Quantum Energy Teleportation without Limit of Distance
aa r X i v : . [ qu a n t - ph ] J a n Quantum Energy Teleportation without Limit of Distance
Masahiro Hotta, ∗ Jiro Matsumoto, † and Go Yusa ‡ Department of Physics, Tohoku University, Sendai 980-8578, Japan (Dated: January 7, 2014)Quantum energy teleportation (QET) is, from the operational viewpoint of distant protocol users,energy transportation via local operations and classical communication. QET has various links tofundamental research fields including black hole physics, the quantum theory of Maxwell’s demon,and quantum entanglement in condensed matter physics. However, the energy that has been ex-tracted using a previous QET protocol is limited by the distance between two protocol users; theupper bound of the energy being inversely proportional to the distance. In this letter, we provethat introducing squeezed vacuum states with local vacuum regions between the two protocol usersovercomes this limitation, allowing energy teleportation over practical distances.
PACS numbers: 03.67.Ac
INTRODUCTION
Quantum fields in the vacuum state accompany spa-tially entangled energy density fluctuations via the non-commutativity of energy density operators. However, theeigenvalue of the total Hamiltonian can be set to zero bydiscarding the zero-point energy, primarily because thisenergy exhibits the fundamental property known as pas-sivity [1] and is of little use; any intended local operationfor extracting the zero-point energy out of a field actu-ally injects energy and excites the vacuum. The zero-point energy, however, can be glimpsed through a spatialregion with negative energy density [2], along with theCasimir effect [3] and the Unruh effect [4]. In such aspatial region, the quantum field can afford to attain alower energy density than the zero value of the vacuumstate because it saves the zero-point energy in the vacuumstate. Of course, we have another region with sufficientpositive energy to ensure that the total energy is greaterthan zero.Recently, quantum information theory has revealedsome exotic aspects of the entangled energy density fluc-tuation of many-body systems in the ground state, andit has been proven that quantum energy teleportation(QET) is possible [5, 6]. QET is, from the opera-tional viewpoint of distant protocol users, energy trans-portation via local operations and classical communica-tion (LOCC). In contrast to the standard protocols ofquantum teleportation [7], QET protocols involve energytransportation. QET is related to the quantum theoryof Maxwell’s demon at low temperatures [8] and the lo-cal cooling problem of many-body quantum systems [9].Furthermore, QET provides insight into the black holeentropy problem [6]. QET can be implemented in var-ious research fields including spin chains [9], harmonicchains [10], and cold trapped ions [11]. Although QEThas not yet been experimentally verified, a realistic ex-periment was recently proposed [12]; the experiment uses1+1-dimensional chiral massless boson fields of quantumHall edge currents [13], and the teleported energy may E ne r g y den s i t y Positive energy density density E A E B Negative energy(b) Long-distance squeezed-state QET0 Classical communicationAlice Bob E ne r g y den s i t y SpacePositive energy density densityby local measurementby local operationEnergy E A injectedEnergy E B extractedNegative energy(a) Vacuum state QET Space FIG. 1. (Color online) Schematic diagram of (a) vacuum statequantum energy teleportation (QET) protocol and (b) long-distance squeezed-state QET. be expected to take values near O (100) µ eV with presenttechnology.Despite