Postselection induced entanglement swapping from a vacuum--excitation entangled state to separate quantum systems
PPostselection induced entanglement swapping from a vacuum–excitation entangledstate to separate quantum systems
Antonio Di Lorenzo
Instituto de F´ısica, Universidade Federal de Uberlˆandia,Av. Jo˜ao Naves de ´Avila 2121, Uberlˆandia, Minas Gerais, 38400-902, Brazil andCNR-IMM-UOS Catania (Universit`a), Consiglio Nazionale delle Ricerche, Via Santa Sofia 64, 95123 Catania, Italy
We show that a single particle in a superposition of different paths can entangle two objects locatedon each path. The entanglement has its maximum visibility for intermediate coupling strengths.In particular, when the two quantum systems with which the particle interacts are detectors thatmeasure its presence and its polarization, the so-called quantum Cheshire cat is realized.
Keywords: Entanglement, Quantum inseparability, Quantum paradoxes
The two most perplexing features of quantum mechan-ics are the interference in a double slit experiment, andthe entanglement of spatially separated systems. In theparadigmatic double slit experiment, a particle, in somesense, follows two paths at the same time, as shown bythe appearance of interference fringes after accumulat-ing many measurements. This feature is very elusive, astrying to measure the presence of the particle on eitherpath destroys the interference. A strong evidence in favorof this ubiquity is that a single particle can induce en-tanglement in two separated quantum systems that havenever mutually interacted and that are placed each ona different path, as if it interacted with both systemssimultaneously. The preceding literature on this topicconsidered only the case of a strong interaction. A re-cent related proposal of Aharonov et al. [1], where theeffect is dubbed a quantum Cheshire cat, on the otherhand, considers the weak coupling limit. Here, we tacklethe problem for an arbitrary coupling. In the follow-ing we shall demonstrate that a particle in a coherentsuperposition of spatially separated paths can induce en-tanglement between two distant meters located one oneach path, and we propose an entanglement indicator toquantify it. We also demonstrate that the optimal cou-plings are not weak, but either strong or intermediate,depending on the dimensions of the Hilbert spaces of themeters.Measurements can be divided approximately into threecategories: strong, intermediate, and weak. Strong mea-surements are the textbook measurements that are de-scribed already in von Neumann book [2]. If a meter ina state | A (cid:105) interacts with a quantum system preparedin an eigenstate | O (cid:105) of the observable ˆ O to be measured,the joint state of the two after the interaction is | A O , O (cid:105) ,where | A O (cid:105) are mutually orthogonal states of the me-ter. The evolution for an arbitrary initial state | ψ (cid:105) ofthe system follows from the linearity of quantum me-chanics. In weak measurements [3], the final states ofthe meter, | A O (cid:105) , instead, are almost indistinguishable, | A O (cid:105) (cid:39) N O ( | ¯ A (cid:105) + gf ( O ) | δ O (cid:105) ), with g an effective cou-pling constant, f a function of O , | δ O (cid:105) a set of statesnot necessarily distinct, and N O a normalization. Per-haps “measurement” is a misleading term, since due to the weak interaction between the system and the meterone cannot infer substantial information about the for-mer from a single trial. However, the statistical analysisof the postselected data — which generally is limited tothe average readout but could be extended to the fullstatistics [4–7] — allows to extract information aboutthe system that is not trivially recovered from standardstrong projective measurements. For instance, weak mea-surements followed by postselection provide a powerfulinference technique, allowing e.g. to reconstruct the un-known wavefunction of a system [8], or the density matrix[9–11]. The coherent quantum nature of the meter wasshown to be of the essence for the peculiar amplificationof the weak measurement [12, 13]. Several experimentalworks have focused on signal amplification [14–17], butthe efficiency of the amplification has been questioneddue to the corresponding decrease in the probability ofpostselection [7, 18–20]. In intermediate measurements,on the other hand, no special form for the output states | A O (cid:105) is postulated, but the evolution U g | A , O (cid:105) is cal-culated by assuming a sensible form for the interaction,depending on a parameter g . For g → ∞ and for g → | Ψ (cid:105) andpostselection | Φ (cid:105) in a pure state for the system, andof an initial pure (and uncorrelated) state for the me-ters | A , B (cid:105) . In a nondemolition measurement, if thephoton is in the left arm, the total state evolves to | L, σ, A , B (cid:105) ; if instead the photon is in the right arm a r X i v : . [ qu a n t - ph ] J un BS1D1 D2V1 V2 Y BS2V3 V4
