τ ) φ (cid:0) x ( τ ) (cid:1)(cid:0) e + i Ω τ σ + + e − i Ω τ σ − (cid:1) , (1)where τ is the proper time of the satellite, η ( τ ) is a weaktime-dependent coupling parameter (which we’ll call thedetector’s “window function”), x ( τ ) is the worldline ofthe satellite, φ ( x ) is the field operator at the spacetimelocation x , and the rest represents the interaction-picturePauli operator σ x ( τ ) for the local qubit with (tunable)energy gap Ω. Roughly speaking, the detector works byinducing oscillations between the two levels at a strengthgoverned by the local value of the field.From now on, we refer to this qubit as a “detector,” al-though the process of “detection” includes only the fieldinteraction (before projective measurement). We wishto examine when two such detectors become entangledthrough their local interactions with the field, so we de-lay classical readout to allow for general quantum post-processing, which may be necessary to show violation ofa Bell inequality [22].The window function η ( τ ) is used to turn the de-tector on and off, but the transitions must be suffi-ciently smooth so as not to excite the field too muchin the process [23]. Beyond this requirement, on physi-cal grounds, our results should not depend on the detailsof the window function as long as it is approximatelytime bounded, so we will always choose η ( τ ) to be pro-portional to a Gaussian, η ( τ ) = η e − ( τ − τ ) / σ , where η = η ( τ ) ≪ | τ − τ | . σ and “off” the rest of thetime and also has a nice analytic form.Without loss of generality, we can set τ = 0. To lowestnontrivial order in η , the qubit after the interaction (butbefore readout) will be found in the state ρ = A | ih | +(1 − A ) | ih | , where A = Z ∞−∞ dτ Z ∞−∞ dτ ′ η ( τ ) η ( τ ′ ) e − i Ω( τ − τ ′ ) D + (cid:0) x ( τ ); x ( τ ′ ) (cid:1) , (2)where D + ( x ; x ′ ) = h φ ( x ) φ ( x ′ ) i is the Wightman functionfor the field, with expectation taken with respect to thestate of the field (assumed to be a zero-mean Gaussianstate, but not necessarily the vacuum). Repeated mea-surement in the {| i , | i} basis for a variety of values ofΩ allows for determination of the state of the detector asa function of Ω [29]. As is clear from Eq. (2), the stateis completely determined by the detector response func-tion D + (cid:0) x ( τ ); x ( τ ′ ) (cid:1) , which is the Wightman functiontaken at two different proper times along the worldlineof the detector [10].We consider two possible universes. The first isMinkowski, ds = dt − P i =1 dx i , with the field in athermal state with temperature T with respect to theinertial trajectory { x i } = (constant). The second is ade Sitter universe, ds = dt − e κt P i =1 dx i , where κ isthe expansion rate, in the conformal vacuum. The con-formal vacuum is the natural choice in this case becauseit is the unique, coordinate-independent vacuum statedictated by the symmetries of the spacetime. Further-more, it can be justified on physical grounds because theconformal vacuum coincides with the massless limit ofthe adiabatic vacuum for de Sitter space [10]. Thus, wecan think of this analysis as applying to the following two ways of adiabatically modifying the Minkowski vacuum:(1) very slowly heating the universe to a temperature T ,and (2) very slowly ramping up the de Sitter expansionrate (from zero) to a final value of κ .The variables { x i } are comoving coordinates, and t iscosmic time. (Since the Minkowski metric is the spe-cial case κ = 0, this terminology carries over to it, aswell.) In both universes, worldlines of constant { x i } areinertial trajectories (geodesics), and intervals of propertime equal those of cosmic time (∆ τ = ∆ t ). In bothcases, the scalar field φ ( x ) is massless and conformallycoupled [10], satisfying [ (cid:3) x + R ( x )] φ ( x ) = 0, where theRicci scalar R ( x ) = 12 κ is a constant proportional tothe expansion rate κ .Gibbons and Hawking [3] showed that the detector re-sponse function for any inertial observer in the de Sit-ter case is exactly the same as that of a detector atrest in a thermal bath of field particles with tempera-ture T = κ/ π in flat spacetime. Thus, a single detectoralone cannot distinguish between the two cases if it for-ever remains on a given inertial trajectory. In both casesconsidered above, the detector is at rest in the comovingframe and thus, D + T (cid:0) x ( τ ); x ( τ ′ ) (cid:1) = − T [ πT ( t − t ′ − iǫ )] , (3)where the subscript T indicates that this is a detector re-sponse function for a thermal state at temperature T .When the satellite begins sending back measurementdata, the reconstructed A (Ω) is found to be consistentwith the detector being at rest in a thermal bath of fieldparticles at a small but nonzero temperature T . If theinhabitants wish to know whether this perceived ther-mality is a result of heating or expansion, though, theymust be more creative.Obviously, they could use astrophysical clues (likewe have done on Earth) and/or Doppler-shift measure-ments [30] to determine whether their universe is expand-ing or not, but we are going to restrict them to using onlysatellite-mounted detectors of the sort described aboveon fixed inertial trajectories. If the detectors are to beuseful, then, they will need more than one.We propose the following alternative that makes use ofentanglement to distinguish the two universes. We imag-ine two satellites, each having many qubits that inter-act locally