The relativistic quantum channel of communication through field quanta
TThe relativistic quantum channel of communication through field quanta
M. Cliche and A. Kempf Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1 (Dated: October 25, 2018)Setups in which a system Alice emits field quanta which a system Bob receives are prototypical forwireless communication and have been extensively studied. In the most basic setup, Alice and Bobare modelled as Unruh-DeWitt detectors for scalar quanta and the only noise in their communicationis due to quantum fluctuations. For this basic setup we here construct the corresponding information-theoretic quantum channel. We calculate the classical channel capacity as a function of the spacetimeseparation and we confirm that the classical as well as the quantum channel capacity are strictlyzero for spacelike separations. We show that this channel can be used to entangle Alice and Bobinstantaneously. Alice and Bob are shown to extract this entanglement from the vacuum through aCasimir-Polder effect.
PACS numbers: 03.67.-a, 03.70.+k, 03.67.Bg
I. INTRODUCTION
The setup in which two quantum systems, Alice andBob, communicate using bosonic field quanta can beviewed as a prototype for wireless communication. Nu-merous aspects of this general setup have been stud-ied in the literature, see e.g. [1]. Here, we focus onthe most basic case, where Alice and Bob are modelledas Unruh-DeWitt detectors, i.e., as point-like two-levelquantum systems that interact through a scalar quantumfield. Our aim is to construct and study the information-theoretic quantum channel, ξ , i.e., the completely posi-tive trace preserving map between the input density ma-trix ρ , in which Alice prepares her detector for the emis-sion, and the output density matrix ρ (cid:48) = ξ ( ρ ) of Bob’sdetector at a later time. This model captures the com-municating of individual q-bits and allows us to studyhow communication and entanglement are impacted byboth relativity and by the unavoidable noise that is dueto the quantum fluctuations of the field.Concretely, we construct the quantum channel andprovide a perturbative expansion for it in terms ofFeynman-like diagrams. We also calculate the classicalchannel capacity of the quantum channel as a function ofthe detectors’ spacetime separation. We then show, to allorders in perturbation theory, that both the classical andthe quantum channel capacities are strictly zero whenBob and Alice are spacelike separated. The impossibilityof superluminal signalling has of course been discussedbefore, see e.g. [2]. What is new here is that we provethe impossibility of superluminal signalling information-theoretically by constructing and studying the quantumchannel. We will then discuss how Alice and Bob canuse the quantum channel to extract entanglement fromthe vacuum. It has been known that Alice and Bob whencoupled to a quantum field can have non-trivial entangle-ment dynamics, see e.g. [3]. It is also known that, due tothe entanglement of the vacuum [4, 5], or the exchange ofvirtual photons [6], two detectors can become entangledeven at spacelike separations, and the speed with whichthis can happen has been discussed. Here, we will show that Alice and Bob can naturally and instantaneouslybecome entangled through the Casimir-Polder effect.To begin, let us denote the overall Hilbert space by H = H (1) ⊗H (2) ⊗H (3) , where the first two Hilbert spacesbelong to the detectors of Alice and Bob respectively andwhere the third Hilbert space is that of the field. Wher-ever necessary to avoid ambiguity we will denote opera-tors O or states | ψ (cid:105) which live in the Hilbert space H ( j ) by a superscript (j), for example, O ( j ) and | ψ ( j ) (cid:105) with j ∈ { , , } . Also, when such operators occur tensoredwith identity operators, such as I (1) ⊗ I (2) ⊗ O (3) , we willoften abbreviate this as, for example, O (3) . The Hamil-tonian of the system is H = H F + H D + H int H F = (cid:90) d x (cid:18) π ( x ) + 12 ( ∇ φ ( x )) + 12 m φ ( x ) (cid:19) H D = (cid:88) j =1 E e | e ( j ) (cid:105)(cid:104) e ( j ) | + E g | g ( j ) (cid:105)(cid:104) g ( j ) | H int = (cid:88) j =1 α j η (cid:16) | e ( j ) (cid:105)(cid:104) g ( j ) | + | g ( j ) (cid:105)(cid:104) e ( j ) | (cid:17) φ ( x j ) (1)where H F is the Hamiltonian of a free field, H D is theHamiltonian of the two detectors, H int is the interactionHamiltonian between the field and the detectors, α j isthe coupling constant of the j ’th detector ( j ∈ { , } ), φ ( x j ) is the field at the point of the j th detector, and m ( j ) := (cid:0) | e ( j ) (cid:105)(cid:104) g ( j ) | + | g ( j ) (cid:105)(cid:104) e ( j ) | (cid:1) is the monopole ma-trix of the j th detector. The function η ( t ) will be used todescribe the continuous switching on and off of the detec-tors within some finite time interval. The use of suitablysmooth switching functions allows one to avoid certaindivergences associated with hard on and off switches, [7].For simplicity we will always choose the same switchingfunction η ( t ) for both detectors. We note that the type ofinteraction term between the detector and the field thatwe use in Eq.1 has been extensively studied in the fieldof quantum field theory in curved space [8].The paper is organized as follows. In Sec. II we showthat causality is manifest in the channel. In Sec. III we a r X i v : . [ qu a n t - ph ] S e p study the properties of the channel and derive a Krausrepresentation, and in Sec. IV we compute explicitly theclassical channel capacity of the channel. In Sec. V wepresent a perturbative expansion of the channel. In Sec.VI we show that the channel can extract entanglementfrom the vacuum and in Sec. VII we compare our channelwith similar models which were analyzed in the quantumoptics framework. In the last section we propose exten-sions. We work with the natural units (cid:126) = c = 1. II. CAUSALITY