Φ Ψ
S1 S2XA B
FIG. 1. The setup is a variant of the Mach-Zehnder interfer-ometer. The source S BS
1, and exits in a coherent superposition ofspatially separated states. In a sense, the particle is simul-taneously in the left and in the right arm, as it can induceentanglement between two quantum systems A and B , as canbe evinced by measuring the cross-moment between two ob-servables X and Y . The entanglement can be observed condi-tionally on the postselection, made by a judicious combinationof a second beam-splitter and of polarization-sensitive detec-tors D j . The local unitary operators V j allow to arbitrarilyset the preparation Ψ and the post-selection Φ, and it has positive (negative) polarization, | Ψ (cid:105) = | R, ±(cid:105) ,the final state is | R, ± , A , B ± (cid:105) . Because of the super-position principle, if the system is in | Ψ (cid:105) = a | L, σ (cid:105) + b | R, + (cid:105) + c | R, −(cid:105) , the final state is a | L, σ, A , B (cid:105) + b | R, + , A , B + (cid:105) + c | R, − , A , B − (cid:105) . Now, if the photonis traced out, the final state of the meters is a mixture, ρ cl.corr. = | a | | A , B (cid:105)(cid:104) A , B | + | b | | A , B + (cid:105)(cid:104) A , B + | + | c | | A , B − (cid:105)(cid:104) A , B − | , that shows only classical correla-tions. If instead the photon is successfully postselectedin a state | Φ (cid:105) , the final state of the meters is entangled,as | F (cid:105) = l | A , B (cid:105) + r + | A , B + (cid:105) + r − | A , B − (cid:105) , (1)where we defined the complex transition amplitudes l = (cid:104) Φ | Π L | Ψ (cid:105) , r ± = (cid:104) Φ | Π R, ± | Ψ (cid:105) , (2)with Π L = (cid:80) ± | L, ±(cid:105)(cid:104) L, ±| the rank-2 projector in theleft arm, and Π R, ± = | R, ±(cid:105)(cid:104) R, ±| the rank-1 projectorson the right arm with polarization ± . The state | F (cid:105) We note that, because of the completeness relation Π L + Π R + + in (1) is not normalized to one, instead (cid:104) F | F (cid:105) = P , theprobability of a successful postselection. If the postse-lection fails, the unnormalized final state of the metersis mixed, ˇ ρ f = ρ cl.corr. − | F (cid:105)(cid:104) F | . Notice how the traceTr(ˇ ρ f ) = P (cid:48) = 1 − P is the probability for the postselec-tion to fail.The entanglement is due to the photon being in a co-herent superposition of states localized in the left andin the right arm, so that, in some sense, it interactswith both meters at the same time. If no postselec-tion would occur, or, more generally, if the prepara-tion or the postselected state would not be a coher-ent superposition of states localized in the left and inthe right arm, the entanglement would not manifest.The situation is analogous to delayed–choice entangle-ment swapping [25], but here there are only three quan-tum systems (the particle and the two pointers), andno preexisting entanglement among them seems to bepresent. As a matter of fact, however, we are in pres-ence of vacuum–excitation entanglement [26–29], whichis swapped to the meters. Indeed, a superposition ofa photon in the left and the right arm can be writtenas a | L,σ , L, − σ , R, + , R, − (cid:105) + b | L,σ , L, − σ , R, + , R, − (cid:105) + c | L,σ , L, − σ , R, + , R, − (cid:105) , having the vacuum state of theelectromagnetic field present in the left or right propa-gating channel, and its excitation, the photon, presentin the right or left propagating channel. The fact thatthe particle is a photon is irrelevant (we are calling it aphoton just to fix the ideas, indeed), the same rationaleapplies to any other particle, which can be consideredan excitation of a quantum field. The issue of whethera single particle is actually entangled is quite debated[30–32], and we shall not address it here. We are con-tent with the uncontroversial