Effects of squeezing on quantum nonlocality of superpositions of coherent states
aa r X i v : . [ qu a n t - ph ] O c t Effects of squeezing on quantum nonlocality of superpositions of coherent states
Chang-Woo Lee and Hyunseok Jeong
Center for Theoretical Physics, Center for Subwavelength Optics and Department of Physics and Astronomy,Seoul National University, Seoul, 151-742, Korea (Dated: December 2, 2018)We analyze effects of squeezing upon superpositions of coherent states (SCSs) and entangledcoherent states (ECSs) for Bell-inequality tests. We find that external squeezing can always increasethe degrees of Bell violations, if the squeezing direction is properly chosen, for the case of photonparity measurements. On the other hand, when photon on/off measurements are used, the squeezingoperation can enhance the degree of Bell violations only for moderate values of amplitudes andsqueezing. We point out that a significant improvement is required over currently available squeezedSCSs in order to directly demonstrate a Bell-inequality violation in a real experiment.
PACS numbers: 03.65.Ud, 42.50.Dv, 42.50.-p
I. INTRODUCTION
Einstein, Podolsky, and Rosen (EPR) questioned com-pleteness of quantum mechanics based on the idea of localrealism [1]. Bell suggested a profound and useful inequal-ity imposed by local hidden variable theories, which re-flects EPR’s idea [2]. A couple of refined versions of Bell’sinequality followed the original one [3, 4], and numerousexperimental demonstrations have also been performed[5, 6]. In these studies, quantum states of light haveplayed a crucial role. Indeed, all Bell inequality tests inwhich the space-like separation between two local par-ties is satisfied have been performed using photons. Inthe meantime, it is worth noting that a loophole-free Bellinequality test is yet to be performed. The major obsta-cle in typical photon-based experiments, where two localparties are separate enough, is probably the detectionloophole [7]. Very recently, a Bell inequality test freefrom the detection loophole was performed using remoteatomic qubits [8], however, it did not satisfy the space-like separation required for a loophole-free Bell test.Recently, various types of continuous-variable stateshave been studied in order to suggest proposals forloophole-free Bell inequality tests [9]. As non-Gaussiancontinuous-variable states have rich structures in thephase space, it is important to explore possibility of ef-ficient Bell inequality tests using those states. Amongnon-Gaussian continuous-variable states, superpositionsof two coherent states (SCSs) [10, 11] in free-travelingoptical fields have been found a very useful tool for fun-damental tests of quantum theory [12, 13, 14, 15, 16, 17]as well as for quantum information applications [18, 19,20, 21, 22, 23]. In particular, they are useful for Bell in-equality tests using various measurements such as photonon/off detection, photon number detection, and homo-dyne detection [12, 13, 14, 15]. Once single-mode SCSsare generated, a 50:50 beam splitter can be used to gen-erate entangled coherent states (ECSs) [24] with whichone can perform Bell-inequality tests [12, 13, 15, 16, 17].Recently, “squeezed” SCSs were generated and de-tected [25, 26, 27], where the size of the states ( α = √ . II. ENTANGLING AND SQUEEZINGSUPERPOSITIONS OF COHERENT STATES