The so-called Fermi problem arises in any system thatis analogous to two atoms communicating via the elec-tromagnetic field, and it has been studied extensively,see e.g. [9]. Consider, in the vacuum, the probability, P F ermi , that a photon is emitted by atom 1 followed bythe absorption of a photon by atom 2. In our model, itis the probability if starting with the state | e (1) (cid:105)| g (2) (cid:105)| (cid:105) to end in the state | g (1) (cid:105)| e (2) (cid:105)| (cid:105) . Using the perturba-tive expansion of the evolution operator U ( t f , t i ) in theinteraction picture, one obtains the transition probability P F ermi = (cid:12)(cid:12)(cid:12) (cid:104) e (1) |(cid:104) g (2) |(cid:104) | U ( t f , t i ) | g (1) (cid:105)| e (2) (cid:105)| (cid:105) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) α α (cid:90) t f t i dt (cid:90) t f t i dt η ( t ) η ( t ) × e i ∆ E ( t − t ) D F ( x ( t ) , x ( t )) (cid:12)(cid:12)(cid:12) + O ( α )(2)where D F ( x − y ) := (cid:104) | T φ ( x ) φ ( y ) | (cid:105) is the Feynmanpropagator and where we defined ∆ E := E e − E g .By choosing the separation between the two detectors | (cid:126)x − (cid:126)x | and a time interval t f − t i in which both detectorsare on we can choose the spacetime windows for emissionand absorption to be time-like or spacelike (or mixed)relative to another. The Fermi problem is the fact thatthis probability amplitude, from Eq.(2), is non-vanishingeven in the case of spacelike separation. Technically, thisis due to the non-vanishing tail of the Feynman propaga-tor outside the lightcone. Hegerfeld and Feynman showedthat in fact no Feynman propagator can identically van-ish outside the lightcone, [10, 11].This reinforces the need to clarify the reason for thenon-vanishing of the Fermi probability in the spacelikeseparated case. As was pointed out in [6], the key to re-solving the puzzle is to take into account that measure-ments on the detectors are local measurements. Namely,Bob performs a measurement only of his detector 2, hedoes not measure Alice’s detector, nor does he measurethe field. This means that Fermi’s probability amplitudeis the amplitude for just one of several processes thatBob cannot distinguish. What should actually vanish forspacelike separations is the sum of the probability ampli-tudes for all processes that depend on the state of Alice. Here, our first aim is to make this argument explicitwithin the information-theoretic framework of quantumchannels. To this end, we notice that Bob’s ignorance ofAlice’ and the field’s state at the late time t f means thatat t f both the state of Alice’s detector and the state of thefield are to be traced over. These traces perform the sumover the probability amplitudes for processes that Bobcannot distinguish. We therefore naturally arrive at thedescription of a quantum channel ξ : ρ (1) → ξ ( ρ (1) ) = ρ (2) (cid:48) . Here, the input is the initial density matrix ρ (1) ofAlice at t i and the output of the channel is Bob’s densitymatrix ρ (2) (cid:48) at t f .We assume that the system starts in the state ρ ( t i ) = ρ (1) ρ (2) ρ (3) , where the initial state of Alice’ detector, ρ (1) ,is arbitrary, the initial state of the Bob’s detector, ρ (2) , isthe ground state and the initial state of the field, ρ (3) , isthe vacuum. The full density matrix evolves according to ρ ( t f ) = U ( t f , t i ) ρ ( t i ) U † ( t f , t i ), where in the interactionpicture U ( t f , t i ) = T e − i R tfti dt (cid:48) H int ( t (cid:48) ) . As always, thetime evolution can be formulated in terms of an infiniteseries of commutators [12]: ρ ( t f ) = ρ (1) ρ (2) ρ (3) + ∞ (cid:88) j =1 (cid:16) ( i ) j (cid:90) t f t i dt ... (cid:90) t j − t i dt j × [[ ... [ ρ (1) ρ (2) ρ (3) , H int ( t n )] , ... ] , H int ( t )] (cid:17) . (3)Then, the trace over detector 1 and the field, which wewill denote T r (1 , , gives the final state ρ (2) ( t f ) = ξ ( ρ (1) )of Bob’s detector: ξ ( ρ (1) ) = ρ (2) + ∞ (cid:88) j =1 (cid:16) ( i ) j (cid:90) t f t i dt ... (cid:90) t j − t i dt j × T r (1 , [[ ... [ ρ (1) ρ (2) ρ (3) , H int ( t n )] , ... ] , H int ( t )] (cid:17) . (4)To prove causality from this starting point, we will usethe following simple lemmas: I) Traces are cyclic and
T r ([ A, B ]) = 0.
II) [ A (1) B (2) C (3) , D (1) E (2) I (3) ] = (cid:110) [ A (1) , D (1) ] (cid:0) B (2) E (2) (cid:1) + (cid:0) D (1) A (1) (cid:1) [ B (2) , E (2) ] (cid:111) C (3) . III) ∃ { R (1) k , S (2) k , T (3) k } such that[[ ... [ A (1) B (2) C (3) , D (1) E (2) ] , ... ] , F (1) G (3) ] = (cid:80) k R (1) k S (2) k T (3) k .Now in Eq.(4), the terms that have a dependence on theinput ρ (1) must have at least one m (1) φ ( x ) which mul-tiplies ρ (1) since otherwise we simply have T r ( ρ (1) ) = 1.In addition, since the trace of commutators vanishes (I),the non-vanishing terms which have a dependence on ρ (1) need to be interacting with at least one m (2) φ ( x ), suchthat all the terms dependent on ρ (1) will be of the form f n (cid:16) ρ (1) (cid:17) = T r (1 , (cid:16) [[ ... [ ρ (1) ρ (2) ρ (3) , m ( j ) φ ( x j )] ,... ] , m ( r ) φ ( x r )] (cid:17) (5)where at least one of the indices { j...r } is equal to 1 and atleast one of the indices is equal to 2, and n is the numberof commutators ( n ≥ φ ( x ) is integratedover time such that the time difference between two φ ( x )is at most t f − t i . If the last index in Eq.