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Kinematic entanglement degradation of fermionic cavity modes Nicolai Friis, Antony R. Lee, David Edward Bruschi, and Jorma Louko ∗ † ‡ § School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom (Dated: January 2012) We analyse the entanglement and the nonlocality of a (1+1)-dimensional massless Dirac field confinedtoacavityonaworldtubethatconsistsofinertialanduniformlyacceleratedsegments,for small accelerations but arbitrarily long traveltimes. The correlations between theaccelerated field modes and the modes in an inertial reference cavity are periodic in the durations of the individual trajectory segments, and degradation of the correlations can be entirely avoided by fine-tuning the individual or relative durations of the segments. Analytic results for selected trajectories are presented. Differences from the corresponding bosonic correlations are identified and extensions to 2 massive fermions are discussed. 1 0 PACSnumbers: 03.67.Mn,04.62.+v,03.65.Yz 2 n a I. INTRODUCTION stationary, as the two Killing vectors do not commute. J For example,inthe (1+1)-dimensionalsetting there is a 3 Oneofthe fundamentalproblemsintheemergingfield uniquemomentatwhichthetwotrajectoriesareparallel, 2 of relativistic quantum information is the degradation of and the trajectories may or may not intersect depending correlationscausedbyacceleratedmotion. Studiesofuni- on their relative spatial location. Yet the analyses men- ] h form acceleration in Minkowski spacetime (see [1–6] for tioned above regard the correlations between observers p a small selection and [7] for a recent review) have re- on the two trajectories as stationary and the relative lo- t- vealed significant differences in the degradation that oc- cationofthetrajectoriesasirrelevant,observingjustthat n curs for bosonic and fermionic fields. There are in par- the spacetimehasaquadrantcausallydisconnectedfrom a ticular clear qualitative differences in the bosonic versus the uniformly-accelerated worldline and noting that the u q fermionicparticle-antiparticleentanglementswapping[5] field modes confined in this quadrant are inaccessible to [ andintheinfiniteaccelerationresidualentanglementand the accelerated observer. While the acceleration horizon nonlocality [6]. that is responsible for this inaccessibility may be seen as 2 The analyses of uniform acceleration mentioned above thebasisoftheUnruheffect[8,9],thehorizonexistsonly v involve two ingredients that make it difficult to com- if the uniform acceleration persists from the asymptotic 6 5 pare the theoretical predictions to experimentally real- past to the asymptotic future. In this setting it is not 7 isable situations. The first is that while the uniformly- clear how to address motion on trajectories that remain 6 accelerated observers are considered to be pointlike and uniformlyacceleratedonlyuptothemomentatwhichlo- 0. perfectly localised on a trajectory of prescribed accelera- calised observers might make their measurements on the 1 tion,thefieldexcitationsareneverthelessusuallytreated quantum state. 1 asdelocalisedfieldmodesofplanewavetype,normalised Bothoftheseconcernshavebeenrecentlyaddressedby 1 in the sense of Dirac rather than Kronecker deltas. This studyingcorrelationsbetweenfieldmodesofarealscalar : v mayseematechnicality,perhapsremediablebyuseofap- field confined in two cavities, one inertial and the other i propriatewavepackets[4],butatpresentitappearsunex- undergoing motion that consists of segments of inertial X plored how localisedobserverswould in practice perform motion and uniform acceleration [10]. In this setting the r measurementstoprobethecorrelationsinthedelocalised field modes are spatially localised in the cavities, and a states. the accelerationeffects canbelocalisedintime bytaking The second concern lies in the time evolution of the the initial and final segments of the accelerated cavity correlations. AninertialtrajectoryinMinkowskispaceis to be inertial. It was found that the entanglement is stationary, in the sense that it is the integral curve of a affected by the acceleration, and in (1+1)-dimensional Minkowski time translation Killing vector. A uniformly- spacetime the mass of the field has a strong effect on acceleratedtrajectoryisalsostationary,inthesensethat the qualitative behaviour on the entanglement. For a it is the integral curve of a boost Killing vector. How- massless Dirichlet field the entanglement is periodic in ever, the combined system of the two trajectories is not the durations of the individual trajectory segments, so that entanglement degradation can be entirely avoided by fine-tuning the durations of the individual segments; further, in the small acceleration limit the degradation ∗ [email protected] can also be avoidedby fine-tuning the relative lengths of † [email protected] ‡ [email protected] the inertial and accelerated segments. § [email protected] In this paper we shall undertake the first steps of in- 2 vestigatingfermionicentanglementinacceleratedcavities II. STATIC CAVITY by adapting the scalar field analysis of [10] to a Dirac fermion. Conceptually, one new