such experimental proposals for various physi-cal systems [9–12], a strong distance limitation has ham-pered experimental verification; in QET over a distance L , the transferred energy E B of 1+1 dimensional mass-less scalar fields is bounded by E B ≤ πL , (1)as long as vacuum-state QET protocols are adopted.Here, the natural unit c = ~ = 1 is adopted. QETthus sends only a small amount of energy over a longdistance L . This vacuum-state QET distance bound ap-pears for two reasons: From an informational viewpoint,the spatial correlations of the zero-point fluctuation, in-cluding quantum entanglement, decay as the distance be-comes large, and hence the amount of information fordistant control of a quantum fluctuation becomes smalland only weak strategies for extracting energy out of thedistant zero-point fluctuation are available. From a phys-ical viewpoint, the localized negative energy induced bya QET protocol cannot be separated from the positiveenergy injected by the measurement device; otherwise,the negative energy excitation would exist without anypositive energy excitations. To make the total energyof the field nonnegative, the negative energy excitationrequires a sufficient amount of positive energy at a closedistance. However, there is an interesting possibility thatavoids this bound by using an exotic quantum state forQET, as we argue in this paper.We propose here a new version of the QET protocolsthat use a squeezed vacuum state in place of the vac-uum state between two protocol users (Fig. 1). In ourproposal, the spatial correlation of the quantum fluctua-tions with zero energy is maintained even if the distancebetween the sender and receiver of QET is very large.The negative energy induced by the extraction of positiveenergy via QET is sustained not by the positive energyinjected by the measurement but instead by the excita-tion energy in the squeezed region of the state. Thus,the bound in Eq. (1) is overcome and long-distance en-ergy teleportation can be achieved. The protocol may beimplemented by adopting a spatial expansion method,which is one strategy for realizing a long-distance corre-lation, thus facilitating experimental verification of QETand potentially contributing to quantum device applica-tions.The paper is organized as follows. In Section 2, a briefreview of vacuum-state QET is provided, and in Section3, the distance bound in Eq. (1) is derived. In Section 4,using a squeezed state, we outline the new protocol thatrealizes QET without the limit of distance. The resultsare summarized in Section 5. BRIEF REVIEW OF VACUUM-STATE QET
Recalling the experimental proposal using quantumHall edge currents [12], let us consider a free masslessscalar quantum field ˆ ϕ in 1+1 dimensions that obeys theequation of motion (cid:0) ∂ t − ∂ x (cid:1) ˆ ϕ = 0 . The general solution is obtained as the sum of a left-moving component ˆ ϕ + ( x + ) and a right-moving compo-nent ˆ ϕ − ( x − ) using the light-cone coordinates x ± = t ± x . For the purposes of our discussion, we can focus solelyon the left mover ˆ ϕ + ( x + ). In quantum Hall systems,ˆ ϕ + ( x + ) describes the charge density fluctuation of theedge [13]. The energy flux operator is given byˆ T ++ ( x + ) =: ˆΠ + ( x + ) , where ˆΠ + ( x + ) = ∂ x + ˆ ϕ + ( x + ), and the total energy oper-ator is calculated asˆ H + = Z ∞−∞ ˆ T ++ ( x + ) dx + , which is clearly a nonnegative operator. The vacuumstate | i is the eigenstate with a zero eigenvalue of ˆ H + .Let us briefly review vacuum-state QET using thisˆ ϕ [6]. Consider two separate experimenters (say, Aliceand Bob) who are able to execute LOCC on this