(5) is 1, using(III) for everything before the last commutator, and (II)to expand the last commutator, f n (cid:0) ρ (1) (cid:1) would simplifyto: f n (cid:16) ρ (1) (cid:17) = (cid:88) k T r (1 , (cid:16) [ R (1) k S (2) k T (3) k , m (1) φ ( x )] (cid:17) = (cid:88) k S (2) k (cid:110) T r (3) (cid:16) T (3) k φ ( x ) (cid:17) T r (1) (cid:16) [ R (1) k , m (1) ] (cid:17) + T r (3) (cid:16) [ T (3) k , φ ( x )] (cid:17) T r (1) (cid:16) m (1) R (1) k (cid:17) (cid:111) = 0 . Thus the non-vanishing contributions of f n (cid:0) ρ (1) (cid:1) mustcome from commutators for which the very last index is2. Now, let us consider the rightmost occurrence of index1 and let us apply (III) to the commutators to the left ofit: f n (cid:16) ρ (1) (cid:17) = (cid:88) k T r (1 , (cid:16) [[ ... [[ R (1) k S (2) k T (3) k , m (1) φ ( x )] , m (2) φ ( x )] ... ] , m (2) φ ( x )] (cid:17) . (6)We can expand the most inner commutators with (II) toobtain:[ R (1) k S (2) k T (3) k , m (1) φ ( x )] =[ R (1) k , m (1) ] (cid:16) S (2) k T (3) k φ ( x ) (cid:17) + m (1) R (1) k (cid:16) S (2) k [ T (3) k , φ ( x )] (cid:17) . (7)Notice that when the first term is back in Eq.(6) it formsan expression of the form (cid:88) k T r (1 , (cid:16) [ R (1) k , m (1) ] × [[ ... [ S (2) k T (3) k φ ( x ) , m (2) φ ( x )] ... ] , m (2) φ ( x )] (cid:17) which implies that after the tracing out of detector 1 thisterm is always absent. Notice also that when the second term is back in Eq.(6), it gives an expression of the form: f n (cid:16) ρ (1) (cid:17) = (cid:88) k T r (1 , (cid:16) m (1) R (1) k × [[ ... [ S (2) k [ T (3) k , φ ( x )] , m (2) φ ( x )] ... ] , m (2) φ ( x )] (cid:17) . (8)Therefore, the term [ T (3) k , φ ( x )] will be multiplied oneach side by some powers of φ ( x ), so there exists a setof operators V (2) k,i,j such that: f n (cid:16) ρ (1) (cid:17) = (cid:88) k,i,j (cid:110) V (2) k,i,j T r (1) (cid:16) m (1) R (1) k (cid:17) × T r (3) (cid:16) φ i ( x )[ T (3) k , φ ( x )] φ j ( x ) (cid:17) (cid:111) . (9)Using cyclicity of the trace (I), this expression can besimplified to: f n (cid:16) ρ (1) (cid:17) = (cid:88) k,i,j (cid:110) V (2) k,i,j T r (1) (cid:16) m (1) R (1) k (cid:17) × T r (3) (cid:16) T (3) k [ φ ( x ) , φ i + j ( x )] (cid:17) (cid:111) . (10)Note that all the information about ρ (1) is contained inthe operators R (1) k . Causality in the channel therefore fol-lows directly from microcausality in quantum field theory[13], namely from the fact that [ φ ( x ) , φ ( y )] | ( x − y ) > = 0(where ( x − y ) = − ( x − y ) + ( (cid:126)x − (cid:126)y ) ). If the twodetectors are spacelike separated during the entire inter-action, ρ (2) ( t f ) does not depend on the state ρ (1) , i.e.,Bob’s detector 2 is not sensitive to the state in whichAlice prepared detector 1. III. NOISE STRUCTURE OF THE CHANNEL
Let us now calculate the precise quantum channel forboth time-like and spacelike separations. Since the evo-lution of the full system is unitary, our channel is neces-sarily described by a CPTP map [14]. Then, as we willshow, assuming detector 2 starts in the ground state, ρ (2) = | g (2) (cid:105)(cid:104) g (2) | , we can write the channel map in thefollowing way, in the basis | e (2) (cid:105) , | g (2) (cid:105) , ξ (cid:16)(cid:16) θ γγ ∗ β (cid:17)(cid:17) = ( ) + (cid:0) P e − P e (cid:1) + θ (cid:0) A − A (cid:1) + β (cid:0) B − B (cid:1) + γ ( CD ) + γ ∗ (cid:0) D ∗ C ∗ (cid:1) (11)where we use θ + β = 1. All terms are space-time scalars.Note that A, B, C and D are causal terms in the sensethat they depend on the input density matrix ρ (1) . Incontrast, P e represents noise in the quantum channelsince its presence does not depend on the input ρ (1) . Toprove Eq.(11), we will use the following properties whichare easy to verify ( k ∈ Z ): i) T r (cid:0) ρ (3) φ k +1 (cid:1) = 0. ii) m k +1 has no diagonal elements,and therefore T r (cid:0) m k +1 M d (cid:1) = 0 where M d is anydiagonal matrix. iii) m k has only diagonal elements,and therefore T r (cid:0) m k M nd (cid:1) = 0 where M nd is anymatrix with no diagonal elements.In a series expansion of the non-causal terms, each or-der has the form ρ (2) m (2) k T r (cid:0) ρ (3) φ ( x ) k (cid:1) . Thus, be-cause of (i) the non-vanishing terms will be propor-tional to ρ (2) m (2)2 k , and because of (iii) we know thatthese are diagonal. Therefore, because we have tracepreservation and because detector 2 starts initially inthe ground state, there cannot be a more general expres-sion for the non-causal terms of Eq.(11). For the causalterms, each order in a series expansion have the form ρ (2) m (2) k T r ( m (1) j ρ (1) ) T r (cid:0) ρ (3) φ ( x ) j φ ( x ) k (cid:1) . Now con-sider the case where the input density matrix ρ (1) is di-agonal, then because of (ii) the non-vanishing terms willhave j even. Using (i), this also means we need k to beeven, hence ρ (2) m (2) k is diagonal following (iii). A simi-lar argument can show that an input density matrix withno diagonal elements cannot have diagonal elements atthe output. Finally, trace preservation, hermiticity andlinearity of the channel are sufficient properties to provethe validity of Eq.(11).From this analysis, we can find a Kraus representa-tion by imposing ξ (cid:0)(cid:0) α γγ ∗ β (cid:1)(cid:1) = (cid:80) k =1 E k (cid:0) α γγ ∗ β (cid:1) E † k and (cid:80) k =1 E † k E k = I where we use E k = ( a k a k a k a k ). Solvingthis nonlinear system of equations is relatively straight-forward as we have more unknowns than equations, so forsimplicity we try to have as many zero matrix elementsas possible. We arrive at a simple representation, in thebasis | e (2) (cid:105)(cid:104) e (1) | , | e (2) (cid:105)(cid:104) g (1) | , | g (2) (cid:105)(cid:104) e (1) | , | g (2) (cid:105)(cid:104) g (1) | : E = (cid:18) C √ − P e − B √ − P e − B (cid:19) E = (cid:32)(cid:113) P e + A − | C | − P e − B