issue with fermions is In this section we quantise the massless Dirac field in that the presence of positive and negative charges allows an inertial cavity and in a uniformly accelerated cavity, a broader range of initial Bell-type states to be consid- establishing the notation and conventions for use in the ered. AnotherconceptualissueisthatinafermionicFock later sections. spacetheentanglementbetweenthecavitiescanbechar- acterisednotjustbythe negativitybutalsobythe violation of the Clauser-Horne-Shimony-Holt(CHSH) version A. Inertial cavity of Bell’s inequality [11, 12], physically interpretable as nonlocality. New technical issues arise from the bound- Let(t,z)bestandardMinkowskicoordinatesin(1+1) ary conditions that are required to keep the fermionic dimensional Minkowski space, and let η denote the field confined in the cavities. µν Minkowski metric, ds2 =η dxµdxν = dt2+dz2. The µν − massless Dirac equation reads We focus this paper on a massless fermion in (1+1) dimensions. In this setting another new technical issue iγµ∂ ψ = 0 , (1) arisesfromazeromodethatispresentinthecavityunder µ boundary conditions that may be considered physically wherethe4 4matricesγµ formtheusualDirac-Clifford preferred. Thiszeromodeneedstoberegularisedinorder algebra, γµ×,γν = 2ηµν. A standard basis of plane to apply usual Fock space techniques. { } wave solutions reads We shall find that the entanglement behaviour of the ψω,ǫ,σ(t,z) = Aω,ǫ,σe−iω(t−ǫz)Uǫ,σ, (2) massless Dirac fermion is broadly similar to that found forthemasslessscalarin[10],inparticularintheperiodic where ω R, ǫ 1, 1 , σ 1, 1 , the constant dependence of the entanglement on the durations of the ∈ ∈ { − } ∈ { − } spinors U satisfy ǫ,σ individual accelerated and inertial segments, and in the propertythatentanglementdegradationcausedbyaccel- α U =ǫU , (3a) 3 ǫ,σ ǫ,σ erated segments can be cancelled in the leading order in γ5U =σU , (3b) the smallaccelerationexpansionbyfine-tuningthe dura- ǫ,σ ǫ,σ tionsoftheinertialsegments. Weshallhoweverfindthat Uǫ†,σUǫ′,σ′ =δǫǫ′δσσ′, (3c) the charge of the fermionic excitations has a quantita- tiveeffectontheentanglement,andthereisinparticular and A is a normalisation constant. Physically, ω ω,ǫ,σ interference between excitations of opposite charge. is the frequency with respect to the Minkowski time, the eigenvalue ǫ of the operator α = γ0γ3 indicates 3 WebegininSec.IIbyquantisingamasslessDiracfield whether the solution is a right-mover (ǫ = 1) or a left- inastaticcavityandinauniformly-acceleratedcavityin mover (ǫ= 1), and σ is the eigenvalue of the helic- (1+1)dimensionalMinkowskispacetime. Wepayspecial ity/chirality−operator γ5 = iγ0γ1γ2γ3 [13]. The right- attentiontotheboundaryconditionsthatarerequiredfor handed(σ =+1)andleft-handed (σ = 1)solutionsare − maintaining unitarity and to the regularisation of a zero decoupled because (1) does not contain a mass term. mode thatarisesunder anarguablynaturalchoiceofthe We encase the field in the inertial cavity a z b, ≤ ≤ boundary conditions. Section III develops the Bogoli- where a and b are positive parameters satisfying a < b. ubov transformation technique for grafting inertial and The inner product reads uniformlyacceleratedtrajectorysegments,presentingthe general building block formalism and giving detailed re- b sults for a trajectory where initial and final inertial seg- ψ(1),ψ(2) = dzψ(†1)ψ(2) , (4) ments are joined by one uniformly accelerated segment. Z The evolution of initially maximally entangled states is (cid:0) (cid:1) a analysed in Sec. IV, and the results for entanglement wheretheintegralisevaluatedonasurfaceofconstantt. are presented in Sec. V. A one-way-trip travel scenario, To ensure unitarity of the time evolution, so that the in which the accelerated cavity undergoes both accelera- inner product (4) is conserved in time, the Hamiltonian tion and deceleration, is analysedin Sec. VI. Section VII must be defined as a self-adjointoperatorby introducing presents a brief discussion and concluding remarks. suitable boundary conditionsat z =a andz =b [14,15]. We specialise to boundary conditions that do not couple Weuseunitsinwhich~=c=1. Complexconjugation right-handed and left-handed spinors. For concreteness, is denoted by an asterisk and Hermitian conjugation by weconsiderfromnowononlyleft-handedspinors,andwe a dagger. O(x) denotes a quantity for which O(x)/x is drop the index σ. The analysis for right-handed spinors bounded as x 0. is similar. → 3 We seek the eigenfunctions of the Hamiltonian in the as above The proper accelerations of the ends are 1/a form and 1/b respectively, and the cavity as a whole is static in the sense that it is dragged along the boost Killing ψ (t,z) = A e iω(t z)U + B e iω(t+z)U , (5) ω ω − − + ω − vector ξ := z∂ +t∂ . At t = 0 the accelerated cavity − t z overlaps precisely with the inertial cavity of Sec. IIA. whereA andB arecomplex-valuedconstants. Itwould ω ω Coordinates convenient for the accelerated cavity are be mathematically possible to maintain unitarity by al- the Rindler coordinates (η,χ), defined in the quadrant lowing probability to flow out through one of the cav- z > t by ity walls and