We introduce two particular types of SCSs, namely,even and odd SCSs, as | SCS ± ( γ ) i = N ± ( γ ) ( | γ i ± |− γ i ) (1) FIG. 1: Entangling and squeezing procedure for (a) anESS and (b) a SECS. The ESS is obtained by single-mode-squeezing | SCS ± ( √ γ ) i and feeding it into a 50:50 beam split-ter, whereas the SECS by feeding | SCS ± ( √ γ ) i into a 50:50beam splitter and two-mode-squeezing it. where N ± are normalization factors, | γ i is a coherentstate of amplitude γ , and γ is assumed to be real forsimplicity without loss of generality. The SCS with theplus (minus) sign between the coherent states in Eq. (1)is called an even (odd) SCS because it contains an even(odd) number of photons regardless of the value of γ .The size of a SCS may be defined by the magnitude ofthe amplitude γ . The ECSs at modes a and b are definedas | Φ ± i = N ± ( | γ i a | γ i b ± |− γ i a |− γ i b ) , | Ψ ± i = N ± ( | γ i a |− γ i b ± |− γ i a | γ i b ) , (2)which can be generated by splitting | SCS ± ( √ γ ) i at a50:50 beam splitter with an appropriate phase. We re-fer to the normalization factor N ± as N ± ( √ γ ) hereafter.Note that | Φ − i and | Ψ − i are maximally entangled ( i.e. ,each of them contains 1 ebit), which in general showstronger Bell violations than | Φ + i and | Ψ + i [13].For Bell inequality tests, we shall use two types of en-tangled states, i.e. , entangled squeezed SCSs (ESSs) andsqueezed ECSs (SECSs). The former can be obtainedby beam-splitting after single-mode-squeezing SCSs, andthe latter by two-mode-squeezing after beam-splitting aSCSs as shown in Fig. 1. The squeezed SCSs (SSCS) andESSs can be represented as | SSCS ± ( γ ) i = S ( s ) | SCS ± ( γ )) i , (3) | ψ ± i = B ab | SSCS ± ( √ γ ) i a | i b , (4)where S a ( s ) = exp (cid:2) s (cid:0) a − a † (cid:1)(cid:3) is the single modesqueezing operator, B ab = exp (cid:2) π (cid:0) a † b − a † b (cid:1)(cid:3) the 50:50beam splitter operator, and a and a † ( b and b † ) thebosonic annihilation and creation operators for mode a (mode b ). The ESSs become the same as | Ψ ± i for thecase of s = 0. The SECSs are (cid:12)(cid:12) Φ s ± (cid:11) = S ab ( s ) | Φ ± i , (cid:12)(cid:12) Ψ s ± (cid:11) = S ab ( s ) | Ψ ± i , (5)where S ab ( s ) = exp (cid:2) s (cid:0) ab − a † b † (cid:1)(cid:3) is the two-modesqueezing operator. We assume that the squeezing pa-rameter s is real for both S a ( s ) and S ab ( s ). The cor-responding state is then squeezed along the real axis in the phase space for s > s < III. VIOLATIONS OF BELL’S INEQUALITYWITH PHOTON PARITY AND ON/OFFMEASUREMENT SCHEMESA. Bell-CHSH inequality with the Wignerfunctions
Banaszek and W´odkiewicz (BW) studied Bell’s in-equality in the phase space, in terms of the Wigner( Q ) function based upon photon number parity (on/off)measurements and the displacement operation [33]. TheWigner function approach is based upon Clauser, Horne,Shimony and Holt (CHSH)’s version of Bell’s inequalitywhile the Q function upon Clauser and Horne (CH)’s [33].The displaced parity operator used for the Bell-CHSH in-equality is P ( α ) = Π even ( α ) − Π odd ( α )= D ( α ) ∞ X n =0 ( | n ih n | − | n +1 ih n +1 | ) D † ( α ) , (6)where D ( α ) = exp( αa † − α ∗ a ) is the displacement oper-ator, and the Bell operator is B CHSH = P a ( α ) ⊗ P b ( β ) + P a ( α ′ ) ⊗ P b ( β )+ P a ( α ) ⊗ P b ( β ′ ) − P a ( α ′ ) ⊗ P b ( β ′ ) . (7)The Wigner functions for state ρ may be obtained bytaking the average of the parity operator P ( α ) as [33, 35] W ( α ) = 2 π Tr [ ρ P ( α )] (8)and for two-mode state ρ ab as W ( α, β ) = (cid:16) π (cid:17) Tr [ ρ ab P a ( α ) ⊗ P b ( α )] . (9)Thus the Bell-CHSH inequality can be represented by theWigner function as | B CHSH | = (cid:16) π (cid:17) | W ( α, β ) + W ( α ′ , β ) + W ( α, β ′ ) − W ( α ′ , β ′ ) | ≤ , (10)where W ( α, β ) is the two-mode Wigner function and werefer to B CHSH = hB CHSH i as the Bell-CHSH function.This inequality can be violated with appropriate mea-surement operators and entangled states, and its maxi-mum value, 2 √