00 0 (cid:33) E = (cid:32) D ∗ √ − P e − A √ − P e − A (cid:33) E = (cid:32) (cid:113) P e + B − | D | − P e − A (cid:33) . (12)There exists no representation with a smaller number ofKraus operator since we verified that the rank of thematrix ( I ( Q ) ⊗ ξ (2) ) | β ( Q, (cid:105)(cid:104) β ( Q, | , where | β ( Q, (cid:105) is themaximally entangled state | β ( Q, (cid:105) = √ ( | e ( Q ) , e (1) (cid:105) + | g ( Q ) , g (1) (cid:105) ) [15], is equal to 4. IV. CHANNEL CAPACITY
The classical channel capacity C (often called the prod-uct state capacity) of a quantum channel ξ is equal to [14] C ( ξ ) = max p j ,ρ j S ξ (cid:88) j p j ρ j − (cid:88) j p j S ( ξ ( ρ j )) (13)where S is the Von Neumann entropy S ( ρ ) := − T r ( ρ ln ρ ). This quantity corresponds to the amountof reliable classical bit we can send through the quantumchannel per use of the channel.Let us first maximize over the input state to obtain: ρ (1)1 = | e (1) (cid:105)(cid:104) e (1) | and ρ (1)2 = | g (1) (cid:105)(cid:104) g (1) | . The maximiza-tion over the probability p gives p = 2 w − P e − BA − Bw − ln(1 − w ) = H ( P e + B ) − H ( P e + A ) A − B (14)where we use the binary entropy H ( p ) := − p ln p − (1 − p ) ln(1 − p ). We finally arrive at the classical channelcapacity C , which we divide by t f − t i to get R , namelythe amount of bits/time which can be sent reliably: R = 1 t f − t i (cid:34) H ( P e + p A + (1 − p ) B ) − p H ( P e + A ) − (1 − p ) H ( P e + B ) (cid:35) . (15)As expected the classical channel capacity is zero forspacelike interactions since in that case A = B = 0.We remark that the channel capacity as a function ofthe spacetime separation is a non-analytic function sinceit identically vanishes outside the lightcone but is a non-trivial function inside. Any analytic function that van-ishes on a finite interval would of course vanish every-where. The occurrence of this non-analyticity may seemsurprising since our quantum channel is mapping in be-tween finite dimensional spaces and therefore appears tobe a matter of mere linear algebra. The non-analyticityarises, of course, from the non-analyticity of the com-mutator [ φ ( x ) , φ ( y )] which originates in the fact that, inthe full system, the field lives in an infinite dimensionalHilbert space. Conversely, if ultraviolet and infrared cut-offs are imposed on the quantum field theory so that itsHilbert space H (3) becomes finite dimensional, this wouldreduce these calculations to linear algebra and will there-fore yield some non-vanishing capacity outside the light-cone. Interestingly, this does not mean that the presenceof a natural UV cutoff in nature would imply a violationof causality. This is because an ultraviolet cutoff impliesthat there is in effect a smallest resolvable length, whichin turn means that the very boundaries of the lightconebecome unsharp. The capacity should decay to essen-tially zero outside the lightcone at a distance from the FIG. 1: Feynman diagrams involved in the quantum channel. D j stands for detector j, j ∈ { , } . lightcone that is about the size of the unsharpness scaleinduced by the UV cutoff. Any candidate quantum grav-ity theory has to reduce to quantum field theory in alimit and most come with a natural UV cutoff, see e.g.,[16]. It should be interesting to check causality for suchtheories by calculating the channel capacity at distancesclose to the light cone.Let us now also consider the quantum channel capacity,[17, 18], i.e., the amount of quantum information whichcan reliably be sent through the channel Q ( ξ ) = lim n →∞ I c ( ξ ⊗ n ) nI c ( ξ ) = max ρ (cid:2) S ( ξ ( ρ )) − S (cid:0) ξ C ( ρ ) (cid:1)(cid:3) (16)where ξ C is the complementary channel. For space- like separated detections, the quantum channel capac-ity is zero since the channel is then anti-degradable:there exists a channel Γ such that Γ (cid:0) ξ C ( ρ ) (cid:1) = ξ ( ρ )[19]. This confirms that superluminal propagation ofclassical or quantum information is not possible. Fortime-like separated detectors, the quantum channel ca-pacity is extremely hard to compute because the chan-nel is not degradable (a degradable channel is such that I c ( ξ ⊗ n ) = nI c ( ξ )). Indeed, a theorem in [15] states thatany channel with input and output of dimension 2 andwith Choi rank (minimum number of Kraus operators)bigger than 2 cannot be degradable. Since the channel weconsider has Choi rank equal to 4, it cannot be degrad-able. We therefore leave open the question of finding anexplicit expression for the quantum channel capacity ofour quantum channel. V. PERTURBATIVE EXPANSION OF THECHANNEL