instantaneously reappear at the other wall; | | physically, this would mean that the spatial surface is t=χsinhη , z =χcoshη , (8) consideredtobeacircle,possiblywithonemarkedpoint. However, we wish to regard the spatial surface as a gen- where 0 < χ < and < η < . The metric reads uine intervalwithtwospatiallyseparatedendpoints, and ds2 = χ2dη2 +∞dχ2. T−h∞e cavity i∞s at a χ b, and we hence specialise to boundary conditions that ensure theboo−stKillingvectoralongwhichthecav≤ityis≤dragged vanishing of the probability current independently at takes the form ξ =∂ . η eachwall. Theboundaryconditiononthe eigenfunctions In Rindler coordinatesthe masslessDiracequation(1) thus reads takes the form [16, 17] ψ¯ωγ3ψω′ z=a = 0 = ψ¯ωγ3ψω′ z=b , (6) i∂ηψ(η,χ) = −iα3(χ∂χ + 21) ψ(η,χ) , (9) where ψ¯(cid:0)=ψ†γ0. (cid:1) (cid:0) (cid:1) and the inner produc(cid:0)t for a field encase(cid:1)d in the acceler- Following the procedure of [14, 15], we find from (5) ated cavity reads and (6) that the self-adjoint extensions of the Hamilto- nian are specified by two independent phases, character- b isingthephaseshiftsofreflectionfromthetwowalls. We ψ(1),ψ(2) = dχψ(†1)ψ(2) , (10) encode these phases in the parameters θ [0,2π) and s [0,1), which can be understood respec∈tively as the (cid:0) (cid:1) Za ∈ normalised sum and difference of the two phases. The wheretheintegralisevaluatedonasurfaceofconstantη. quantum theories then fall into two qualitatively differ- Working as in Sec. IIA, we find that the orthonormal ent cases,the generic case 0<s<1 and the special case energy eigenfunctions are s=0. In the generic case 0 < s < 1, the orthonormal eigen- e iΩnη χ iΩnU +eiθ χ −iΩnU − + functions are a a − ψn(η,χ)= (cid:18)(cid:16) (cid:17) (cid:16) (cid:17) (cid:19), e iωn(t z+a)U + eiθe iωn(t+z a)U 2χln(b/a) − − + − − ψn(t,z)= √2δ − , (7a) b p (11a) (n + s)π (n+s)π Ω = , (11b) ωn = , (7b) n ln(b/a) δ where n Z and δ := b a. Note that ω = 0 for where n Z. The parameters θ and s have the same n ∈ − 6 ∈ all n, andpositive (respectively negative)frequencies are meaning and values as above: we consider the micro- obtained for n 0 (n < 0). A Fock space quantisation physical build of the cavity walls not to be affected by ≥ can be performed in a standard manner [13]. their acceleration. For s = 0 a Fock space quantisation 6 The special case s = 0 corresponds to assuming that can be performed in a standard manner. For s = 0 the the two walls are of identical physical build. In this case mode n = 0 is again a zero mode, and we consider the ω = 0 for n = 0 but ω = 0. It follows that a Fock s=0 quantum theory to be defined as the limit s 0 . n 0 + 6 6 → quantisation can proceed as usual for the n = 0 modes, 6 but n = 0 is a zero mode that does not admit a Fock spacequantisation. Inwhatfollowsweconsiderthes=0 III. GRAFTING TRAJECTORY SEGMENTS quantum theory to be defined by first quantising with s > 0 and at the end taking the limit s 0+. All our We now turn to a cavity whose trajectory consists of → entanglementmeasureswillbeseentoremainwelldefined inertial and uniformly accelerated segments. in this limit. The prototype cavity configuration is shown in Fig. 1. Two cavities, referred to as Alice and Rob, are initially inertial and in the configuration described in Sec. IIA. B. Uniformly accelerated cavity At t = 0, Rob’s cavity begins to accelerate to the right, following the Killing vector ξ = ∂ . The acceleration η Weconsideracavitywhoseendsmoveontheworldlines ends at Rindler time η = η , and the duration of the 1 z = √a2+t2 and z = √b2+t2, where the notation is accelerationinpropertime measuredatthe centreofthe 4 ct and the inner product in (15) is evaluated on the surface Alice t=0. Note that A is unitary by construction. We shall be working perturbatively in the limit where Rob the acceleration of Rob’s cavity is small. To implement Η = Η 1 this, we follow [10] and introduce the dimensionless pa- III rameterh:=2δ/(a+b),satisfying0<h<2. Physically, h is the product of the cavity’s length δ and the accel- II eration at the centre of the cavity. Expanding (15) in a Maclaurin series in h, we find z a b I A=A(0) + A(1) + A(2) + O(h3) , (16) where the superscript indicates the power of h and the explicit expressions for A(0), A(1) and A(2) read FIG.1. Space-timediagramofcavitymotionisshown. Rob’s cavity is at rest initially (Region I), then undergoes a period A(0) =δ , (17a) mn mn of uniform acceleration from t = 0 to η = η1 (Region II) and isthereafteragaininertial(RegionIII). Alice’scavityoverlaps A(1) =0, (17b) nn withRob’scavityinRegionIandremainsinertialthroughout. ( 1)m+n 1 (m+n+2s) A(1) = − − h, (m=n) mn 2π2(m n)3 6 (cid:2) (cid:3)− cavityisτ := 1(a+b)η . Werefertothe threesegments (17c) 1 2 1 of Rob’s trajectory as Regions I, II and III. Alice remains 1 π2(n+s)2 A(2) = + h2, (17d) inertial throughout. nn − 96 240 (cid:18) (cid:19) We shall discuss the evolution in Rob’s cavity in two steps: first from Region I to Region II and then from ( 1)m+n+1 A(2) = − (m+s)2+3(n+s)2 Region II to Region III. We then use the evolution to mn 8π2(m n)4 relate the operatorsandthe vacuumof RegionI to those (cid:2) − (cid:3)(cid:2) +8(m+s)(n+s) h2. (m=n) (17e) in