Loophole-free Bell test for continuous variables via wave and particle correlations
Se-Wan Ji, Jaewan Kim, Hai-Woong Lee, M. S. Zubairy, Hyunchul Nha
aa r X i v : . [ qu a n t - ph ] O c t Loophole-free Bell test for continuous variables via wave and particle correlations
Se-Wan Ji , , Jaewan Kim , Hai-Woong Lee , M. S. Zubairy , and Hyunchul Nha , , ∗ Department of Physics, Texas A & M University at Qatar, Doha, Qatar School of Computational Sciences, Korea Institute for Advanced Study, Seoul 130-012, Korea Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea Department of Physics and Institute of Quantum Studies,Texas A& M University, College Station, TX 77843, USA (Dated: November 8, 2018)We derive two classes of multi-mode Bell inequalities under local realistic assumptions, which areviolated only by the entangled states negative under partial transposition in accordance with thePeres conjecture. Remarkably, the failure of local realism can be manifested by exploiting waveand particle correlations of readily accessible continuous-variable states, with very large violation ofinequalities insensitive to detector-efficiency, which makes a strong case for a loophole-free test.
PACS numbers: 03.65.Ud, 03.65.Ta, 03.67.Mn, 42.50.Dv
Introduction —Whether there exists a strong correla-tion that no local realistic theories can grasp has been anissue of crucial importance since the Einstein-Podolsky-Rosen (EPR) argument [1], which was later cast into anexperimentally testable form by J. S. Bell [2]. The Belltest not only provides an opportunity to look into fun-damental aspects of quantum mechanics, but also can beused for practical applications in quantum informationscience, e.g., the security test for quantum cryptography[3] and the entanglement witness. Numerous experimen-tal data have been obtained to date in support of quan-tum mechanics, however, there still remain some impor-tant issues to resolve. First, no experiment ever closedboth the locality and the detector-efficiency loopholes toconclusively rule out local hidden variable (LHV) theo-ries [4, 5]. Second, although the original EPR argumentconsidered the correlation of continuous variables (posi-tion and momentum), almost all experiments were so farperformed for discrete variables (e.g. spin-1/2 states [6]).The Bell test using continuous variables (CVs) can pro-vide a new insight into quantum world via their enrichedstructure in infinite dimension. Furthermore, the CV Belltest is considered practically desirable for a loophole-freetest because the measurement scheme (homodyne detec-tion) is highly efficient. However, the proposals so farhave not been made to take the merits of CVs fully. Tobegin with, the EPR state (two-mode squeezed state) isnot adequate as such for the CV Bell test due to a non-negative distribution in phase space, admitting a LHVdescription [7]. It was thus suggested to exploit the cor-relation of discrete nature, photon-number parity [8] orpseudo-spin observables [9], which are hard to implementdue to inefficient photon counting. In order to utilizethe merit of homodyne detection, a different approach,i.e. transforming a nonnegative distribution to a non-positive one by photon subtraction, was proposed butthe violation of Bell inequality was very small [10]. Thissmall violation may be attributed to the binning pro-cess that converts CV data to binary ones; binning is used to adopt the Bell-inequalities typically establishedfor discrete variables [2], leading undesirably to the lossof information on CV correlation. Remarkably, Acin etal. found some CV states that maximally violate multi-mode Bell inequalities under the binning and proposeda 3-mode state for a loophole-free test, which however isnot very practicable in current technologies [11].Therefore, it is important both fundamentally andpractically to have Bell-inequalities that can directlyprobe CV correlations in full capacity [12]. Recently,such an inequality was derived by Cavalcanti et