Using perturbation theory, we can find explicit expres-sions for the terms P e , A, B, C and D in the weak cou-pling regime ( α j (cid:28) | (cid:105) : P e (∆ E ) = α (cid:90) t f t i dt (cid:90) t f t i dt η ( t ) η ( t ) (cid:104) | φ ( x ( t )) φ ( x ( t )) | (cid:105) e − i ∆ E ( t − t ) + O (cid:0) α (cid:1) (17) A (∆ E ) = 2( α α ) (cid:90) t f t i dt (cid:90) t t i dt (cid:90) t t i dt (cid:90) t t i dt (cid:110) η ( t ) η ( t ) η ( t ) η ( t ) cos (∆ E ( t − t )) × [ φ ( x ( t )) , φ ( x ( t ))] (cid:104) e − i ∆ E ( t − t ) (cid:104) | φ ( x ( t )) φ ( x ( t )) | (cid:105) − e i ∆ E ( t − t ) (cid:104) | φ ( x ( t )) φ ( x ( t )) | (cid:105) (cid:105) + ( t ↔ t ) + ( t ↔ t ) + iη ( t ) η ( t ) η ( t ) η ( t ) sin (∆ E ( t − t )) × [ φ ( x ( t )) , φ ( x ( t ))] (cid:104) e − i ∆ E ( t − t ) (cid:104) | φ ( x ( t )) φ ( x ( t )) | (cid:105) + e i ∆ E ( t − t ) (cid:104) | φ ( x ( t )) φ ( x ( t )) | (cid:105) (cid:105)(cid:111) + O (cid:0) α (cid:1) (18) B (∆ E ) = A ( − ∆ E )+4( α α ) (cid:90) t f t i dt (cid:90) t t i dt (cid:90) t t i dt (cid:90) t t i dt (cid:110) η ( t ) η ( t ) η ( t ) η ( t ) × sin (∆ E ( t − t )) sin (∆ E ( t − t )) [ φ ( x ( t )) , φ ( x ( t ))][ φ ( x ( t )) , φ ( x ( t ))] (cid:111) + O (cid:0) α (cid:1) (19) C (∆ E ) = α α (cid:90) t f t i dt (cid:90) t t i dt η ( t ) η ( t ) e i ∆ E ( t − t ) [ φ ( x ( t )) , φ ( x ( t ))] + O (cid:0) α (cid:1) (20) D (∆ E ) = − α α (cid:90) t f t i dt (cid:90) t t i dt η ( t ) η ( t ) e i ∆ E ( t + t ) [ φ ( x ( t )) , φ ( x ( t ))] + O (cid:0) α (cid:1) . (21)We can picture the perturbative expansion with Feynmandiagrams [13], see Fig.(1) (the expressions of Eq.(17)-(21) are represented by the first diagram of their respective se- FIG. 2: Classical channel capacity as a function of time ( t f − t i ) with | (cid:126)x − (cid:126)x | = 1 and ∆ E = 1. The arrow points to thelightcone t f − t i = | (cid:126)x − (cid:126)x | . ries). A connection between the two detectors representsa photon emission/absorption process and a connectionbetween a detector and itself (a loop) represents a quan-tum field fluctuation. The terms { A, B } have an evennumber of connections between the detectors while theterms { C, D } have an odd number of connections. Theonly distinction between A and B is the input state atdetector 1: the excited state for A and the ground statefor B . Thus, the causal connections of A are resonantwhile the causal connections of B are not resonant. Asimilar argument is also true for C and D , the connec-tions of C are resonant while the connections of D arenot resonant.Using Eq.(17)-(19) along with Eq.(15), we can numeri-cally evaluate the classical channel capacity as a functionof time for inertial detectors in Minkowsky spacetime,for example, for a massless field, see Fig.(2). The ar-row points to the threshold when the spacetime windowsin which the detectors are switched on start to becomepartially time-like. VI. CREATION OF ENTANGLEMENT IN THECHANNEL
Two detectors that interact with a quantum field haveaccess to a renewable source of entanglement. It hasbeen shown in [4, 5] that detectors coupled to a masslessquantum field can become entangled even when space-like separated. The entanglement was found to appearto propagate in quantum fields at a speed which dependson the switching functions η ( t ) and on the energy gap∆ E . The speed of propagation was found to be largerthan the speed of light for suitable η ( t ) and ∆ E . A re-lated analysis was also conducted in an expanding space-time [20]. In this section we follow up on these results byshowing that the two detectors will in fact automaticallyand instantaneously become entangled, namely throughwhat is essentially the Casimir effect. We find that theCasimir effect entangles significantly which is encourag- ing for experimental verification.In this section, we switch from the interaction pictureto the Schr¨odinger picture and we assume detectors atrest in Minkowsky spacetime separated by a fixed dis-tance (cid:126)L := (cid:126)x − (cid:126)x . This allows us to use perturbationtheory for time-independent perturbations [21]. We ob-tain the new ground state | e g,new (cid:105) = | e g (cid:105) + (cid:88) k (cid:54) = g | e k (cid:105) (cid:104) e k | H int | e g (cid:105) E g − E k + (cid:88) k (cid:54) = g (cid:88) l (cid:54) = g | e k (cid:105) (cid:104) e k | H int | e l (cid:105)(cid:104) e l | H int | e g (cid:105) ( E g − E k )( E g − E l ) − | e g (cid:105) (cid:88) k (cid:54) = g |(cid:104) e k | H int | e g (cid:105)| ( E k − E g ) + O ( α ) (22)where | e k (cid:105) are the eigenstates of the free Hamiltonianand we used the fact that in our case (cid:104) e g | H int | e g (cid:105) = 0.To regularize the ultraviolet, we give a spatial extent toour detectors: H int = (cid:88) j =1 α j (cid:16) | e ( j ) (cid:105)(cid:104) g ( j ) | + | g ( j ) (cid:105)(cid:104) e ( j ) | (cid:17) (cid:90) d xf j ( (cid:126)x ) φ ( (cid:126)x ) . (23)Here, the functions f j ( (cid:126)x ) describe the smearing of thedetectors, and for simplicity we choose f ( (cid:126)x ) = f ( (cid:126)x − (cid:126)L ).Our initial ground state is | e g (cid:105) = | g (1) (cid:105)| g (2) (cid:105)| (cid:105) , and usingEq.