Region III. 6 (cid:3) The expressions (17) show that the small h expansion of A. Region I to Region II Amn is not uniform in the indices, but we have verified thattheexpansionmaintainstheunitarityofAperturba- tivelytoorderh2andtheproductsoftheorderhmatrices Consider the Dirac field in Rob’s cavity. In Regions intheunitarityidentitiesareunconditionallyconvergent. I and II we may expand the field using the solutions (7) The perturbative unitarity of A persists in the limit and (11) respectively as s 0 . Had we set s=0 atthe outset anddropped the + → I: ψ = a ψ + b ψ , (12a) zero mode from the system by hand, the resulting trun- n n †n n catedAwouldnotbe perturbativelyunitary toorderh2. n 0 n<0 X≥ X II: ψ = a ψ + b ψ , (12b) m m †m m m 0 m<0 B. Region I to Region III X≥ X where the nonvanishinbg anbticommutbatorbs are InRegionIII,weexpandthe DiracfieldinRob’s cavity I: am,a†n = bm,b†n =δmn, (13a) as II: (cid:8)a ,a (cid:9)=(cid:8)b ,b (cid:9) =δ . (13b) m †n m †n mn III: ψ = a˜ ψ˜ + ˜b ψ˜ , (18) n n †n n Matching the ex(cid:8)pbansibon(cid:9)s (12n)bat tb=o0, we have the Bo- nX≥0 nX<0 goliubov transformation where the mode functions ψ˜ are as in (7) but (t,z) are n ψ = A ψ , ψ = A ψ , (14) replaced by the Minkowski coordinates (t˜,z˜) adapted to m mn n n ∗mn m the cavity’s new rest frame, with the surface t˜= 0 coin- n m X X wherebthe elements of the Bogoliubov coeffibcient matrix ciding with η = η1. The nonvanishing anticommutators are A=(A ) are given by mn III: a˜ ,a˜ = ˜b ,˜b =δ . (19) Amn = ψn,ψm (15) m †n m †n mn (cid:8) (cid:9) n o (cid:0) (cid:1) b 5 TheBogoliubovtransformationbetweentheRegionIand It follows from (12a), (18) and (20) that the creation Region III modes can then be written as andannihilationoperatorsinRegionsIandIIIarerelated by ψ˜ = ψ , ψ = ψ˜ , (20) m Xn Amn n n Xm A∗mn m n≥0: an =(ψn,ψ)= a˜mAmn + ˜b†mAmn , where the coefficient matrix A = (Amn) has the decom- mX≥0 mX<0 (28a) position =A G(η )A (21) n<0: b†n =(ψn,ψ)= a˜mAmn + ˜b†mAmn . A † 1 m 0 m<0 X≥ X (28b) andG(η )isthediagonalmatrixwhosediagonalelements 1 are Using (26) and (28a), the condition a 0 =0 reads n | i G (η )=exp(iΩ η )=:G . (22) nn 1 n 1 n a˜ + ˜b eW ˜0 =0. (29) The roleofG(η1) in (21) is to compensate for the phases mAmn †mAmn! that the modes ψm develop between η = 0 and η = η1, mX≥0 mX<0 (cid:12) (cid:11) and the matrix A = A 1 in (21) arises from matching (cid:12) † − From the anticommutators (19) it follows that Region II to Regibon III at η =η1. Note that is unitary A by construction. [W , a˜ ] = V ˜b , (30a) Expanding in a Maclaurin series in h as m − mq †q A Xq<0 = (0) + (1) + (2) + O(h3) , (23) [W , [W , a˜m]] = 0. (30b) A A A A where the superscript again indicates the power of h, we Using (30) and the Hadamard lemma, obtain from (16) and (21) eABe A = B + [A, B] + 1[A, [A, B]] + ..., − 2 (0) =G(η ), (24a) (31) 1 A (1) =G(η1)A(1)+ A(1) †G(η1), (24b) (29) reduces to A A(2) =G(η1)A(2)+(cid:0)A(2)(cid:1)†G(η1)+ A(1) †G(η1)A(1). AmnVmq = −Aqn (n ≥ 0, q < 0) . (32) (24c) m 0 (cid:0) (cid:1) (cid:0) (cid:1) X≥ Note that the diagonal elements of (1) are vanishing. Asimilarcomputationshowsthattheconditionbn 0 = | i A 0 reduces to Unitarity of implies the perturbative relations A V = (n < 0, p 0) . (33) 0=G(η1)∗ (1)+ (1) †G(η1), (25a) A∗mn pm A∗pn ≥ A A m<0 X 0=G(η1)∗ (2)+(cid:0) (2)(cid:1)†G(η1)+ (1) † (1), (25b) Iftheblockof wherebothindicesarenon-negativeis A A A A A invertible, Eq. (32) determines V uniquely. Similarly, if which will be useful(cid:0)below(cid:1). (cid:0) (cid:1) the block of where both indices are negative is invert- A ible, Eq. (33) determines V uniquely. If both blocks are invertible,itcanbe verifiedusing unitarityof thatthe C. Operators and vacua A ensuing two expressions for V are equivalent. Working perturbatively in h, the invertibility assumptions hold, We denote the Fock vacua in Regions I and III by 0 and ˜0 respectively. To relate the two, we mimic |thei and using (23) and (24) we find bosonic case [18] and make the ansatz V = V(1) + O(h2) (34) (cid:12) (cid:11) (cid:12) 0 = NeW ˜0 , (26) | i where where (cid:12) (cid:11) (cid:12) V(1) = (1)G = (1) G (p 0, q <0). (35) pq −Aqp p∗ Apq∗ q ≥ W = V a˜ ˜b (27) pq †p †q We shall show in Section IV that the normalisation con- p 0,q<0 ≥X stant N has the small h expansion and the coefficient matrix V = (V ) and the normalisa- pq tionconstantN areto bedetermined. Note thatthetwo N = 1 − 12 |Vpq|2 +O(h3). (36) indices of V take values in disjoint sets. p,q X 6 IV. EVOLUTION OF ENTANGLED STATES for k <0, so that the superscript indicates the sign of ± the charge. From (37) we obtain In this section we apply the results of Section III to the evolution of Bell-type quantum states between the two cavities whichareinitially maximally entangled. We eW ˜0 = ˜0 + Vpq ˜1p + ˜1q − shall work perturbatively to quadratic order in h. p,q (cid:12) (cid:11) (cid:12) (cid:11) X (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:12) (cid:12) (cid:12) Focusing first on Rob’s cavity only, we write out in 1 V V (1 δ )(1 δ ) − 2 pq ij − pi − qj × Sec. IVA the Region I vacuum and the Region I states p,q,i,j X