al. [13],however, its test appears demanding as it requires at least10-mode entangled states. Although the case was im-proved to use 5-mode states by optimizing the functionalform of the inequalities [14], it is necessary to obtain Bellinequalities that can reveal nonlocality for a broad classof CV entangled states including, desirably, easily acces-sible ones. In this Letter, we derive two classes of Bellinequalities by using the Cauchy inequality under localrealistic conditions. We show that these inequalities canbe violated only by the quantum entangled states thatare negative under partial transposition (NPT), in accor-dance with the Peres conjecture [15–17]. Remarkably, theviolation of our inequalities occurs at all levels of n -mode( n ≥ Bell inequality —We first show how a Bell inequalitycan be derived from LHV descriptions. Let r j be a realrandom variable at two parties j = 1 ,
2. The LHV theoryaccounts for the correlation of r and r by h r r i = Z dλρ ( λ ) r ( λ ) r ( λ ) , (1)where it is assumed that the local values r ( λ ) and r ( λ )are predetermined (realism) independent of each party(locality). The realistic values r ( λ ) and r ( λ ) can beidentified if the hidden variable λ , with the probabilisticdistribution ρ ( λ ), is revealed. This can be extended tocomplex variables C and C that essentially representtwo real random variables at each site, as h C C i = Z dλρ ( λ ) C ( λ ) C ( λ ) , (2)which refer to four correlations collectively [Cf. Eq. (9)].We impose no conditions on random variables that maybe bounded/unbounded and continuous/discrete.One can use the Cauchy inequality to obtain the upperbound of the correlation as |h C C i| = (cid:12)(cid:12)(cid:12)(cid:12)Z dλρ ( λ ) C p ( λ ) C q ( λ ) C − p ( λ ) C − q ( λ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z dλρ ( λ ) | C p ( λ ) C q ( λ ) | Z dλρ ( λ ) (cid:12)(cid:12)(cid:12) C − p ( λ ) C − q ( λ ) (cid:12)(cid:12)(cid:12) = h| C | p | C | q ih| C | − p ) | C | − q ) i , where the last line again follows in view of the LHVdescription in Eq. (1), with real numbers p, q ∈ [0 , |h C C i| ≤ h| C | p | C | q ih| C | − p ) | C | − q ) i . (3)If one starts with the complex conjugate C ∗ instead of C , another inequality similarly emerges, |h C C ∗ i| ≤ h| C | p | C | q ih| C | − p ) | C | − q ) i . (4)In (3) and (4), the correlation of C and C is boundedfrom above. We particularly note that considering C and C as complex amplitudes, the upper bound—theproduct of h| C | p | C | q i and h| C | − p ) | C | − q ) i —refers to the (fractional-order) “intensity” correlations.Now, we want to know if the inequalities (3–4) canbe violated by quantum systems. We first discuss howthe correlations in our inequalities can be experimentallytested. The left-hand side (LHS) of each inequality refersto the correlation of complex amplitudes C j ≡ C jx + iC jy ( j = 1 , C j ≡ ˆ C jx + i ˆ C jy ( ˆ C jx , ˆ C jy : Hermitian). On the other hand, the right-hand side (RHS) of each inequality refers to the intensitycorrelation. An intensity can generally be expressed intwo different forms, which, importantly, are not distin-guished from each other in classical descriptions. Firstis to represent a complex variable by its real and imag-inary parts, C ≡ C x + iC y , leading to | C | = C x + C y .Second is to represent the intensity as the product of theoriginal variable and its conjugate, | C | = C ∗ C . The for-mer indicates the correspondence to quantum operator as | C | . = ˆ C x + ˆ C y and the latter | C | . = ˆ C † ˆ C . Two unequal observables ( ˆ C x + ˆ C y and ˆ C † ˆ C ) in quan-tum domain usually carry distinguished physical con-texts. The distinction may be particularly related to thewave-particle duality in quantum optics, as addressed be-low. The classical LHV descriptions, however, disallowthe violation of inequalities (3–4) regardless of intensityobservables. Here we particularly focus on the second ap-proach, | C | . = ˆ C † ˆ C . For simplicity, let { p, q } = { , } ,and then, we have only two distinct cases.