fieldin the vacuum state. Alice stays in the spatial region[ x A , x A ] and Bob in [ x B , x B ], with x A < x B . Bob’sregion is located to the right of Alice’s region, and the dis-tance between them is L = x B − x A . Assume that theinitial state is the vacuum state | ih | . The field possesseszero-point fluctuation, and its nontrivial correlation isinduced by vacuum-state entanglement. Hence, if Al-ice obtains information about a local fluctuation aroundher through a measurement, she simultaneously obtainssome information about a local fluctuation around Bobvia the correlation. Although the average value of theenergy density in Bob’s region remains zero after Alice’smeasurement, Bob’s local field in the post-measurementstate carries positive or negative energy, depending on Al-ice’s measurement result. When the result indicates thepositive-energy case, Bob can extract energy from thefield after receiving the information from Alice. At time t = 0, Alice instantaneously conducts a general measure-ment [14] in [ x A , x A ]. Although several measurementsare useful for realistic QET experiments [15], let us con-sider a simple measurement model with a one-bit output,to grasp the essence of QET. Alice prepares a qubit Q in |−i , which is the down eigenstate of the third Paulimatrix ˆ σ Q , as a probe of ˆΠ + ( x + ) in | i . The interactionbetween the two states such that H int ( t ) = δ ( t ) (cid:16) ˆ A − π (cid:17) ⊗ ˆ σ Q , where ˆ σ Q is the second Pauli matrix of Q , and ˆ A is alocal Hermitian operator defined asˆ A = Z ∞−∞ g A ( x ) ˆΠ + ( x ) dx, with the real function g A ( x ) localized in [ x A , x A ]. Afterswitching off the interaction, a projection measurementof ˆ σ Q is performed for Q , and Alice obtains a binarymeasurement result of µ = 0 , − µ +1 of ˆ σ Q . The (unnormalized) post-measurement state of the field is given by ˆ M µA | i , whereˆ M µA are the measurement operators and are explicitlycomputed [6] as ˆ M A = cos (cid:16) ˆ A − π (cid:17) , (2)ˆ M A = sin (cid:16) ˆ A − π (cid:17) . (3)A straightforward computation shows that the emergenceprobability of each result µ is the same: p µ = h | ˆΞ µA | i =1 /
2. The post-measurement state for each µ is calculatedas ˆ ρ µ = 1 p µ ˆ M µA | ih | ˆ M † µA . As a result of the passivity, this measurement injects anaverage energy of E A = X µ p µ Tr h ˆ H + ˆ ρ µ i − Tr h ˆ H + | ih | i into the field and generates a positive energy wave packet.The injected energy is computed as E A = Z ∞ | e g A ( ω ) | ω dω π . Here, e g A ( ω ) is the Fourier transform of g A ( x ). Becauseˆ M µA includes only the left-mover operator ˆΠ + ( x ), thewave packet moves to the left, i.e., further away fromBob. Thus, Bob cannot receive the energy through directpropagation of the wave packet. Note that the field inBob’s region is in a local vacuum state with zero energy,although a wave packet with E A exists far from Bob.Assuming that Bob receives the information of µ fromAlice at time t = T , Bob then performs an instantaneousunitary operation dependent on µ given byˆ U µB = exp (cid:16) iθ µ ˆ B (cid:17) , (4)where θ µ is a µ -dependent real parameter fixed so as tomaximize the amount of teleported energy [6], andˆ B = Z ∞−∞ g B ( x ) ˆΠ + ( x ) dx with the real function g B ( x ) localized in [ x B , x B ]. Thepost-operation state is computed asˆ ρ QET = X µ ˆ U µB exp (cid:16) − iT ˆ H + (cid:17) ˆ M µA | ih | ˆ M † µA × exp (cid:16) iT ˆ H + (cid:17) ˆ U † µB . It can then be verified that the total energy decreasesduring this local operation by Bob. This implies thata positive amount of teleported energy E B is extracted from the