with a single (anti-)particle in terms of Region III excitations on the Region III vacuum. In Sec. IVB we address ˜1p + ˜1i + ˜1q − ˜1j −+O(h3), (40) × an entangled state where one field mode is controlled by (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) Alice and one by Rob. In Sec. IVC we addressa state of (cid:12) (cid:12) (cid:12) (cid:12) the type analysedin[5]where the entanglementbetween wherethe orderingofthe single-particlekets encodesthe Alice and Rob is in the charge of the field modes. orderingofthefermioncreationoperators. Itfollowsthat the normalisation constant N is given by (36), and (26) gives A. Rob’s cavity: vacuum and single-particle states Considerthe RegionIvacuum 0 inRob’scavity. We | i sehxaciltlautsioen(s26o)vetrotehxepRreesgsiotnhisIIIsvtaatceuuinmter˜0m.s of Region III |0i= 1 − 12 |Vpq|2 ˜0 + Vpq ˜1p + ˜1q − (cid:16) Xp,q (cid:17)(cid:12) (cid:11) Xp,q (cid:12) (cid:11) (cid:12) (cid:11) We expand the exponential in (26) a(cid:12)(cid:12)s (cid:11) − 21 VpqVij(1−(cid:12)δpi)(1−δqj)× (cid:12) (cid:12) p,q,i,j X eW =1 + Vpqa˜†p˜b†q ˜1p + ˜1i + ˜1q − ˜1j −+O(h3). (41) p,q × X (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) + 12 VpqVija˜†p˜b†qa˜†i˜b†j + O(h3). (37) (cid:12) (cid:12) (cid:12) (cid:12) p,q,i,j X We denote the Region III single-particle states by Consider then in Rob’s cavity the state with exactly ˜1k + :=a˜†k ˜0 (38) one Region I particle, |1ki− := b†k|0i for k < 0 or for k 0 and by (cid:12) (cid:11) (cid:12) (cid:11) |1ki+ :=a†k|0i for k ≥ 0. Acting on the Region I ≥ (cid:12) (cid:12) vacuum (41) by (28b) and the Hermitian conjugate of (28a) respectively, we find ˜1k − :=˜b†k ˜0 (39) (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:12) k <0: |1ki− = VpqApk ˜1q −+ Amk" 1− 12 |Vpq|2 ˜1m −+ Vpq(1−δmq) ˜1p + ˜1q − ˜1m − Xp,q (cid:12) (cid:11) mX<0 (cid:16) Xp,q (cid:17)(cid:12) (cid:11) Xp,q (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) − 12 VpqVij(1−δpi)(1−δqj)(1−δmq)(1−δmj) ˜1p + ˜1i + ˜1q − ˜1j − ˜1m −#+O(h3), (42a) p,q,i,j X (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) k >0: |1ki+ =− VpqA∗qk ˜1p ++ A∗mk" 1− 12 |Vpq|2 ˜1m ++ Vpq(1−δmp) ˜1m + ˜1p + ˜1q − Xp,q (cid:12) (cid:11) mX≥0 (cid:16) Xp,q (cid:17)(cid:12) (cid:11) Xp,q (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) − 21 VpqVij(1−δpi)(1−δqj)(1−δmp)(1−δmi) ˜1m + ˜1p + ˜1i + ˜1q − ˜1j −#+O(h3). (42b) p,q,i,j X (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 7 B. Entangled two-mode states C. States with entanglement between opposite charges We wish to consider a Region I state where one field mode is controlledby Alice and one by Rob. Concretely, We finally consider the Region I state we take φ±init AR+ = √12 0kˆ A|0kiR ± 1kˆ κA|1ki+R , χ±init AR = √12 |1ki+A|1k′i−R ± |1k′i−A|1ki+R , (cid:12) (cid:11) (cid:16)(cid:12) (cid:11) (cid:12) (cid:11) ((cid:17)43a) (cid:12)(cid:12) (cid:11) (cid:16) (cid:17)(47) (cid:12) (cid:12) (cid:12) φ±init AR = √12 0kˆ A|0kiR ± 1kˆ κA|1ki−R , wherethemeaningofthesubscriptsandsuperscriptsisas (cid:12)(cid:12) (cid:11) − (cid:16)(cid:12)(cid:12) (cid:11) (cid:12)(cid:12) (cid:11) ((cid:17)43b) IdnestchriisbsetdatfeorAElicqe. a(4n3d),Rionbdiecaacthinhgavtheaatcckes≥st0oabnodthko′f<th0e. where the subscripts A andR refer to the cavity andthe modes k andk ,andthe entanglementis inthe charge of ′ superscripts indicatewhetherthemodehaspositiveor the field modes, similarly to the states considered in [5]. negative freq±uency, so that κ = + for kˆ 0 and κ = We form the reduced density matrix to order h2 as in for kˆ<0. Furthermore, we consider the t≥wo particle ba−- Sec. IVB, but now the partial tracing over Rob’s modes sis stateofthe twomode Hilbertspace,correspondingto excludes both mode k and mode k′. The relevantmatrix oneexcitationeachinthemodeskînAlice’scavityandk elements take the form in Rob’s cavity, to be ordered as in (43). As pointed out inRef.[19],makingsuchachoicecanleadtoambiguities Tr¬k,k′|1k′i−−h1k′| = fk−′ ˜0k+ ˜0k′ −−˜0k′ +˜0k inthe entanglement. Inourcase,the ambiguityamounts to a relative phase shift of π, i.e., a sign change, in (43), + 1−fk−′−fk−+ A(k1k)′ 2(cid:12)(cid:12) ˜0k(cid:11)+(cid:12)(cid:12) ˜1k(cid:11)′ −(cid:10)−˜1k(cid:12)(cid:12)′ (cid:10)+˜0k(cid:12)(cid:12) which does not affect the amount of entanglement. In other words, the states (43) are pure, bipartite, maxi- + (cid:0)fk−− A(k1k)′ 2 (cid:12)(cid:12) ˜1k+(cid:12)(cid:12) ˜1(cid:1)k(cid:12)(cid:12)′ −−(cid:11)˜1(cid:12)(cid:12)k′ +(cid:11)˜1k(cid:10) (cid:12)(cid:12) (cid:10) (cid:12)(cid:12) mmoaldlye kenitnanRgolbed’ssctaavtietsy.of mode kˆ in Alice’s cavity and + (cid:0) G(cid:12)(cid:12)kGk∗′(cid:12)(cid:12)A(cid:1)(q1k(cid:12)(cid:12))∗A(cid:11)(q1k(cid:12)(cid:12))′ ˜0k(cid:11)+(cid:10)˜0k′(cid:12)(cid:12)−(cid:10)−˜1(cid:12)(cid:12)k′ +˜1k We formthe density matrixforeachofthe states(43), (cid:16)Xq<0 (cid:12) (cid:11) (cid:12) (cid:11) (cid:10) (cid:12) (cid:10) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) express the density matrix in terms of Rob’s Region III + h.c. , (48a) basis to order h2 using (41) and (42), and take the par- (cid:17) tmiaoldterakc.eAolvleorfaRllobo’fsRmoobd’sesmeoxdceepstekxcaerpettthhuesrreefgearrednecde