fact that the two distantmeters get entangled without having interacted, and weshall not debate whether this entanglement was swappedfrom the single-particle entanglement or whether it wascreated by a nonlocal interaction.How to detect the entanglement between the meters?Let us consider an unnormalized average of the form m = (cid:104) F | ˆ X A ˆ X B | F (cid:105) , with ˆ X A an observable of the meter A and ˆ X B an observable of the meter B . We have m = m cl + m ent + m l.i. , where m cl = | l | (cid:104) A | ˆ X A | A (cid:105)(cid:104) B | ˆ X B | B (cid:105) + (cid:88) ± | r ± | (cid:104) A | ˆ X A | A (cid:105)(cid:104) B ± | ˆ X B | B ± (cid:105) (3)is the classical part, m ent = (cid:88) ± (cid:60) (cid:16) l ∗ r ± (cid:104) A | ˆ X A | A (cid:105)(cid:104) B | ˆ X B | B ± (cid:105) (cid:17) (4) Π R − = , l + r + + r − = (cid:104) Φ | Ψ (cid:105) . It is customary to define the weakvalues L w = l/ ( l + r + + r − ) and Σ w = ( r + − r − ) / ( l + r + + r − ),associated, respectively, to the operators Π L and σ R = Π R, + − Π R, − . However, we prefer to use transition amplitudes, whichare always well behaved. is the contribution from the interference between the twometers, i.e. from their entanglement, and m l.i. = 2 (cid:60) (cid:16) r + ∗ r − (cid:104) A | ˆ X A | A (cid:105)(cid:104) B + | ˆ X B | B − (cid:105) (cid:17) (5)is the contribution from the local interference in the me-ter B . If either ˆ X A or ˆ X B is the identity, in the strongcoupling limit, the contribution from the entanglementvanishes. Therefore, one needs to consider two nontriv-ial operators, as in the case of Bell inequalities. In theweak coupling limit, however, entanglement contributesto the average m even if it is a local average, because (cid:104) A | A (cid:105) (cid:39) (cid:104) B ± | B (cid:105) (cid:39) et al. [1], instead, adopts the same scheme (the authorsare apparently unaware of this), but with a weak cou-pling.If it is possible to make a measurement on the me-ters that projects their states into arbitrary combinations α | A (cid:105) + β | A (cid:105) and α | B (cid:105) + β | B + (cid:105) + γ | B − (cid:105) , then one couldcheck the violation of a Bell-like inequality [36–38], or,better, one could use the criteria discussed by Peres [39]and Horodecki et al. [40], as the entanglement is betweena two-level system and a three-level system. In this case,the maximum entanglement is achieved for a strong in-teraction, so that {| A (cid:105) , | A (cid:105)} and {| B (cid:105) , | B ± (cid:105)} form or-thogonal bases. Thus, we have reached a first partial con-clusion: if the meters have a finite-dimensional Hilbertspace, whose relevant two– and three–dimensional sub-spaces can be probed projectively along any basis, thenit is possible to observe the entanglement induced by thepostselection already in the strong coupling regime.However, if the meters have an infinite dimensionalHilbert space, the task of making projective measure-ments on α | A (cid:105) + β | A (cid:105) and α | B (cid:105) + β | B + (cid:105) + γ | B − (cid:105) maybe a practical impossibility. Furthermore, unwanted ex-ternal influences can drive the states of the meters awayfrom the simple two– and three–dimensional subspacesspanned by these bases. Therefore another criterion forentanglement should be used. Our goal is to find ob-servables ˆ o A and ˆ o B such that m cl = m l.i = 0 and m ent (cid:54) = 0, so that m works as an unambiguous indi-cator for entanglement. A sufficient condition is that (cid:104) A | ˆ X A | A (cid:105) = (cid:104) B | ˆ X B | B (cid:105) = 0, i.e. if the particle isnot in the left (respectively, right) arm, the expectationvalue of ˆ X A (resp., ˆ X B ) is zero. We note, however, thatthe observed value of X A may not be zero, as we are notrequiring a strong measurement —which implies that thestate | A (cid:105) is an eigenstate of ˆ X A with null eigenvalue—thus quantum statistical fluctuations and environmentalnoise can yield a nonzero result in an individual trial.We indicate with x and y the pointers of the meters,whose initial states have the representation (cid:104) x | A (cid:105) = φ ( x ), (cid:104) y | B (cid:105) = φ ( y ). By pointers, we mean that, in thestrong coupling regime, observing x and y gives unam-biguous information about the presence of the particle inthe left arm and the value of its polarization in the right