field in the local vacuum state as negative workby Bob’s operation: E B = Tr " ˆ H + X µ p µ exp (cid:16) − iT ˆ H + (cid:17) ˆ ρ µ exp (cid:16) iT ˆ H + (cid:17) − Tr h ˆ H + ˆ ρ QET i > . Simultaneously, a wave packet with negative energy − E B is generated in Bob’s region and begins to move to theleft. It is useful at this point to define the two-pointcorrelation function C µAB as C µAB = h | ˆΞ µA ˆ B ′ ( T ) | i /p µ , (5)with ˆΞ µA = ˆ M † µA ˆ M µA and ˆ B ′ ( T ) = Z ∞−∞ g B ( x − T ) ∂ x ˆΠ + ( x ) dx. Note that C µAB is real because of the operator locality: h ˆΞ µA , ˆ B ′ ( T ) i = 0. To maximize the extracted energy, wefix the real parameter θ µ as θ µ = 2 C µAB G B , where G B = R ∞−∞ ( ∂ x g B ( x )) dx . This implies that pos-itive energy is extracted from the field in the local vac-uum state as negative work by Bob’s operation. Theteleported energy E B can be evaluated as E B = 1 G B X µ p µ C µAB . (6)From the correlation function, h | ˆΠ + ( x B ) ˆΠ + ( x A ) | i = − π ( x B − x A − iǫ ) , (7)we can evaluate C µAB in a straightforward manner, and E B can be explicitly computed [6] as E B = (cid:16)R ∞−∞ R ∞−∞ g B ( x B ) g A ( x A )( x B − x A + T ) dx B dx A (cid:17) π R ∞−∞ ( ∂ x g B ( x )) dx exp (cid:16) π R ∞ | e g A ( ω ) | ωdω (cid:17) . (8)The teleported energy E B is not larger than E A becauseof the nonnegative property of the total Hamiltonian. DISTANCE BOUND FOR VACUUM-STATE QET
For vacuum-state QET, when the distance L (= x B − x A ) between Alice and Bob increases, E B in Eq. (8)decreases as E B ∝ /L . This damping behavior can beslightly improved to E B ∝ /L by replacing ˆ B in Eq. (4)with ˆ B = R ∞−∞ ˜ g B ( x ) ˆ ϕ ( x ) dx . A natural question thenarises: To what extent does another QET protocol im-prove this long-distance behavior of E B ? As mentionedabove, a stringent bound on the long-distance damping of E B is given by Eq. (1) for any vacuum-state QET. Thisbound is based on Flanagan’s theorem [16], which assertsthe following. Consider a nonnegative continuous func-tion ξ ( x ) with ξ ( x → ±∞ ) = 0 and define a Hermitianoperator as ˆ H ξ = Z ∞−∞ ξ ( x ) ˆ T ++ ( x ) dx. The inequalityTr h ˆ H ξ ˆ ρ i ≥ − π Z ∞−∞ (cid:16) ∂ x p ξ ( x ) (cid:17) dx holds for an arbitrary state ˆ ρ , and applying this theoremto vacuum-state QET yields Eq. (1).Let ˆ ρ QET denote the post-operation state following anarbitrary vacuum-state QET. Assume that a wave packetwith negative energy − E B is generated in [ x B , x B ].In the intermediate region between Alice and Bob,[ x A , x B ], the average energy density vanishes. In theregion to the left of Alice, ( −∞ , x A ], a wave packet ex-ists with positive energy E A . Let us impose the val-ues ξ ( x ) = 0 for x ∈ ( −∞ , x A ] and ξ ( x ) = 1 for x ∈ [ x B , x B ] on ξ ( x ). In the region [ x B , ∞ ), it issufficient to assume that ξ ( x ) slowly decreases to 0. Re-sultantly, Tr h ˆ H ξ ˆ ρ QET i = − E B for an arbitrary ξ ( x ) satisfying the above conditions.Thus, E B ≤ π inf ξ ( x ) Z ∞−∞ (cid:16) ∂ x p ξ ( x ) (cid:17) dx must be satisfied. The infimum of the ξ ( x ) satisfyingthe above boundary conditions is then obtained, using avariation method, from the function ξ opt ( x ) obeying ξ opt ( x ) = ( x/L ) for x ∈ [ x A , x B ]. This results in the inequality ofEq. (1). Note that a spatial region with negative energycan appear only when another region with sufficient pos-itive energy exists. If an excitation with a fixed negativeenergy could be separated from a positive-energy exci-tation by an infinitely large distance, then the positive-energy excitation at the spatial infinity will not influ-ence the negative-energy excitation due to the localityof quantum field theory. This leads to an apparent con-tradiction in the nonnegativity of the total energy in a broad region surrounding the negative-energy excitation.Thus, in vacuum-state QET, the negative-energy excita-tion generated by Bob should be located in the neigh-borhood of the positive-energy excitation generated byAlice. This fact yields the distance bound of Eq. (1). LONG-DISTANCE SQUEEZED-STATE QET