Tr¬k,k′|1ki++h1k| = fk+ ˜0k+ ˜0k′ −−˜0k′ +˜0k aascceenlevriarotinomn.enTth,etorewlehviacnhtipnafortrimalattrioanceissolfosRtodbu’semtoattrhixe + 1−fk+′−fk++ A(k1k)′ 2(cid:12)(cid:12) ˜1(cid:11)k+(cid:12)(cid:12) ˜0k(cid:11)′ −(cid:10)−˜0k(cid:12)(cid:12)′(cid:10)+˜1(cid:12)(cid:12)k keleme0n,tcsordreepspenodndoinngtthoe(s4i3gan),owf ethfiendmode label k. For + (cid:0)fk+′− A(k1k)′ 2 (cid:12)(cid:12) ˜1k+(cid:12)(cid:12) ˜1(cid:1)k(cid:12)(cid:12)′−−(cid:11) ˜1(cid:12)(cid:12)k′ +(cid:11)˜1k(cid:10) (cid:12)(cid:12) (cid:10) (cid:12)(cid:12) ≥ Tr¬k|0kih0k|=(1−fk−) ˜0k ˜0k +fk− ˜1k++˜1(k44,a) − (cid:16)(cid:0)Xp≥0G(cid:12)(cid:12)kGk∗′(cid:12)(cid:12)A(cid:1)(p1k(cid:12)(cid:12))∗A(cid:11)(p1k(cid:12)(cid:12))′(cid:12)˜0(cid:11)k(cid:11)+(cid:10)(cid:12)˜0k′(cid:12)(cid:12)(cid:11)−(cid:10)−(cid:10)˜1(cid:12)(cid:12)k′(cid:12)+(cid:10)˜1k(cid:12) (cid:12) (cid:11)(cid:10) (cid:12) (cid:12) (cid:11) (cid:10) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) + h.c. , (48b) Tr 0 +1 = G + (2) ˜0 +˜1 , (44b) ¬k| ki h k| k Akk k k (cid:17) Tr¬k|1ki++h1k|=((cid:16)1−fk+) ˜1k(cid:17)+(cid:12)(cid:12)+˜1(cid:11)k(cid:10)+f(cid:12)(cid:12)k+ ˜0k ˜0(k44,c) Tr¬k,k′|1ki+−h1k′|=(cid:16)Gk∗Gk∗′(cid:12)A(k1k)′(cid:12)2 where we have used (25a) a(cid:12)(cid:12)nd(cid:11)(35(cid:10)) a(cid:12)(cid:12)nd int(cid:12)(cid:12)rodu(cid:11)c(cid:10)ed(cid:12)(cid:12)the +A∗kkA∗k′k′ ˜1k+ ˜0k′−−˜1(cid:12)k′ +˜0(cid:12)k , (48c) abbreviations (cid:17)(cid:12) (cid:11) (cid:12) (cid:11) (cid:10) (cid:12) (cid:10) (cid:12) where in (48c) (cid:12) (cid:12)is kept on(cid:12)ly to(cid:12)order h2 in the fk+ := A(p1k) 2, fk− := A(q1k) 2. (45) small h expansiAon∗kkA∗k′k′ p 0 q<0 For k <0, correXs≥po(cid:12)(cid:12)ndin(cid:12)(cid:12)g to (43b), wXe fin(cid:12)(cid:12)d sim(cid:12)(cid:12) ilarly A∗kkA∗k′k′ = Gk∗Gk∗′ +Gk∗′A(k2k)∗+Gk∗A(k2′)k∗′ +O(h3). (49) Tr¬k|0kih0k|=(1−fk+) ˜0k ˜0k +fk+ ˜1k−−˜1k , (cid:12) (cid:11)(cid:10) (cid:12) (cid:12) (cid:11) (cid:10) (4(cid:12)6a) (cid:12) (cid:12) (cid:12) (cid:12) V. ENTANGLEMENT DEGRADATION AND Tr¬k|0ki−h1k|= Gk∗+A(k2k)∗ ˜0k −˜1k , (46b) NONLOCALITY Tr¬k|1ki−−h1k|=(cid:16)(1−fk−) ˜1k−(cid:1)−(cid:12)(cid:12) ˜1k(cid:11) (cid:10)+f(cid:12)(cid:12)k− ˜0k ˜0k . We are now in a position to study the entanglement (46c) (cid:12) (cid:11) (cid:10) (cid:12) (cid:12) (cid:11)(cid:10) (cid:12) and the nonlocality of our states in Region III. (cid:12) (cid:12) (cid:12) (cid:12) 8 A. Entanglement of two-mode states Consider the states φ±init AR+ and φ±init AR (43), in which Alice and Rob controlone mode each. W−e shall (cid:12) (cid:11) (cid:12) (cid:11) quantify the entanglem(cid:12)ent by the negat(cid:12)ivity [20–22] and the nonlocality by a possible violation of the CHSH inequality [11, 12]. Thenegativity [ρ]isanentanglementmonotonethat N quantifies howstronglythe partialtransposeofa density operator ρ fails to be positive. It is defined as the sum of the absolute values of the negative eigenvalues λ of (1 T )ρ, R ⊗ N[ρ] = |λ| , (50) FIG. 2. The plot shows fk/h2 as a function of u := λX<0 21η1/ln(b/a) = hτ1/(cid:2)4δatanh(h/2)(cid:3), over the full period 0 ≤ u ≤ 1. The solid curve (black) is for s = 0 with k = ±1. where (1 T ) denotes the transpose in one of the two ⊗ R The dashed, dash-dotted and dotted curves are respectively subsystems (which wehavetakento be Robwithout loss for s= 1, s= 1 and s= 3, for k =1 (blue) above the solid 4 2 4 of generality). The negativity is a useful measure for curveand for k=−1 (red) below thesolid curve. our system because all the entangled states that it fails to detect are necessarily bound entangled, that is, these states cannot be distilled [23], and a system with two with period 2δ(h/2)−1atanh(h/2), which is the proper fermionic modes cannot be bound entangled. timemeasuredatthecentreofRob’scavitybetweensend- We work perturbatively in h. The unperturbed part ing and recapturing a light ray that is allowedto bounce of (1⊗TR)ρ±AR has the triply degenerate eigenvalue 12 off each wall once. fk is non-negative, and it vanishes and the nondeg±enerate eigenvalue 1. In a perturba- only at integer multiples of the period. f is not even in −2 k tive treatment the positive eigenvalues remain positive k forgenericvaluesofs,butitisevenink inthelimiting and the only correction to the negativity comes from case s = 0 in which the spectrum is symmetric between the perturbative correction to the negative eigenvalue. positive and negative charges. f diverges at large k k Astraightforwardcomputationusing(44)and(46)shows proportionally to k2, and the domain of validity of o|ur| that the leadingcorrectionto the negativitycomes inor- perturbative analysis is k h 1. Plots for k = 1 are der h2, and to this order the negativity formula reads shown in Fig. 2. | | ≪ ± Another, potentially useful measure of entanglement N[ρ±AR±] = 21(1−fk) (51) for the states at hand would be the concurrence [25]. While a perturbative computation of the concurrence where fk :=fk++fk−. fk can be expressed as would as such be feasible, we have verified that obtain- ing the leading order correctionwould requireexpanding fk = ∞ E1k−p−1 2 A(k1p) 2 tehxepasntsaitoens.