(22) the new ground state | e g,new (cid:105) is | e g,new (cid:105) = (cid:34) | g (1) (cid:105)| g (2) (cid:105) (cid:32) − (cid:0) α + α (cid:1) S (∆ E ) (cid:33) − α | e (1) (cid:105)| g (2) (cid:105) Q (3)1 (∆ E ) − α | g (1) (cid:105)| e (2) (cid:105) Q (3)2 (∆ E )+ α α | e (1) (cid:105)| e (2) (cid:105) R (∆ E, L ) + ... (cid:35) | (cid:105) (24)where we use the definitions: Q (3) j (∆ E ) := (cid:90) d p (2 π ) (cid:82) d xf j ( (cid:126)x ) e − i(cid:126)p · (cid:126)x a † (cid:126)p (cid:112) E p ( E p + ∆ E ) (25) R (∆ E, L ) := (cid:90) d p (2 π ) e i(cid:126)p · (cid:126)L (cid:12)(cid:12)(cid:82) d xf ( (cid:126)x ) e − i(cid:126)p · (cid:126)x (cid:12)(cid:12) E p ( E p + ∆ E )(∆ E ) (26) S (∆ E ) := (cid:90) d p (2 π ) (cid:12)(cid:12)(cid:82) d xf ( (cid:126)x ) e − i(cid:126)p · (cid:126)x (cid:12)(cid:12) E p ( E p + ∆ E ) . (27)The resulting state is clearly entangled because it is apure state which cannot be written in a tensor productform. Let us now ask whether this is indeed an entangledstate from the point of view of the detectors. To see thiswe need to trace out the field, leaving the remaining sys-tem in a mixed state ρ g,new,d := T r (3) ( | e g,new (cid:105)(cid:104) e g,new | ) ρ g,new,d = α R (∆ E,L )0 α S (∆ E ) α T (∆ E,L ) 00 α T (∆ E,L ) α S (∆ E ) 0 α R (∆ E,L ) 0 0 1 − α S (∆ E ) + O ( α ) (28)where the matrix is written in the basis | e (1) e (2) (cid:105) , | e (1) g (2) (cid:105) , | g (1) e (2) (cid:105) , | g (1) g (2) (cid:105) , we assumedfor simplicity α = α = α and we use the followingdefinition: T (∆ E, L ) := (cid:90) d p (2 π ) e i(cid:126)p · (cid:126)L (cid:12)(cid:12)(cid:82) d xf ( (cid:126)x ) e − i(cid:126)p · (cid:126)x (cid:12)(cid:12) E p ( E p + ∆ E ) . (29)To measure the entanglement of the mixed state, weuse the negativity [22], which is twice the absolute valueof the sum of the negative eigenvalues of the partial trans-pose of the density matrix. We find the negativity N forthe density matrix ρ g,new,d : N (∆ E, L ) = 2 α max ( | R (∆ E, L ) | − S (∆ E ) , . (30)For simplicity, we analyse this expression when thesmearing functions are gaussian f ( (cid:126)x ) = e − | (cid:126)x − (cid:126)x | X (2 π ) / ∆ X (31)so the size of the detectors is about ∆ X . Such smearingfunctions could be physically implemented by putting thedetectors in a quantum harmonic potential. Even if gaus-sian smearing functions have a finite probability for thedetectors to overlap, we are only looking at the regimewhere L ∆ E → Lm → L ∆ X → ∞ , and in thisregime the overlap is insignificant. In fact, in this regimeall the smearing functions have the same effect, namelyto create an effective momentum cutoff. Thus our re-sults would not change for detectors which are delocal-ized within a region of space which has compact support.If ∆ E (cid:29) m like for the case of a massless field, we arriveat N ≈ α π max (cid:18) π L ∆ E − ln (cid:18) E ∆ X (cid:19) , (cid:19) . (32)Similarly if ∆ E (cid:28) m we have N ≈ α π max (cid:18) π L ∆ E − ln (cid:18) m ∆ X (cid:19) , (cid:19) . (33)We therefore see that the ground state of the interact-ing theory is entangled from the point of view of thedetectors if L < π E ln(1 / ∆ E ∆ X ) when ∆ E (cid:29) m and if L < π E ln(1 /m ∆ X ) when ∆ E (cid:28) m . To estimate how long it takes to extract entanglementfrom the vacuum, we use the adiabatic theorem. We as-sume the system starts in the ground state of the free the-ory, | e g (cid:105) = | g (1) (cid:105)| g (2) (cid:105)| (cid:105) . Then, the interaction Hamilto-nian is smoothly turned on using the switching function η ( t ). For the system to remain in the ground state, weneed η ( t ) to increase slowly enough such that the pertur-bation is adiabatic. Following the validity condition foradiabatic behaviour [23, 24], we needmax t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:104) e k | ˙ H ( t ) | e g (cid:105) E g ( t ) − E k ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:28) min t | E g ( t ) − E k ( t ) | (34)to hold for any energy level E k . A rigorous use of theadiabatic theorem requires normalized eigenstates, so letus put our system in a large box of volume V = L IR .This procedure creates an infrared cut-off and normalizesthe eigenstates of the free Hamiltonian. Hence, in ourcase, if we retain only the dominant order, the adiabaticcondition translates to:max t | ˙ η ( t ) | (cid:28) (cid:34) m + 3 (cid:18) πL IR (cid:19) (cid:35) / /α × (cid:115) m + 3 (cid:18) πL IR (cid:19) + ∆ E . (35)Thus, for a massive field, it is always possible to adia-batically turn on the interaction, and since the groundstate of the interacting theory is entangled, there will bean instantaneous creation of entanglement. If the field ismassless, there still is instantaneous creation of entangle-ment, for any finite size of box to which we confine oursystem. Therefore, while Alice and Bob cannot exchangeclassical or quantum information faster than the speed oflight, their ability to extract entanglement by interactingwith the vacuum is not bounded by any finite speed.From Eq.35 we notice that in order to obtain thefull amount of entanglement from the ground state,the system needs an interval of time of the order of(max t | ˙ η ( t ) | ) − . This entanglement could either be usedin computations or swapped to other quantum systemsfor distillation. After the entanglement is used up, thedetector - field interaction may be switched off and thesystem can be put back in the ground state of the freetheory | e g (cid:105) = | g (1) (cid:105)| g (2) (cid:105)| (cid:105) , e.g., by cooling. Thus, Aliceand Bob can extract entanglement by interacting withthe field in a cyclic and therefore sustainable way. How-ever, we also see that the extraction of a large amount ofentanglement from the vacuum by this method will costa large amount of time. Interestingly, the amount of timeneeded is determined in a similar way to how the speedof adiabatic quantum computation is determined. Recallthat the closeness of eigenvalues determines how fast spe-cific states such as the ground state can be reached viaan adiabatic approach or through cooling, see e.g. [25].The reason why the finite rate of entanglement extractiondoes not lead to a finite speed of entanglement “propaga-tion”, is that there is no threshold: negativity, indicatingentanglement between the detectors, arises