2, is known as Cirel’son’s bound [34].Using Eqs. (1) and (8), the Wigner functions for theeven and odd SCSs can be calculated as W SCS ± ( α ) = N ± h W √ γ ( α ) + W −√ γ ( α ) ± X √ γ ( α ) i , (11)where W γ ( α ) = 2 e − | α − γ | /π is the Wignerfunction of coherent state | γ i and X γ ( α ) =2 e − | α | cos [4 Im ( α ∗ γ )] /π . The Wigner functionsof the ESSs can be obtained using Eqs. (4) and (9), andthey can also be expressed as W ψ ± ( α, β ) = W SCS ± (cid:18) α s − β s √ (cid:19) W (cid:18) α + β √ (cid:19) , (12)where W ( α ) is the Wigner function of the vacuum andthe superscript s is used to indicate α s = α cosh s + α ∗ sinh s = e s Re α + i e − s Im α. (13)for an arbitrary complex number α . The two-modeWigner functions for the ECSs are calculated in the samemanner as [13] W Φ ± ( α, β ) = N ± h W γ ( α ) W γ ( β ) + W − γ ( α ) W − γ ( β ) ± X γ ( α ) X γ ( β ) ∓ Y γ ( α ) Y γ ( β ) i ,W Ψ ± ( α, β ) = N ± h W γ ( α ) W − γ ( β ) + W − γ ( α ) W γ ( β ) ± X γ ( α ) X γ ( β ) ± Y γ ( α ) Y γ ( β ) i , (14)where Y γ ( α ) = 2 e − | α | sin [4 Im ( α ∗ γ )] /π . The Wignerfunctions for SECSs are then W Φ s ± ( α, β ) = W Φ ± (˜ α s , ˜ β s ) ,W Ψ s ± ( α, β ) = W Ψ ± (˜ α s , ˜ β s ) , (15)where˜ α s = α cosh s + β ∗ sinh s, ˜ β s = β cosh s + α ∗ sinh s. (16)Note that when s = 0, W Ψ s ± ( α, β ) = W ψ ± ( α, β ) and W Φ s ± ( α, β ) = W ψ ± ( − β, α ).It is known that the Bell violation for an ECS ap-proaches Cirel’son’s bound [34] when the amplitude γ becomes large [13]. Figure 2 shows that a couple of char-acteristic properties in common when squeezing is ap-plied to the states being considered. The squeezing op-eration increases the degree of the Bell violation up tosome extent for small γ , but has a tendency of degradingit for squeezing in the specific direction for larger γ . Forexample, the squeezing in both the real and imaginarydirections in the phase space, enhances violation for ψ + ( ψ − ) until γ reaches around 0.5 (1.0). On the other hand,for larger values of γ squeezing in the real direction ( i.e. , s >
0) decreases the degree of violation while squeez-ing in the imaginary direction ( i.e. , s <
0) increases theviolation.In fact, for larger γ , squeezing along the real axis makesthe interference fringes less sharp and this could be re-lated to the decrease of the Bell violations. In the caseof Φ s ± with large γ , since ˜ α s ( ˜ β s ) → α ′ ( − α ′∗ ) as s → −∞ where α ′ = e − s ( α − β ∗ ), and hence W Φ s ± ( α, β ) → FIG. 2: (Color online) Optimized Bell-CHSH function B = | B CHSH | max for (a) ψ + (b) ψ − (c) Φ s + (d) Φ s − (e) Ψ s + (f) Ψ s − for parity measurements. The split line in each graph indi-cates no squeezing ( s = 0). Note that the plots for ψ ± aresimilar to the ones for Φ s ± , and that B ’s of Ψ s ± are rather sim-ilar and symmetric to the ones Φ s ± with respect to s = 0 line.One can observe that for small γ squeezing in any directioncan enhance Bell violations, whereas for large γ squeezing inspecific direction only can enhance them. In any case, squeez-ing causes Bell violations to increase monotonically from thenon-squeezed values and converge to specific ones. W Φ ± ( α ′ , − α ′∗ ) which is the very condition when maxi-mum violations occur for W Φ ± ( α, β ). But as s → ∞ ,the interference part X γ (˜ α s ) X γ ( ˜ β s ) − Y γ (˜ α s ) Y γ ( ˜ β s ) inthe Wigner function fades out, which may play a cru-cial role in degrading the Bell violations. The case ofΨ s ± can be explained in a similar way. Therefore, in thecase of photon parity measurements, squeezing in a well-chosen quadrature direction can enhance Bell violationsof tested states, though its contribution gets slighter asthe amplitudes of the states grow larger. B. Bell-CH inequality with the Q functions The operator used for tests of the Bell-CH inequalityis B CH = Q a ( α ) ⊗ Q b ( β ) + Q a ( α ′ ) ⊗ Q b ( β )+ Q a ( α ) ⊗ Q b ( β ′ ) − Q a ( α ′ ) ⊗ Q b ( β ′ ) − Q a ( α ) ⊗ I b − I a ⊗ Q b ( β ) , (17)where Q ( α ) = D ( α ) | ih | D † ( α ) (18)is a displaced “no photon” operator and I is the identityoperator. Subsequently, the Bell-CH function B CH = hB CH i is given in terms of Q representation as B CH = π (cid:2) Q ab ( α, β ) + Q ab ( α ′ , β ) + Q ab ( α, β ′ ) − Q ab ( α ′ , β ′ ) (cid:3) − π (cid:2) Q a ( α ) + Q b ( β ) (cid:3) , (19)where Q a ( α ) and Q b ( β ) are marginal Q functions in thecorresponding modes. As implied above, the Q functionsof single-mode state ρ and two-mode state ρ ab can beobtained using the operator Q ( α ) as (1 /π )Tr[ ρ Q ( α )] and(1 /π ) Tr[ ρ ab Q a ( α ) ⊗ Q b ( β )], respectively [35]. The Q functions for the SSCS are then given as Q SSCS ± ( α ) = N ± (cid:2) Q + √ γ ( α )+ Q −√ γ ( α ) ± Q X √ γ ( α ) (cid:3) , (20)subsequently for ESSs as Q ψ ± ( α, β ) = Q SSCS ± (cid:18) α − β √ (cid:19) Q (cid:18) α + β √ (cid:19) , (21)where Q ± γ ( α ) = cos θ Q ± γ − s ( α s ) , (22) Q X γ ( α ) = cos θ Q ( α s ) e −| γ − s | cos [2 Im( α ∗ s γ s )] , (23)where α s = α cos( θ/
2) + α ∗ sin( θ/ γ − s = γ cos( θ/ − γ ∗ sin( θ/ θ/ − (cid:0) tanh s (cid:1) , Q γ ( α ) = e −| α − γ | /π ,and Q ( α ) = e −| α | /π . Note that as s → ∞ ( −∞ ), α s →√ α ] (cid:0) i √ α ] (cid:1) .In the meantime, the Q functions for SECSs are Q Φ s ± ( α, β ) = N ± (cid:2) Q ++ ( α, β ) + Q −− ( α, β ) ± Q XY+ ( α, β ) (cid:3) , (24) Q Ψ s ± ( α, β ) = N ± (cid:2) Q + − ( α, β ) + Q − + ( α, β ) ± Q XY − ( α, β ) (cid:3) , (25)where Q ±± ( α, β ) = cos θ Q ± γ ∓ s (˜ α s ) Q ± γ ∓ s ( ˜ β s ) ,Q ±∓ ( α, β ) = cos θ Q ± γ ∓ s (˜ α s ) Q ∓ γ ± s ( ˜ β s ) , (26) Q XY ± ( α, β ) = cos θ Q (˜ α s ) Q ( ˜ β s ) e − | γ ∓ s | × cos [2 Im( α ∗ s γ s ± β ∗ s γ s )] , (27) with ˜ α s = α cos( θ/
2) + β ∗ sin( θ/
2) and ˜ β s = β cos( θ/
2) + α ∗ sin( θ/ C. Bell-CHSH inequality with on/offmeasurements
One can test the Bell-CHSH inequality by the followingdisplaced “on/off ” measurement operator O ( α ) = Π on ( α ) − Π off ( α )= D † ( α ) ∞ X n =1 | n ih n | − | ih | ! D ( α ) , (28)which assigns +1 or –1 to each measured result depend-ing on whether (any) photons are detected or not at adetector such as an avalanche photodiode. Then the Bell-CHSH inequality can be represented in the same way asdone in Eq. (10) just with P replaced with O , so that theBell-CHSH function becomes B CHSH = A ( α, β ) + A ( α ′ , β ) + A ( α, β ′ ) − A ( α ′ , β ′ ) (29)with A ( α, β ) = 1 − π Q a ( − α ) − π Q b ( − β )+ 4 π Q ab ( − α, − β ) , (30)where the Q functions are the ones obtained in the pre-vious subsection. It is worth noting that this Bell-CHSHfunction is related to the previous Bell-CH function as B CHSH ( α, β ) = 4 B CH ( − α, − β ) + 2 . (31)A Bell inequality test with photon on/off measure-ments is obviously more feasible than that of photonnumber parity measurements. However, if the averagephoton number of the state under consideration is toolarge, Bell violations cannot be observed using photonon/off measurements because the possibility of getting a“off” result approaches zero [13]. Because of this, Bellviolations for ESSs and SECSs shown in Fig. 3 show dif-ferent behaviors compared to the cases of photon paritymeasurements. In the case of “+” states (Φ + , Ψ + ( ψ + )),quadrature squeezing in any direction increases Bell vi-olations only for small γ . Meanwhile, in the case of “–”states (Φ − , Ψ − ( ψ − )), squeezing in specific direction in-creases the violations only for γ &
1, whereas it is notany desirable for violations for small γ . In any case,large squeezing in any quadrature direction causes Bellviolations to eventually vanish. This is different from thecases for the parity measurements where large values ofsqueezing cause the Bell functions to converge to certainvalues (smaller or larger than the ones in the cases of nosqueezing). FIG. 3: (Color online) Optimized Bell-CHSH function B = | B CHSH | max for (a) ψ + (b) ψ − (c) Φ s + (d) Φ s − for on/off mea-surements. The lowest two plots are for (e) ψ + (thick), Φ s + (thin), Ψ s + (dashed) respectively with γ = 0 . ψ − (thick), Φ s − (thin), Ψ s − (dashed) with γ = 1 .