(42) to order h4. We have not pursued this p= X−∞(cid:12) (cid:12) (cid:12) (cid:12) We nowturnto nonlocality,asquantifiedby the viola- (cid:12) (cid:12) (cid:12) (cid:12) =2 Q(2k+s,1) Q(2k+s,E ) h2 (52) tion of the CHSH inequality [11, 12] 1 − where (cid:2) (cid:3) 2, (55) |hBCHSHiρ| ≤ 2 1 Q(α,z):= Re α2 Li (z) Li (z2) where CHSH is the bipartite observable π4 6 − 64 6 B (cid:20) (cid:18) (cid:19) 1 CHSH :=a σ (b+b′) σ+a′ σ (b b′) σ, +Li4(z) Li4(z2) , (53) B · ⊗ · · ⊗ − · (56) − 16 (cid:21) Li is the polylogarithm [24] and a, a′, bandb′ areunit vectorsinR3,andσ is the vector of the Pauli matrices. The inequality (55) is satisfied by iπη iπhτ alllocalrealistictheories,butquantummechanicsallows 1 1 E :=exp =exp . (54) 1 ln(b/a) 2δatanh(h/2) theleft-handsidetotakevaluesupto2√2. Theviolation (cid:18) (cid:19) (cid:18) (cid:19) of (55) is hence a sufficient (although not necessary [6, We see from (51) that acceleration does degrade the 26]) condition for the quantum state to be entangled. initially maximal entanglement, and the degradation is To look for violations of (55), we proceed as in [6], determined by the function f (52). f is periodic in τ noting that the maximum value of the left-hand-side in k k 1 9 the state ρ is given by [12] proportional to A(1) 2 in (61): this interference term is kk′ nonvanishing iff k and k have different parity, and when ′ = 2√µ +µ , (57) (cid:12) (cid:12) hBmaxiρ 1 2 itisnonvanishing(cid:12),itd(cid:12)iminishesthedegradationeffect. In the charge-symmetric special case of s=0 and k = k, where µ1 and µ2 are the two largest eigenvalues of the the degradationcoincides with that found in (51) for−th′e matrix U(ρ)=TTT and the elements of the correlation ρ ρ two-mode states (43). matrix T = (t ) are given by t = Tr[ρσ σ ]. In our ij ij i j ⊗ scenario 1 f 0 0 VI. ONE-WAY JOURNEY k U(ρ±AR±)= −00 1−0fk 1 0f  + O(h4), (58) OuranalysisfortheRobtrajectorythatcomprisesRe- 4 − k   gions I, II and III can be generalised in a straightforward and working to order h2 we hence find way to any trajectory obtained by grafting inertial and uniformly-acceleratedsegments,witharbitrarydurations hBmaxiρ±AR± = 2√2 1− 21fk . (59) and proper accelerations. The only delicate point is that the phase conventions of our mode functions distinguish (cid:0) (cid:1) The acceleration thus degrades the initially maximal vi- the left boundary of the cavity from the right boundary, olation of the CHSH inequality, and the degradation is and in Sec. IIIA we set up the Bogoliubov transforma- again determined by the function f . k tion from Minkowski to Rindler assuming that the acceleration is to the right. It follows that the Bogoliubov transformation from Minkowski to leftward-accelerating B. Entanglement between opposite charges Rindler is obtained from that in Sec. IIIA by inserting theappropriatephasefactors,A ( 1)m+nA ,and mn mn We finally turn to the entanglement between opposite in the expansions (17) this amount→s to−the replacement chaErxgperseisnsinthgetshteatdeenχsi±itnyitmAaRtr(ix47i)n. the Region III basis, h→As−ahn.example,considertheRobcavitytrajectorythat (cid:12) (cid:11) tracingoverRob’sun(cid:12)observedmodesandworkingpertur- starts inertial, accelerates to the right for proper time τ 1 batively to order h2, we find that the only nonvanishing as above, coasts inertially for proper time τ and finally 2 elements ofthe reduceddensitymatrixarewithina6 6 performs a braking manoeuvre that is the reverse of the × block. Partially transposing Rob’s subsystem replaces initial acceleration, ending in an inertial state that has the last lines in (48a) and (48b) by their respective con- vanishing velocity with respect the initial inertial state. jugates and shifts the particle-antiparticle off-diagonals Denotingthemodefunctionsinthe finalinertialstateby (48c) awayfrom the diagonal. The only nonvanishing el- ψ , and writing ements of the partial transpose are thus within an 8 8 n × block, which decomposes further into two 3 3 blocks that correspond respectively to (48a) and (48×b) and the ee ψm = Bmnψn , (62) n 2 2 block X × we find ee (1) 2 ±21 G G (1) 02+ GkGk′(cid:12)Akk′(cid:12)0 +AkkAk′k′!, Bm(1n) 2 = E1m−n−1 2 (E1E2)m−n−1 2 A(m1)n 2 (63) k∗ k∗′ Akk′ A∗kkA∗k′k′ (cid:12) (cid:12) wh(cid:12)ereE(cid:12) :=(cid:12)exp(iπτ /δ(cid:12)).(cid:12)Forthetwo-mo(cid:12)de(cid:12)initi(cid:12)alstates (60) (cid:12) (cid:12)2 (cid:12) 2 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)(cid:12) (cid:12)(cid:12) φ±init AR+ and φ±init AR (43), the negativity and the wheretheoff-diagonalcomponentsarekeptonlytoorder maximum violation of the−CHSH inequality hence read (cid:12) (cid:11) (cid:12) (cid:11) h2 in their small h expansion (49). (cid:12)respectively (cid:12) The only negative eigenvalue comes from the 2 2 block (60). We find that the negativity is given by × N[ρ±AR±] = 21 1−fk , (64a) (cid:16) (cid:17) N[ρ±χ]= 12 − 14 A(k1p) 2 − 41 A(k1′)p 2 hBmaxiρ±AR± = 2√2 1ee− 12fk , (64b) = 21 − 14(pXf6=kk+′(cid:12)(cid:12) fk′)(cid:12)(cid:12)+ 12 E1kpX−6=kk′(cid:12)(cid:12)−1 2(cid:12)(cid:12) A(k1k)′ 2. where (cid:16) ee (cid:17) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (61) fk = ∞ Bk(1p) 2 p= The entanglement is hence again degraded by the accel- ee X−∞(cid:12)(cid:12) (cid:12)(cid:12) eration, and the degradation has the same periodicity in =2 2Q(2k+s,1) 2Q(2k+s,E )+Q(2k+s,E ) 1 2 τ asinthecasesconsideredabove. Thedegradationnow − d1epends however on k and k′ not just through the indi- (cid:2)−2Q(2k+s,E1E2)+Q(2k+s,E12E2) h2 . (65) vidual functions fk and fk′ but also through the term (cid:3) 10 anexample,weidentifiedastatewheretheentanglement degradation contains a contribution due to interference between excitations of opposite charge. Compared with bosons, working in a fermionic Fock space led both to technical simplifications and to technical complications. A technical simplification was that the relevant reduced density matrices act in a lower- dimensionalHilbertspacebecauseofthefermionicstatis- tics, and this made it possible to quantify the entangle- mentnotjustintermsofthenegativitybutalsointerms of the CHSH inequality. It would further be possible to investigate the concurrence, although doing so would require pushing the perturbative low-accelerationexpan- sion to a higher order than we have done in this paper. A technical complication was that when the boundary conditions at the cavity walls were chosen in an ar- guably natural way that preserves charge conjugation symmetry, the spectrum contained a zero mode. This FIG. 3. The plot shows feek as a function of u := zero mode could not be consistently omitted by hand, hτ1/(cid:2)4δatanh(h/2)(cid:3) and v := τ2/(2δ) over the full period but we were able to regularise the zero mode by treating 0 ≤ u ≤ 1 and 0 ≤ v ≤ 1, for s = 0 and k = 1. Note the charge-symmetric boundary conditions as a limiting the zeroes at u≡0 mod 1 and at u+v≡0 mod 1. case of charge-nonsymmetric boundary conditions. All our entanglement measures remained manifestly well defined when the regulator was removed. The negativity in the state χ±init AR (47) reads Another technical complication occurring for fermions (cid:12) (cid:11) is the ambiguity [19] in the choice of the basis of the N[ρ±χ]= 12 − 14 fk+fk′(cid:12) two-fermion Hilbert space in (48). An alternative valid + 21 E1k(cid:16)−eek′ −e1e2(cid:17)(E1E2)k−k′ −1 2 A(k1k)′ 2. csihnogilceepoafrtbiacsleiskiestsobinta(i4n8e)d, wbyhicrehvaemrsionugnttshetooardcehranofgethoef (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)(66) the signs in the off-diagonalelements of (48a) and (48b). (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) While our treatment does not remove this ambiguity, all The degradation caused by acceleration is thus again ofourresultsfortheentanglementandthenonlocalityof periodicinτ1 withperiod2δ(h/2)−1atanh(h/2),anditis these states are independent of the chosen convention. periodic in τ with period 2δ. The degradation vanishes 2 Our analysis contained two significant limitations. iffE =1orE E =1,sothatanydegradationcausedby 1 1 2 First,whileourBogoliubovtransformationtechniquecan the accelerated segments can be cancelled by fine-tuning be applied to arbitrarily complicated graftings of iner- the duration of the inertial segment, to the order h2 in tialanduniformlyacceleratedcavitytrajectorysegments, which we are working. A plot off is shown in Fig. 3. the treatment is perturbative in the accelerations and k hence valid only in the small acceleration limit. We ee werethusnotabletoaddressthelargeaccelerationlimit, VII. CONCLUSIONS in which striking qualitative differences between bosonic and fermionic entanglement have been found for field We have analysed the entanglement degradation for modes that are not confined in cavities [2–6]. a massless Dirac field between two cavities in (1 + 1)- Second, a massless fermion in a (1 + 1)-dimensional dimensionalMinkowskispacetime,onecavityinertialand cavity is unlikely to be a good model for systems realis- the othermovingalonga trajectorythatconsistsofiner- able in a laboratory. A fermion in a linearly-accelerated tial and uniformly accelerated segments. Working in the rectangular cavity in (3+1) dimensions can be reduced approximationof small accelerationsbut arbitrarilylong tothe(1+1)-dimensionalcasebyseparationofvariables, travel times, we found that the degradation is qualita- but forgeneric fieldmodes the transversequantumnum- tively similar to that found in [10] for a massless scalar bers then contribute to the effective (1+1)-dimensional field with Dirichlet boundary conditions. The degrada- mass;further,anyforeseeableexperimentwouldpresum- tion is periodic in the durationsof the individual inertial ably need to use fermions that have a positive mass al- and accelerated segments, and we identified a travel sce- readyin(3+1)dimensionsbeforethereduction. Itwould nario where the degradation caused by accelerated seg- be possible to analyseour(1+1)-dimensionalsystemfor ments can be undone by fine-tuning the duration of an amassivefermion,andweanticipatethatthemasswould inertialsegment. Thepresenceofchargeallowshowevera enhance the magnitude of the entanglement degradation widerrangeofinitialstatesofinteresttobeanalysed. As as in the bosonic situation [10]. A detailed analysis of a