A long-distance QET is expected to open new doorsin the development of quantum devices. Thus, it is im-portant to raise the question, can the distance bound inEq. (1) be overcome by some means? Interestingly, aloophole can be found. Here, we propose the use of asqueezed state between Alice and Bob instead of the vac-uum state (Fig. 1) to achieve a long-distance QET. Forsimplicity, let us set x A = − ( L + T ) / − d and x B + T = ( L + T ) / d. Consider a non-decreasing C function f ( x ) such that f ( x ) = x + l/ x ≤ − d and f ( x ) = x − l/ x ≥ d . Here, l is a length parameter, and because ∂ x f ( x ) ≥
0, the parameter l satisfies l ≤ L + T = 2 d. In the following analysis, any f ( x ) satisfying these con-ditions can be applied. A typical example of f ( x ) thatexactly satisfies these conditions is provided as a C -classodd function under the x → − x transformation as fol-lows. A coordinate value ¯ x (0 < ¯ x < d ) and a positiveparameter Λ satisfying 0 < Λ < [2( d − ¯ x )] − can be usedto define f ( x ) as f ( x ) = (1 −
2Λ ( d − ¯ x )) x (11)for 0 ≤ x ≤ ¯ x and f ( x ) = x − l/ x − d ) (12)for ¯ x ≤ x ≤ d . In this case, the shift is given by l = 2Λ (cid:0) d − ¯ x (cid:1) . When Λ → [2( d − ¯ x )] − and ¯ x → d , the parameter l approaches its upper limit, L + T . From any f ( x ), itis possible to define a complete set of mode functions { v ω ( x ) } of ˆ ϕ + ( x + ) using the relation [2] v ω ( x ) = 1 √ πω exp ( − iωf ( x )) . (13)The orthonormality of the modes are proven by calculat-ing the inner products( v ω , v ω ′ ) = i Z ∞−∞ v ω ( x ) ∗ ∂ x v ω ′ ( x ) dx. Through a change of coordinates x ′ = f ( x ), it is verifiedthat ( v ω , v ω ′ ) = ( u ω , u ω ′ ) = δ ( ω − ω ′ ) , where u ω ( x ) = 1 √ πω e − iωx . (14)The completeness of { v ω ( x ) } is trivial because v ω ( x ) hasa one-to-one correspondence with u ω ( x ) due to the mono-tonicity of f ( x ) and the fact that { u ω ( x ) } is complete.Then, ˆ ϕ + ( x + ) can be expanded in terms of this modefunction asˆ ϕ + ( x + ) = Z ∞ (cid:16) ˆ f ω v ω ( x + ) + ˆ f † ω v ∗ ω ( x + ) (cid:17) dω, (15)where ˆ f † ω and ˆ f ω are creation and annihilation opera-tors, respectively, satisfying h ˆ f ω , ˆ f † ω ′ i = δ ( ω − ω ′ ) and h ˆ f ω , ˆ f ω ′ i = 0 [2]. Let us introduce a quantum state | f i such that ˆ f ω | f i = 0 (16)for all ω . Because v ω ( x ) is not a superposition of u ω ( x )with positive frequency, the state | f i is a squeezed vac-uum state of the form | f i ∝ exp (cid:18)Z ∞ dω Z ∞ dω ′ γ ωω ′ ˆ a † ω ˆ a † ω ′ (cid:19) | i , where ˆ a † ω is a creation operator corresponding to u ω ( x ).For an arbitrary | f i , the two-point correlation function h f | ˆΠ + ( x ) ˆΠ + ( x ′ ) | f i is evaluated as h f | ˆΠ + ( x ) ˆΠ + ( x ′ ) | f i = − ∂ x f ( x ) ∂ x ′ f ( x ′ )4 π ( f ( x ) − f ( x ′ ) − iǫ ) . Thus, for x A ≤ x A and x ′ A ≤ x A , h f | ˆΠ + ( x A ) ˆΠ + ( x ′ A ) | f i = h | ˆΠ + ( x A ) ˆΠ + ( x ′ A ) | i holds. Because | f i is a Gaussian state completely spec-ified by h f | ˆΠ + ( x ) ˆΠ + ( x ′ ) | f i , the above relation impliesthat the quantum fluctuation in ( −∞ , x A ] is the same as the zero-point fluctuation of | ih | . This