arm. Think of the Stern-Gerlach apparatus, where theposition of a spot on the screen is the pointer revealingthe value of the spin of the atom. The meters are assumedto be unbiased, so that the initial averages of the pointers x and y with | φ ( x ) φ ( y ) | are null. We shall consider thepointers in units of the initial uncertainties ∆ A and ∆ B ,i.e. (cid:82) dx x | φ ( x ) | = (cid:82) dy y | φ ( y ) | = 1. For simplic-ity, we assume the von Neumann model of measurement.In this model, after the interaction with a particle, thewave functions of the meters become (cid:104) x | A (cid:105) = φ ( x − g A ), (cid:104) y | B ± (cid:105) = φ ( y ∓ g B ), with g A , g B the dimensionless cou-pling constants. The final state of the meters, in thepointers representation, is (cid:104) x, y | F (cid:105) = φ ( x, y ).In optics, it is possible to realize arbitrary couplings g A , g B , respectively, by using a refractive crystal thatdislocates the beam along the x axis by an amount δx = g A ∆ x , and a birefringent crystal of appropriate length,so that the beams exiting the latter have a separation δy = g B ∆ y , with ∆ x , ∆ y the variance of the input beam.As the entanglement indicator, we shall consider thecross-moment (cid:104) xy (cid:105)P = (cid:104) F | ˆ x ˆ y | F (cid:105) , to which only the en-tanglement terms give a nonzero contribution, (cid:104) xy (cid:105)P = 2 (cid:88) ± (cid:60) (cid:20) l ∗ r ± (cid:90) dx xφ ( x ) φ ∗ ( x − g A ) × (cid:90) dy yφ ( y ∓ g B ) φ ∗ ( y ) (cid:21) . (6)It may happen that the two contributions from the φ ( y + g B ) and the φ ( y − g B ) wave function are presentbut they cancel out. In this case, entanglement may bedetected by using another cross-moment, as we shall dis-cuss elsewhere [41].When the postselection fails, which happens withprobability P (cid:48) = 1 − P , the entanglement indicatoris (cid:104) xy (cid:105) f P (cid:48) = Tr(ˆ x ˆ y ˇ ρ f ) = Tr(ˆ x ˆ yρ cl.corr ) − (cid:104) F | ˆ x ˆ y | F (cid:105) = −(cid:104) xy (cid:105)P , where we used the fact that Tr(ˆ x ˆ yρ cl.corr ) = 0,i.e., there is no entanglement contribution to classicalcorrelations. Therefore, we can use all the experimen-tal data by defining the entanglement indicator as fol-lows: In the j -th trial, if the postselection is success-ful, consider the product c j = x j y j , otherwise, consider c j = − x j y j ; sum the c j and divide by the number oftrials; the value C = 2 (cid:104) xy (cid:105)P is thus obtained, allowingto establish whether a Cheshire cat is observed ( C (cid:54) = 0)or not ( C = 0). Formally, if we assign a binary variable τ = ± τ = +1representing a successful postselection in E , and with τ = − C = (cid:104) τ xy (cid:105) a , (7)which provides the signature for the entanglement be-tween the meter measuring the presence of the photonand the meter measuring its polarization in the two armsof the interferometer. The index a is a reminder that theaverage in (7) is made over all the experimental data, notonly on the ones obtained for a successful postselection.So far, we have provided exact results. In the weakcoupling limit, a shift of g A is small compared to therange over which φ varies appreciably, so that one canapproximate φ ( x − g A ) (cid:39) φ ( x ) − g A dφ ( x ) /dx , etc.As id/dx represents the momentum operator, it is pos-sible to approximate the overlap integrals with appro-priate combinations of the initial averages of ˆ x ˆ p x , etc.[42] . In the strong coupling limit, instead, the wavefunctions φ ( x ) and φ ( x − g A ) have a negligible overlap, φ ∗ ( x − g A ) φ ( x ) (cid:39)
0, etc., so that the interference termsdisappear. Precisely, the overlap terms of interest in (6)behave asymptotically as (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) dx xφ ∗ ( x ) φ ( x − g A ) (cid:12)(cid:12)(cid:12)(cid:12) ≈ (cid:40) g A for g A (cid:28) g A | φ ( g A ) | for g A (cid:29) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) dy yφ ∗ ( y ) φ ( y ∓ g B ) (cid:12)(cid:12)(cid:12)(cid:12) ≈ (cid:40) g B for g B (cid:28) g B | φ ( ± g B ) | for g B (cid:29) Theentanglement indicator C vanishes bi-linearly in the cou-plings g A g B for a weak measurement, and it vanishes as g A g B | φ ( g A ) || φ ( g B ) | for a strong measurement. There-fore, there must be an optimal intermediate couplingstrength for which the entanglement is not only present,but it gives a maximum