immediatelyas their interaction with the field is switched on.We will now show that the underlying reason why Al-ice and Bob are entangled when in the ground state ofthe interacting theory is that this ground state is a statein which Alice and Bob are attracted to another throughthe exchange of virtual photons. This exchange inter-action is in effect the scalar field version of the Casimir Polder force [26], which is known to be the relativisticgeneralization of the van der Waals force between atomsor molecules.Let us now derive the Casimir force between Alice andBob for point-like detectors f ( (cid:126)x ) = δ ( (cid:126)x − (cid:126)x ). To thisend, we calculate the energy of the new ground state withtime independent perturbation theory [21], and renor-malize using δ ˜ E g ( L ) := δE g ( L ) − lim L →∞ δE g ( L ). Theresult of the calculation is: δ ˜ E g ( L, ∆ E ) = (cid:88) n (cid:54) = g (cid:88) k (cid:54) = g (cid:88) l (cid:54) = g (cid:104) e g | H int | e n (cid:105)(cid:104) e n | H int | e k (cid:105)(cid:104) e k | H int | e l (cid:105)(cid:104) e l | H int | e g (cid:105) ( E g − E n )( E g − E k )( E g − E l ) + O ( α )= − α (cid:34) E (cid:12)(cid:12)(cid:12) (cid:90) d p (2 π ) e − i(cid:126)p · ( (cid:126)x − (cid:126)x ) E p ( E p + ∆ E ) (cid:12)(cid:12)(cid:12) + (cid:90) d p (2 π ) (cid:90) d p (2 π ) e − i ( (cid:126)p − (cid:126)p ) · ( (cid:126)x − (cid:126)x ) E p E p ( E p + E p ) (cid:18) E p + ∆ E + 1 E p + ∆ E (cid:19) +2 (cid:90) d p (2 π ) (cid:90) d p (2 π ) e − i ( (cid:126)p − (cid:126)p ) · ( (cid:126)x − (cid:126)x ) E p E p ( E p + E p + 2∆ E )( E p + ∆ E )( E p + ∆ E )+ (cid:90) d p (2 π ) (cid:90) d p (2 π ) e − i ( (cid:126)p − (cid:126)p ) · ( (cid:126)x − (cid:126)x ) E p E p ( E p + E p + 2∆ E ) (cid:18) E p + ∆ E ) + 1( E p + ∆ E ) (cid:19) (cid:35) + O ( α ) . (36)The ground state energy is lowered because of the inter-action, causing Alice and Bob to attract each other withthe Casimir force F C = − ∂δ ˜ E g ( L ) ∂L . For a massless field, δ ˜ E ( L, ∆ E ) ∼ − α L ∆ E in the limit L ∆ E → ∞ . For com-parison, the electromagnetic Casimir-Polder energy [26],scales as ∼ − L − for large distances.Note that so far we did not need to specify the detec-tors’ mass since we assumed their position to be fixed.Considering now the dynamics of Alice and Bob due tothe Casimir force, it is clear that if their mass is smallenough, their acceleration could be strong enough to be-come non-adiabatic. In this case, their motion wouldcause the system to evolve non-adiabatically and there-fore to become excited. The Casimir force would there-fore no longer be simply the derivative of the ground stateenergy, − ∂δ ˜ E g ( L ) ∂L because the state of the system wouldno longer be the ground state. To stay in the regimewhere the Casimir force is the derivative of the Casimirenergy the detectors can move toward each other at amaximum speed v which needs to be small enough suchthat the perturbation is adiabatic. Thus, when our sys-tem is in a large box, the validity condition for adiabaticbehaviour of Eq.(34) translates to v (cid:28) ∆ E / α √ √ . VII. RELATED MODELS
A quantum channel modelled by an atom interact-ing with a photon has recently been analysed in [27].The model uses an atom-photon interaction given bythe Jaynes-Cumming interaction Hamiltonian H JC = α (cid:16) | g (cid:105)(cid:104) e | ⊗ a † k + | e (cid:105)(cid:104) g | ⊗ a k (cid:17) where a k and a † k are the an-nihilation and creation operator for a single mode k . Asimilar Hamiltonian was also used in [28] to model aquantized cavity mode kicked by a stream of two-levelatoms. This interaction Hamiltonian has a natural quan-tum field generalization, the Glauber scalar detector [29],which can be used to model two detectors interactingwith a quantum scalar field H GS = (cid:88) j =1 α j η (cid:16) | g ( j ) (cid:105)(cid:104) e ( j ) | φ − ( x j ) + | e ( j ) (cid:105)(cid:104) g ( j ) | φ + ( x j ) (cid:17) (37)where φ + ( x ) = (cid:82) d p (2 π ) √ E p e − ipx a (cid:126)p and φ − ( x ) = φ + † ( x )are respectively the positive and negative frequency partof the field. While this detector is not sensitive to thequantum fluctuations of the field, i.e., in our notation, P e = 0, this detector model allows non-local effects, see[30]. We can confirm the non-locality by using in ourchannel Glauber detectors instead of Unruh-DeWitt de-tectors. To this end, we use Eq.37 in Eq.(4). We seethat then terms that are dependent on ρ (1) are no longernecessarily proportional to [ φ ( x ) , φ ( x )]. Using the per-turbative expansion of the channel in Sec.V shows thatnon-causal terms appear already in the O ( α ) order: ξ ( ρ (1) ) = | g (2) (cid:105)(cid:104) g (2) |− α α (cid:90) t f t i dt (cid:90) t t i dt (cid:104) η ( t ) η ( t ) × e i ∆ E ( t − t ) | e (2) (cid:105)(cid:104) g (2) |(cid:104) e (1) | ρ (1) | g (1) (cid:105)× D ( x ( t ) − x ( t )) + c.c. (cid:105) + O (cid:0) α (cid:1) . (38)Here, D ( x − y ) := (cid:104) | φ + ( x ) φ − ( y ) | (cid:105) . Since the correla-tor D ( x − y ) is not vanishing outside the lightcone, de-tector 2 would indeed be influenced by detector 1 as soonas the interaction is turned on even if the detectors arespacelike separated. It may be interesting to see if simi-lar effectively non-local detectors, such as the one in [31],behaves causally or non-causally under our channel pic-ture. VIII. OUTLOOK