indicates that( −∞ , x A ] is a local-vacuum-state region with zero en-ergy. The term “zero energy” is used here to indicatenot only that the average value of the energy densityin this region vanishes but also that all correlations be-tween the energy density operators are the same as thosein the vacuum state. Similarly, for x B ≥ x B + T and x ′ B ≥ x B + T , h f | ˆΠ + ( x B ) ˆΠ + ( x ′ B ) | f i = h | ˆΠ + ( x B ) ˆΠ + ( x ′ B ) | i holds, implying that [ x B + T, ∞ ) is also a local-vacuum-state region with zero energy. After time T has elapsed,the local-vacuum-state region moves to [ x B , ∞ ) becauseof the left-moving evolution.Consider the case of a large L . At time t = 0, Al-ice (who stays in the zero-energy region of ( −∞ , x A ])applies the measurements in Eqs. (2) and (3) to ˆ ϕ in the state | f i . The measurement result µ is sent toBob (who stays in the zero-energy region of [ x B , ∞ )) at t = T . During communication time T , the information of µ jumps across the long-distance region ( x A , x B ) witha positive finite energy E C , evaluated as E C = 148 π Z x B + Tx A ( ∂ x ln ( ∂ x f ( x ))) dx. (17)The excitation energy E C is so large that it can affordto maintain the negative energy − E B generated by Bobbecause E C is placed near to − E B . Hence, the positiveenergy E A injected by Alice can be separated far from − E B , and a long-distance QET becomes possible. Since f ( x ) is a nonsingular C function, E C is finite unless itexactly attains ∂ x f ( x ) = 0 with l = 2 d . At t = T ,Bob is able to extract the teleported energy from ˆ ϕ byperforming the same operation as in Eq. (4) on the localzero-point fluctuation. Note that f ( x B ) − f ( x A ) = x B − x A − l. This means the effective distance for the correlation be-tween the two points in the state | f i is much less thanthe physical distance. By simply replacing Eq. (7) with h f | ˆΠ + ( x B ) ˆΠ + ( x A ) | f i = − π ( x B − x A − l − iǫ ) , (18)the amount of teleported energy E Bf can be evaluatedas E Bf = (cid:16)R ∞−∞ R ∞−∞ g B ( x B ) g A ( x A )( x B − x A + T − l ) dx B dx A (cid:17) π R ∞−∞ ( ∂ x g B ( x )) dx exp (cid:16) π R ∞ | e g A ( ω ) | ωdω (cid:17) . (19)The difference between Eq. (8) and Eq. (19) is just theappearance of l in the correlation function between thetwo separate regions of the numerator integral. It should A’x + x + B’E A E Bf μ FIG. 2. (Color online) Schematic showing abrupt expansionof the space where quantum fluctuation of the field becomesseverely stretched. The upper figure depicts quantum fluc-tuation in the vacuum state before the expansion, while thelower figure depicts the stretched quantum fluctuation, whichcan be described by the modified mode function v ω ( x + ) afterthe expansion. be noted that l can, in principle, take a large value sat-isfying l ≤ L + T . By taking an L -dependent squeezedstate | f i such that l ∼ L + T, (20)the long-distance damping of E Bf behaves not as O ( L − )but as O ( L ) as the distance L increases. Therefore,this squeezed-state QET indeed overcomes the distancebound in Eq. (1). Note that E Bf for an L -independent | f i with a fixed l again exhibits the original dampingbehavior of O ( L − ) when L ≫ l , as it should. The energydistribution of the final state of the protocol is depictedin Fig. 1.The essence of QET without the limit of distance is asfollows. The distance dependence of E B for vacuum-stateQET comes from C µAB as defined in Eq. (5). This cor-relation is generated by the vacuum-state entanglementbetween local zero-point fluctuations in the regions of Al-ice and Bob. If we supply two local quantum fluctuationsof ˆ ϕ that are far away from each other and with the sameentanglement and correlations as those of the local zero-point fluctuations of two close regions, the QET remainseffective, independent of the distance, because C µAB isthe same. However, to sustain the negative-energy ex-citation generated by Bob, additional positive energy ofthe field must be placed near the negative-energy exci-tation. In this new protocol, the negative energy is sus-tained by the positive energy of the squeezed region. Asdepicted in Fig. 2, such a long distance correlation is in-deed realized by an abrupt expansion of the space overwhich the quantum field ˆ ϕ exists, analogous to cosmolog-ical inflation in general relativity. The upper diagram inFig. 2 depicts the vacuum fluctuation of ˆ ϕ in the state-preparation region, which is equipped on the right-handside of B , before moving left to the original QET exper-imental region of A and B . Assume that, if we performthe vacuum QET from A ′ to B ′ , the distance between A ′ and B ′ in Fig. 2 is so small that the amount of teleported energy attains a large value of E Bf . The lower diagram inFig. 2 shows the sudden expansion of the small subspacebetween A ′ and B ′ , which generates local excitation ofˆ ϕ due to severe stretching of the field modes. Note thatthe stretched mode function can be described by v ω ( x + )in a similar way to expanding Universe models includingcosmological inflation [2, 17]. The expansion is specifiedby the metric ds = g µν dx µ dx ν = dt − ( a ( t, x ) dx ) , (21)the scale factor of which obeys a ( t, x ) = 1 outside thestretching region. Let us assume that the expansion isvery rapid so that we can regard it instantaneous, and theexpansion happens at time t = 0. The initial conditionof the scale factor is a ( t = − , x ) = 1, and the equationof motion of ˆ ϕ in the expansion is given by ∂ µ (cid:0) √− gg µν ∂ ν ˆ ϕ (cid:1) = 0 . (22)When we consider the plane-wave mode function u ω ( x )in Eq. (14) just before the rapid expansion, the formof the mode function remains unchanged under the in-stantaneous expansion of space at the coordinate ( t, x ).However, from Eq. (21), the correct physical distance co-ordinate X should be obtained by X = X ( x ) = Z x a (+0 , x ′ ) dx ′ . Thus, the mode function after the expansion is computedin terms of the physical coordinate ( t, X ) as v ω ( X ) = 1 √ πω e − iωx ( X ) , where x ( X ) is the inverse function of X ( x ). This indeedreproduces the squeezed mode function in Eq. (13) bydefining f ( X ) = x ( X ). Thus, f ( x ) in Eq. (13) is com-puted from the relation f − ( x ) = Z x a (+0 , x ′ ) dx ′ , where f − is the inverse function of f . Hence, thequantum state after the expansion is equivalent to thesqueezed vacuum state given by Eq. (16). Since thefield is still in a local vacuum state outside the expandedregion, the correlation between A ′ and B ′ remains un-changed. Hence, if QET from A ′ to B ′ is executed, wewould be able to teleport the same amount of energy E Bf , although the distance would become very large.After the sudden expansion, the long-range correlatedquantum fluctuation moves left into the experimental re-gion including A and B and can be used for the originallong-distance QET from A to B with teleported energy E Bf .Such spatial expansion, for example, may be performedin quantum Hall edge current systems [12]. Recall that (a) (b)1 2 4 53 12 3 45 FIG. 3. (Color online) Schematics of quantum Hall edge cur-rent (red lines) (a) before and (b) after the expansion. the current [red line and arrows in Fig. 3(a)] is well de-scribed by a chiral (unidirectional) massless field in