with the scalar field. (Having many detectorsallows access to many copies of the same state.) We as-sume that the satellites have no initial entanglement witheach other and that the qubits each begin in the groundstate. After interacting with the field, measurement isdelayed to allow for general quantum operations (localto each satellite) on the multitude of qubits on board. Inthe end, however, the only data that can be transmittedback to the home planet are measurement results, plusinformation about the postprocessing and the particularmeasurements performed.In an attempt to be as simple as possible, we analyzethe case of two inertial detectors, a and b , on the comov-ing trajectories x = ± L/ x = x = 0). Due tothe homogeneity and isotropy of space in both scenarios,this case is remarkably general—but not entirely so sinceone could imagine the detectors in motion with respect toeach other (beyond the relative motion generated by anyexpansion). For simplicity, we’ll also require that the twodetectors have synchronized local clocks with τ a,b = t ,equal resonant frequencies Ω a,b = Ω, and identical win-dow functions η a,b ( τ ) = η e − τ / σ . Finally, we desirethat L ≫ σ so that the detector-field interactions canbe considered noncausal events [31]. As we shall see,these restrictions will still allow the inhabitants, locatedat x i = 0, to distinguish expansion from heating.By spatial symmetry, each detector alone must respondusing the detector response function from Eq. (3) andthus provides no useful information. The only hope,then, is in the correlations between the detectors. Wewill focus on those correlations that signal the presenceof entanglement of the detectors after interaction with thefield. For a pair of qubits, the negativity [24] of a stateis nonzero if and only if the systems are entangled [25].Since we have access to (by assumption) multiple copiesof an entangled state of pairs of qubits, a local measure-ment protocol (on the many copies of the state) alwaysexists to verify entanglement by showing a violation of aBell inequality [22, 26]. This can be verified by a thirdparty using classical data received from both satellites.We will focus on finding the regimes in which entan-glement is nonzero, rather than on the magnitude of theentanglement for two reasons. First, the amount of ex-tractable entanglement is small enough to be impracticalas a resource and will depend on the details of the de-tector coupling. Second, we are primarily interested inunderstanding a qualitative difference between the quan-tum behavior of curved and flat spacetime; examiningentanglement ensures that this is a genuinely quantummechanical effect [14].An analogous calculation to Reznik’s [16] shows thatthe negativity of the joint state of the qubits is N =max (cid:0) | X | − A, (cid:1) , where A is the individual detector re-sponse from Eq. (2), while X is defined as X = − Z ∞−∞ dt Z t −∞ dt ′ η ( t ) η ( t ′ ) e i Ω( t + t ′ ) × (cid:2) D + (cid:0) x a ( t ); x b ( t ′ ) (cid:1) + D + (cid:0) x b ( t ); x a ( t ′ ) (cid:1)(cid:3) = − Z t ′ 0. Both X and A can be evalu-ated analytically: X = − e − L σ − σ Ω σ erfi (cid:0) L σ (cid:1) L √ π , (6) A = e − σ Ω − √ πσ Ω erfc( σ Ω)4 π , (7)where σ is the width of the window function (the timefor which the detector is turned on), and the subscriptsindicate that these are the Minkowski vacuum results,with erfi( z ) = − i erf( iz ) and erfc( z ) = 1 − erf( z ), whereerf( z ) is the error function. In the Minkowski vacuumcase, the detectors become entangled if and only if | X | >A . This region in the L -Ω plane is above the slantedblack line in Fig. 1.Let’s see what happens with a nonzero temperature.Since we are interested in the possibility that the per-ceived thermality is due to de Sitter expansion, we havea restriction on the temperature, which sets the scale forthe cosmic horizon L H = κ − = (2 πT ) − . If observersare to exist at all, this horizon must be much larger thantheir typical scale of experience, which can’t be muchsmaller than σ if the detector is to be useful to them.(Consider how useful a “detector” that operates on thescale of the Hubble time would be for humans.) Thus, forde Sitter expansion even to be a possibility, we requirethat T ≪ σ − .In both cases, the detector response function is givenby Eq. (3), while the Wightman function to be used in X in the thermal case is [28] D +th (cid:0) x a ( t ); x b ( t ′ ) (cid:1) = T πL × n coth (cid:2) πT ( L − y ) (cid:3) + coth (cid:2) πT ( L + y ) (cid:3)o (8)and in the de Sitter case is [10] D +dS (cid:0) x a ( t ); x b ( t ′ ) (cid:1) = (cid:18) − π (cid:19) (cid:20) sinh ( πT y ) π T − e πT x L (cid:21) − , (9)where x = t + t ′ , and y = t − t ′ − iǫ in both. One canverify that in both cases, taking L → T → T (as T → D +th and D +dS , respectively, about x = y = 0.Although the radius of convergence of the Taylor seriesis finite, for any reasonable detector setup, we are re-quiring that L ≫ σ . Since the nearest pole is either O ( L ) or O ( T − ) away, the Gaussian window function,whose width is much smaller than either L or T − , willregularize, within the integral, any reasonably truncatedTaylor approximation to the Wightman function. Thisresults in a valid asymptotic series for X in either case,as T → 0. The integral in Eq. (2) can be done similarlyby writing D + T = D +0 + ∆ D + T (noting that the pole at y = 0 has been eliminated in ∆ D + T ) and calculating thetemperature-dependent correction to Eq. (7). Numericalchecks of particular cases verify that these approxima-tions are valid. The results are presented in Fig. 1. (cid:144) Σ Σ W H horizon L entangledin all 3 casesentangledonly if T = ø FIG. 1: Entanglement profile for detector pairs in severaluniverses— σ is detection time, Ω is detector resonance fre-quency, L is detector separation. The slanted black line is theentanglement cutoff in the Minkowski vacuum case (entangledabove, separable below). The solid red curve is the thermalMinkowski cutoff, and the dashed blue curve is the de Sittervacuum cutoff, both with perceived local temperatures satis-fying 2 πT = 10 − σ − . The de Sitter horizon distance (10 σ )is given by the dotted green line. The red star indicates oneparticular detector setup that could be used to distinguishexpansion from heating. Several points are in order here. First, detectors seeanything at all in the Minkowski vacuum case because the time-energy uncertainty relation, ∆ t ∆ E & , impliesthat a detector operating for a finite time has a nonzeroprobability A of becoming excited, even when the fieldis in the vacuum state. Entanglement exists when virtualparticle exchange dominates over local noise. When themagnitude of the exchange amplitude | X | exceeds A ,the detectors become entangled [16, 25]. Because of howboth functions scale with Ω and L , in the vacuum caseone can always reduce the local noise below | X | by suffi-ciently increasing Ω. In the thermal and de Sitter cases,the local noise profile A fails to decrease fast enough forlarge Ω, resulting in a maximum entangling frequencyfor a given L , as well as a maximum separation beyondwhich entanglement is impossible, regardless of Ω.What does this mean for our curious planetary inhabi-tants? Let’s assume they have two satellites, with detec-tors of the sort we’ve been using, located on comovingtrajectories as described above, with κ − < L < κ − so that in the de Sitter case they would be outside ofeach other’s cosmic horizon but within that of the homeplanet (so they can still send messages to it, as describedin Fig. 2). The satellites are programmed to interact thefield locally with qubits having a resonant frequency thatwill lead to entanglement in the thermal case and to aseparable state in the de Sitter case (e.g., the red starin Fig. 1). After the interactions, they each run a localmeasurement protocol that implements one side of a testof Bell inequality violation, after which they send databack to the home planet for analysis. If thermality is aresult of expansion, there will be no entanglement, but ifit is a result of heating in flat spacetime, then the entan-glement can be verified upon receipt of the transmissionsfrom both satellites. Because this effect only manifestswhen the detectors pass beyond each others’ cosmic hori-zons (in the de Sitter case), a third party is required tomake the determination.We have demonstrated that while expansion and heat-ing give rise to the same (thermal) signature in a singleinertial particle detector, for certain choices of detectorparameters, a heated field in flat spacetime is able to en-tangle detector pairs that the conformal vacuum in theassociated de Sitter universe cannot. Thus, the universescan be distinguished by their entangling power . Two de-tectors are required and must be beyond each others’ cos-mic horizons (in the de Sitter case) to see the effect. Al-though, if present, the entanglement is exceedingly small,in principle its presence can always be determined byclassical communication of local measurement data to athird party, as long as the verifier is able to receive mes-sages from both detectors. These results are contraryto the intuition that “curvature generates entanglement”between field modes [8], since from it one would expect a larger entangled region in the de Sitter case. The abilityof the field to swap its entanglement to local detectors isan operational question, though, and for this setup, thevacuum in a curved spacetime has less entangling power t M x M a b FIG. 2: Spacetime diagram in Minkowski coordinates in therest frame of the home planet (circle). Null rays travel at45 degrees and light dotted lines represent geodesics in deSitter space. Messages sent from detectors a or b ( ⋆ ) neverreach the other detector because of the Hubble expansion ofthe universe. However, the home planet can receive and ana-lyze the messages, differentiating the entanglement scenariosdepicted in Fig. 1 . than a corresponding heated field in flat spacetime, eventhough both produce the same local detector response.We thank John Preskill, Sean Carroll, Gerard Milburn,and Carl Caves for invaluable discussions, comments, andguidance. N.C.M. thanks the faculty and staff of the Cal-tech Institute for Quantum Information for their hospi-tality during his visits, which allowed this work to cometo fruition. Both G.V.S. and N.C.M. acknowledge sup-port from the National Science Foundation, with N.C.M.also supported by the U.S. Department of Defense. ∗ Electronic address: [email protected] † Electronic address: [email protected][1] J. D. Bekenstein, Phys. Rev. D , 2333 (1973).[2] S. W. Hawking, Comm. Math. Phys. , 199 (1975).[3] G. W. Gibbons and S. W. Hawking, Phys. Rev. D ,2738 (1977).[4] R. Bousso, Rev. Mod. Phys. , 825 (2002).[5] S. Hawking, J. Maldacena, and A. 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