The type of quantum channel that we here consideredcould be useful, for example, in the context of imple-mentations of quantum networks, where photons carryquantum information in between atoms that possess ef-fectively two levels, see e.g., [32]. But it should also bestraightforward to generalize our study to detectors withany number of energy levels. The number and spacingof the energy levels of the detectors should translate intoan effective alphabet size. This should also allow one togeneralize the results of [33], where it was first shown howquantum noise imposes a natural bound to the capacityof an otherwise noiseless bosonic channel. The analysisof [33] employed the time-energy uncertainty principleto describe the limit to the distinguishability of photonsof energy difference ∆ E in an observation time ∆ t . Itshould be interesting to re-analyze these results withinthe present information-theoretic framework of the quan-tum channel in which all effects of quantum noise are built in from the start.It should also be interesting to generalize our model toyield a new approach to analyzing the setup of [34], whereAlice and Bob are inertial observers which are exchang-ing modes of a quantum field, while Eve is acceleratingand tries to intercept the message. It was shown therethat, because of the Unruh effect, it is always possiblefor Alice and Bob to communicate privately. To showthis, the approach to the Unruh effect using Bogoliubovtransformations was used. Generalizing our setup, onemay use Unruh-DeWitt detectors, which are known toallow a more flexible description of the Unruh effect. Forexample, Eve would not have to accelerate uniformly andcould indeed take an arbitrary trajectory.The channel which we studied here should also begeneralizable to curved spacetimes to study, for example,the impact of spacetime expansion and horizons. Finally,let us recall that, in the presence of a suitable naturalultraviolet cutoff, the density of degrees of freedomin quantum fields is finite, see e.g. [35]. It shouldbe interesting to investigate how this finite density ofdegrees of freedom translates into a finite informationcarrying capacity of quantum fields, in the concretesense of the capacity of quantum channels. Indeed,the quantum channel that we investigated here can beinterpreted as describing one detector which imprintsinformation in a quantum field, and a second detectorreading out this information. The approach thereforeallows one to ask questions such as, how write and readcycles can be optimized, how much information is left inthe field after a cycle, or how much quantum or classicalinformation can maximally be written into and retrievedfrom a quantum field in some finite region of spacetime. Acknowledgments
The authors whish to thank Ralf Sch¨utzhold for valu-able comments and suggestions. M.C. acknowledges sup-port from the NSERC PGS program. A.K. acknowledgessupport from CFI, OIT, the Discovery and Canada Re-search Chair programs of NSERC and is grateful for thevery kind hospitality at the University of Queenslandduring the early stages of this work. [1] C.M. Caves and P. D. Drummond, Rev. Mod. Phys. ,481-537 (1994).[2] Y. Aharonov, B. Reznik, and A. Stern, Phys. Rev. Lett. , 2190-2193 (1998).[3] S.-Y. Lin and B. L. Hu, Phys. Rev. D , 085020 (2009).[4] B. Reznik, Found. Phys. , 167 (2003).[5] B. Reznik, A. Retzker and J. Silman, Phys. Rev. A ,042104 (2005).[6] J. D. Franson, Journal of Modern Optics , 2117 - 2140 (2008).[7] A. Satz, Class. Quantum Grav. , 1719-1731 (2008).[8] N. D. Birrell and P. C. W. Davies, Quantum fields incurved space , Cambridge University Press (1982).[9] E. A. Power and T. Thirunamachandran, Phys. Rev. A , 3395-3408 (1997).[10] G. C. Hegerfeldt, Phys. Rev. D. , 3320-3321 (1974).[11] R.P. Feynman, Dirac Memorial Lecture, The reason forantiparticles , Cambridge University Press (1987). [12] J. D. Franson and M. M. Donegan, Phys. Rev. A ,052107 (2002).[13] M. E. Peskin and D. V. Schroeder, An Introduction toQuantum Field Theory , Westview Press (1995).[14] M. A. Nielson and I. L. Chuang,
Quantum computationand Quantum Information , Cambridge University Press(2000).[15] T. S. Cubitt, M. B. Ruskai and G. Smith, Journal ofMathematical Physics , 102104 (2008).[16] C. Rovelli, Quantum Gravity , Cambridge UniversityPress (2004).[17] S. Lloyd, Phys. Rev. A , 1613 (1997).[18] H. Barnum, M. A. Nielsen and B. Schumacher, Phys.Rev. A , 4153 (1998).[19] I. Devetak and P. W. Shor, Commun. Math. Phys. ,4153 (1998).[20] G. Ver Steeg and N. C. Menicucci, Phys. Rev. D ,044027 (2009).[21] C. Cohen-Tannoudji, B. Diu and F. Laloe, Mecaniquequantique , Hermann (1973).[22] G. Vidal and R. F. Werner, Phys. Rev. A , 032314(2002).[23] M.S. Sarandy, L.-A. Wu and D.A. Lidar, Quantum. In-form. Proc. , 331 (2004). [24] L. I. Schiff, Quantum Mechanics , McGraw-Hill (1955).[25] E. Farhi, J. Goldstone, S. Gutmann, and M.Sipser,
Quantum computation by adiabatic evolution ,arXiv:quant-ph/0001106 (2000).[26] H. B. G. Casimir and D. Polder, Phys. Rev. , 360-372(1948).[27] X. Chen, The capacity of transmitting atomic qubit withlight , arXiv:0802.2327 (2008).[28] G.J. Milburn, Phys. Rev. A , 744-749 (1987).[29] R.J. Glauber, Phys. Rev. , 2529 (1963); Phys. Rev. , 2766 (1963).[30] F. Buscemi and G. Compagno, Non-local quantumfield correlations and detection processes in QFT ,arXiv:0904.3238v1 (2009).[31] F. Costa and F. Piazza,
Modelling a Particle Detector inField Theory , arXiv:0805.0806 (2008).[32] J. I. Cirac, P. Zoller, H. J. Kimble and H. Mabuchi, Phys.Rev. Lett. , 3221 (1997).[33] J. I. Bowen, IEEE Trans. Inform. Theory , 230 (1967).[34] K. Bradler, P. Hayden and P. Panangaden, J. High En-ergy Phys. , 074 (2009).[35] A. Kempf, Phys. Rev. D69