contribution to C . Thus, we need an expression working for any couplingstrength, in order to determine the optimal one. We shallconsider the initial state of the meters to be Gaussian, φ ( x ) ∝ exp( − x / C = g A g B w A w B (cid:60) [Tr( Eσ R ρ Π L )] , (9)with E a mixed postselection state, ρ a mixed prepa-ration state, σ R = (cid:80) ± ±| R, ±(cid:105)(cid:104) R, ±| the local spinoperator in the right path, and w A = exp (cid:0) − g A / (cid:1) , w B = exp (cid:0) − g B / (cid:1) . As a function of the preparation andthe postselection, the extremal values of the Cheshire catparameter is C max = g A g B w A w B / |C| is a non-monotonous function ofthe coupling constants. As we noted earlier, it goes tozero both in the weak coupling limit g A,B →
0, and in thestrong coupling limit g A,B → ∞ . Its extremal value, asa function of the couplings, is reached for g A = g B = 2,yielding C extr = 4 e − (cid:60) [Tr( Eσ R ρ Π L )], which, as a func-tion of the preparation and postselection, has an absolutemaximum C extr = e − . Therefore, the criterion C (cid:54) = 0does not require a very weak coupling, but it reachesits optimum when the coupling strengths are twice theinitial uncertainty of the meters. Hence the optimal mea-surement is neither strong nor weak, but intermediate.We stress that we have so far assumed that the read-out of the meters is projective and errorless, the only uncertainty ∆ A , ∆ B coming from the initial preparationof the meters. When external noise is accounted for, letus call its square variance ν A , ν B , the criterion to observeunambiguously a Cheshire cat is that ν A ν B (cid:28) C . A nec-essary condition is that ν A (cid:28) ∆ A and ν B (cid:28) ∆ B , i.e. theresolution of the readout must be much smaller than theinitial uncertainty. By using once more the analogy withthe Stern-Gerlach apparatus, this means that the inputbeam of silver atoms may have a waist ∆ (cid:29) g , where g isthe deflection due to the magnetic field gradient, but thesize of the spot on the screen created by each individualatom should be ν (cid:28) ∆.We conclude by comparing our criterion to the oneused in Ref. [1]. The preparation and postselection werechosen by the authors of Ref. [1] in such a way that, inthe weak coupling limit, the average outputs take the spe-cial values lim g A → (cid:104) x/g A (cid:105) = 1 and lim g B → (cid:104) y/g B (cid:105) = 1.From this it was inferred that the photon is in the leftarm, while its polarization is in the right arm. Thisphenomenon is called a quantum Cheshire cat, in thesense that a physical property can be separated from itscarrier. The interpretation attributing to the averageslim g A → (cid:104) x/g A (cid:105) = 1 and lim g B → (cid:104) y/g B (cid:105) = 1 the meaningof having one particle on one path and its polarizationon the other path is problematic. Indeed, it has beenestablished since a long time[12] that the averages (cid:104) x (cid:105) and (cid:104) y (cid:105) should not be interpreted literally as represent-ing a value of the measured observable of the system.One should give these averages no more meaning thanthey have: they represent the average positions of point-ers that have interacted with a quantum system. Theirstatistics differs from the classical statistics because oftheir own quantum nature, which leads to interference.In the present case, the interference is between two spa-tially separated meters, i.e. it manifests as entanglement.While in the weak coupling limit, as discussed above, en-tanglement does contribute to the local averages (cid:104) x (cid:105) and (cid:104) y (cid:105) , it is very difficult to unscramble the entanglementcontribution in the latter two quantities. By contrast,the quantity C proposed here comes exclusively from theentanglement, and it is well defined for any couplingstrength. As such, it is better suited to characterize thepresence of quantum correlations between the meters.In conclusion, we have demonstrated that the two fun-damental aspects of quantum mechanics, coherence andentanglement, concur in the variant of the Mach–Zehnderinterferometer proposed by Ref. [1]. The phenomenonseems to confirm the point of view that a single particlecan be entangled with the vacuum, as separate quantumsystems get entangled by interacting simultaneously withthe single particle. ACKNOWLEDGMENTS
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