0. The plots ofΨ s ± not presented here are similar to those of Φ s ± provided thesign of s is altered as in the parity measurement case. Notethat for small γ , squeezing “+” states increases B up to someextent (e), and that for large γ , squeezing “–” states in spe-cific direction only contributes to maximal values of B CHSH ’s(f).
IV. ESTIMATION OF BELL VIOLATIONSWITH REALISTIC STATES
We are also interested in whether a recently generatedSSCS [25], which can be immediately used to generatean ESS, may be used for tests of Bell’s inequality. Thesize of the generated SSCS, an “even” one, was as largeas γ = √ . | φ i = p / | i + p / | i . (32)This state is a very good approximation of an ideal SSCS | SSCS + i = S ( s ) | SCS + ( α ) i (33) FIG. 4: Optimized Bell-CHSH function B = | B CHSH | max vs.fidelity F of ρ exp with respect to | φ i for the case of (a) photonparity measurements and (b) photon on/off measurements.Dotted line in each plot indicates the local realistic bound forBell-CHSH inequality. Bell violation for the case of paritymeasurements can be observed when the fidelity approaches92% while that for on/off measurements case cannot be ob-served until the fidelity goes over 99%. where s = 0 . α = √ . |h φ | SSCS + i| ≈ | ψ + i with γ = p . /
2. When this ideal two-mode stateis used to obtain the Bell function B CHSH , its optimizedvalue is 2.419 (2.033) with photon number parity (on/off)measurements. In the meantime, state | φ i shows a Bellinequality violation as large as B CHSH = 2 .
401 (2 . ρ exp ,which is degraded by experimental imperfections such asnon-unit efficiencies, noises, and errors related to mea-suring devices, we use the following Wigner function inRef. [25], W exp ( x, p ) = exp (cid:0) − x /α − p /β (cid:1) π √ αβ (cid:20)(cid:16) − δα (1 − ν ) α − β ) (cid:17) + (cid:16) δα (1 − ν ) α − β ) (cid:17) (cid:21) ( δ (cid:20) x α + αν p β (cid:21) + 2 δ " − δ αν − β ) β ( α − β ) ! x α + αν p β (cid:21) + δ ( αν − β ) β ( α − β ) (cid:20) x α − αν p β (cid:21) + " − δ αν − β ) β ( α − β ) ! + δ " ( αν − β ) β ( α − β ) ) , (34)where x = √ α ] , p = √ α ] in our case, andthe four parameters α, β, ν, δ are defined by gain andvarious imperfection parameters [37]. However, since theBell function depends very sensitively on such imperfec-tion parameters, we assume perfect measuring deviceswith no errors, in which case α → g, β → α − ( g − /g, ν → /g, δ → , (35)where g is an optical parametric amplifier gain describingthe two-photon number state, and then the fidelity F = h φ | ρ exp | φ i depends only on g . Note also that for test-ing the on/off measurement case, we can transform theabove Wigner function into the Q function simply by justreplacing the parameters α, β, δ by α + 1 , β + 1 , αα +1 δ .As can be seen in Fig. 4, in order for ρ exp to show Bellviolations, the fidelity should be improved up to around92% in the case of parity measurements. However, wenote again that the violations are possible only when allthe experimental imperfections nearly vanish, which isextremely demanding. When on/off measurements areused, the fidelity should be even more improved up to atleast 99% to show a Bell violation. V. REMARKS
We have studied how squeezing influences the degree ofBell inequality violations of several different beam-split- entangled SCSs. It has been found that squeezing canalways increase Bell violations, given the squeezing di-rection is properly chosen, for the case of photon paritymeasurements. On the other hand, in the case of thephoton on/off measurements, squeezing can enhance Bellviolations only for well-chosen values of amplitudes andsqueezing. Therefore, it should be noted that for certainmeasurement schemes, the squeezing action is not alwayshelpful in enhancing Bell violations of entangled states oflight.In order to demonstrate a Bell violation in a real ex-periment, a significant improvement is required over thecurrently available SSCS. For example, the fidelity of thegenerated state should be improved up to 92% even whenall the other conditions including the efficiency of photonparity measurements are ideal. There are ongoing effortsto effectively generate high-fidelity SSCSs using currentlyavailable experimental resources [38]. It would be a morerealistic target to perform homodyne tomography to re-construct a generated SSCS and “indirectly” show a Bellviolation using Eq. (7).
Acknowledgments
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