onedimension [13]. A local extrusion of bulk electrons towardthe outside [Fig. 3(b)] can be experimentally obtained bydynamically controlling the electron density in the de-pleted region [the gray region in Fig. 3]. Such operationsare commonly used in field-effect transistors at frequen-cies up to the subterahertz regime. Since marked points1–5 on the current trajectory in Fig. 3(a) are separatedas in Fig. 3(b) after the extrusion, the transition de-scribes an expansion of space in which the massless fieldexists. The field satisfies Eq. (22) in terms of a metric g µν induced by the extrusion, creating a continuously param-eterized multi-mode squeezed state. Generation of sucha squeezed state is well known in research of quantumfields in curved spacetime, such as inflationary universes[17], and provides an interesting squeezing method evenin condensed matter physics. Finally, it should be notedthat high-precision squeezed-state generation is not re-quired for long-distance QET. However, it is imperativethat a very long detour path with length l for the edgecurrent be inserted between two very-close local vacuumregions just as A ′ and B ′ in Fig. 2. This achieves thesame amount of E B in Eq. (19). This spatial expansionmethod is one strategy for realizing long-distance corre-lation, thus facilitating experimental verification of QETand potentially contributing to quantum device applica-tions. SUMMARY
In this paper, we have pointed out that vacuum-stateQET suffers from the distance bound in Eq. (1). Thebound is a severe obstacle to the implementation of long-distance QET in nanophysics. To overcome the bound onthe distance L between the sender and receiver of QET,we proposed a new QET protocol that adopts a squeezedvacuum state defined by Eq. (16), which corresponds tothe mode function in Eq. (13) with a C function f ( x )defined by Eqs. (9)–(12). The measurement of Alice andthe local operation of Bob are the same as those of the vacuum-state QET and are given by Eqs. (2), (3), and(4). By taking an L -dependent squeezed state | f i suchthat the parameter l of f ( x ) satisfies Eq. (20), the long-distance damping of E Bf behaves as O ( L ). Therefore,QET without the limit of distance can be attained. Long-distance QET may be experimentally verified by adopt-ing a spatial expansion method in quantum Hall edgecurrents, which is a new scheme to create a continuouslyparameterized multi-mode squeezed state in condensedmatter physics. Acknowledgments
G. Y. is supported by a Grant-in-Aid for ScientificResearch (No. 24241039) from the Ministry of Educa-tion, Culture, Sports, Science and Technology (MEXT),Japan. ∗ [email protected] † [email protected] ‡ [email protected][1] W. Pusz and S. L. Woronowicz, Commun. Math. Phys. , 273 (1978).[2] N. D. Birrell and P. C. W. Davies, Quantum Fields inCurved Space (Cambridge Univ. Press, 1982).[3] H. B. G. Casimir, Proc. Kon. Nederland. Akad. Weten-sch.
B51 , 793 (1948).[4] W. G. Unruh and R. M. Wald, Phys. Rev. D , 1047(1984).[5] M. Hotta, Phys. Rev. D , 045006 (2008).[6] M. Hotta, Phys. Rev. D , 044025 (2010).[7] C. H. Bennett, G. Brassard, C. Cr´epeau, R. Jozsa, A.Peres, and W. K. Wootters, Phys. Rev. Lett. , 1895(1993).[8] M. R. Frey, K. Gerlach, and M. Hotta, J. Phys. A: Math.Theor. , 455304 (2013).[9] M. Hotta, Phys. Lett. A , 5671 (2008).[10] Y. Nambu and M. Hotta, Phys. Rev. A , 042329(2010).[11] M. Hotta, Phys. Rev. A , 042323 (2009).[12] G. Yusa, W. Izumida, and M. Hotta, Phys. Rev. A ,032336 (2011).[13] D. Yoshioka, The Quantum Hall Effect (Springer, Berlin,2002).[14] For information on general measurement, see M. A.Nielsen and I. L. Chuang,