(i) p = q = 1: One class of inequalities follows from (4), (cid:12)(cid:12)(cid:12) h ˆ C ˆ C † i (cid:12)(cid:12)(cid:12) ≤ h ˆ C † ˆ C ˆ C † ˆ C i : 1st − inequality . (5)There is another inequality from (3), (cid:12)(cid:12)(cid:12) h ˆ C ˆ C i (cid:12)(cid:12)(cid:12) ≤h ˆ C † ˆ C ˆ C † ˆ C i , which is, however, never violated by anyquantum states as shown below.(ii) p = 0 , q = 1 : Another class follows from (3) (cid:12)(cid:12)(cid:12) h ˆ C ˆ C i (cid:12)(cid:12)(cid:12) ≤ h ˆ C † ˆ C ih ˆ C † ˆ C i : 2nd − inequality . (6)The other inequality from (4), (cid:12)(cid:12)(cid:12) h ˆ C ˆ C † i (cid:12)(cid:12)(cid:12) ≤h ˆ C † ˆ C ih ˆ C † ˆ C i , is never violated as shown below. Peres Conjecture —We now prove that only NPT en-tangled states can violate the Bell inequalities (5) and (6)regardless of ˆ C and ˆ C . First, note that for any operatorˆ f , the positive operator ˆ f † ˆ f must give h ˆ f † ˆ f i ≥ h ˆ f † ˆ f i PT ≥ D ˆ O A ˆ O B E ρ PT = D ˆ O A ˆ O †∗ B E ρ , (7)where ˆ O A and ˆ O B are operators acting on subsystems A and B , respectively, with PT taken for B [17]. The sym-bol ∗ denotes complex conjugation of matrix elements.First, taking ˆ f = a + b ˆ C ˆ C , the condition h ˆ f † ˆ f i ≥ a and b , which gives (cid:12)(cid:12)(cid:12) h ˆ C ˆ C i (cid:12)(cid:12)(cid:12) ≤ h ˆ C † ˆ C ˆ C † ˆ C i for all quantum states—therefore, no violation at all. In contrast, the PT condi-tion h ˆ f † ˆ f i PT ≥ f = a + b ˆ C ˆ C ∗ gives the inequal-ity (5). That is, if the Bell inequality (5) is violated, thestate must be NPT. Secondly, with ˆ f = a ˆ C + b ˆ C , thecondition h ˆ f † ˆ f i ≥ (cid:12)(cid:12)(cid:12) h ˆ C ˆ C † i (cid:12)(cid:12)(cid:12) ≤ h ˆ C † ˆ C ih ˆ C † ˆ C i for all quantum states. In contrast, its PT version withˆ f = a ˆ C + b ˆ C ∗ gives the inequality (6), so its violationagain confirms NPT entanglement. The Peres conjecture[15–17] that only NPT entanglement is incompatible withLHV descriptions is thus supported in our framework. CV case —Let us first apply the inequalities (5) and (6)to two-mode CV states. The simplest among all possibletests is to take ˆ C j = ˆ a j ( j = 1 , a j is the anni-hilation operator describing the field amplitude of mode j . It can be decomposed into two Hermitian operators,ˆ a j = ˆ X j + i ˆ Y j ( j = 1 , X j ≡ (ˆ a j + ˆ a † j ) andˆ Y j ≡ i (ˆ a j − ˆ a † j ) are two orthogonal quadrature ampli-tudes. Thus, to test the 1st-inequality (5), which reads |h ˆ a ˆ a † i| ≤ h ˆ N ˆ N i : 1st − inequality , (8)with ˆ N j = ˆ a † j ˆ a j ( j = 1 , |h ˆ a ˆ a † i| can be measured in 4 segments importantly by local homodyne measurements, |h ˆ a ˆ a † i| = (cid:16) h ˆ X ˆ X i + h ˆ Y ˆ Y i (cid:17) + (cid:16) h ˆ X ˆ Y i − h ˆ Y ˆ X i (cid:17) . (9)On the other hand, the intensity correlation h ˆ N ˆ N i canbe measured by photon counting at each mode. Thereare some broad classes of two-mode states that violate theinequality (8). The most practically feasible among themis the single-photon entangled state, | Ψ s i = cos θ | , i +sin θe − iφ | , i , giving |h ˆ a ˆ a † i| = sin θ and h ˆ N ˆ N i =0. In view of (8), the degree of violation can be measuredby the deviation of the ratio LHSRHS = |h ˆ a ˆ a † i| h ˆ N ˆ N i from unity,which becomes infinite in this case.We note that the same inequality (8) was derived alsoby Hillery and Zubairy [18], but within a distinct contextof separability condition along a different route for non-Gaussian entanglement [19, 20]. The proposed schemesto test the inequality (8) in [18, 19] considered collec-tive measurements from the SU(2) algebra, not localones as proposed here, thus unsuitable for nonlocalitytest. A closely-related Bell inequality was also derivedby Cavalcanti et al. in [13], where the intensity corre-lation h| C | | C | i in (4) was addressed by the squared-quadrature correlation, h ( ˆ X + ˆ Y )( ˆ X + ˆ Y ) i . Thus,they considered only the wave-like correlations in bothsides [21], and it is known that no violation occurs fortwo-mode states within their framework [15]. In contrast,our inequality incorporates two distinct aspects of fieldcorrelations in (8), the wave-like (LHS) and the particle-like (RHS) correlation. In this sense, our approach em-phasizes the role of the wave-particle dual aspects in man-ifesting the failure of local realism [22].Let us turn our attention to the 2nd-inequality (6).Again, by the substitution ˆ C j = ˆ a j ( j = 1 , |h ˆ a ˆ a i| ≤ h ˆ N ih ˆ N i : 2nd − inequality , (10)which also appeared as a separability condition in[18]. In this case, the optical EPR state (two-mode squeezed state), | TMSS i = e r ( a † a † − a a ) | , i = P ∞ n =0 tanh n r cosh r | n, n i , violates the inequality regardless of r (degree of squeezing). The degree of violation measuredby |h a a i| h a † a ih a † a i − − r − r >
0) increases with r decreasing and becomes extremely large as r → Loophole-free test —Let us now address how our testscan avoid both the locality and the detector-efficiencyloopholes. First, to enforce a strict locality condition, arandom-number generator yielding R = 0 , , and 2 canbe used at each observer to choose local measurementsettings, similar to the method of [4]. In the balancedhomodyne detection, the local oscillator (LO) with ad-justable phase is mixed with a signal at a 50:50 beam-splitter (BS). In our case, an electronic attenuator can beput between the LO and the BS to reduce/unblock theLO. For R = 0 ( R = 1) case, the LO phase is adjustedto X ( Y ) quadrature with the attenuator off (homodynedetection). For R = 2, the attenuator turns on to reducethe LO, measuring the signal intensity (photon counting)as below. If R is randomly generated at the last instantwhen the signal impinges on the BS, the time-like com-munication between two observers can be ruled out [4].Second, we consider a full LHV model including allnon-detection events to address the detection-loopholeissue. For the case that the real or the imaginary partof C j ≡ C jx + iC jy ( j = 1 ,
2) is undetected, one mayassign a fixed value 0 to such events and the inequal-ity (3) still holds. The intensities of the RHS can be de-composed as h| C j | i = p j,D h| C j | i D + (1 − p j,D ) h| C j | i U for the 2nd-inequality (6), where h| C j | i D,U denotesthe intensity average for detected/undetected ensembles,with p j,D the detection probability in photon count-ing. A consequence of nonideal efficiency η < h| C j | i U ≤ h| C j | i D , which can be proved within clas-sical description. In turn, it gives h| C j | i U ≤ h| C j | i ,where h| C j | i = h C jx + C jy i is the total intensity aver-age that can be alternatively measured via homodynedetection. Therefore, a full LHV inequality leads to |h ˆ a ˆ a i| ≤ Q j =1 , h p j,D h ˆ N j i D + (1 − p j,D ) h ˆ X j + ˆ Y j i i .To enhance the detection probability p j,D , one may mixthe signal with LO (amplitude ∼ β ) at a beam splitterand measure the intensity sum S of two outputs. In eachevent, the signal intensity is assigned the value S − h I i LO where h I i LO is the LO intensity average that can be sep-arately measured. Our LHV inequalities are still validwith the LO field included as another (predetermined)random variable, and p j,D rapidly approaches 1 by in-creasing β for any η and squeezing r . In this case, thecontribution of the second term h ˆ X j + ˆ Y j i is negligible.The photodetection with efficiency η is practicallyequivalent to the ideal detection after the signal ˆ a j ismixed with a vacuum ˆ v j at a beam-splitter of trans-missivity √ η . Namely, the observed signal ˆ a oj is ex-pressed by ˆ a oj = √ η ˆ a j + √ − η ˆ v j ( j = 1 , h ˆ N o ˆ N o i = η h ˆ N ˆ N i and h ˆ N o ih ˆ N o i = η h ˆ N ih ˆ N i inthe above-mentioned intensity measurement. In the bal-anced homodyne detection to measure quadrature ampli-tudes ˆ X j and ˆ Y j at each mode, the same model applies,ˆ a oj = √ η ˆ a j + √ − η ˆ v j , in the limit of large-intensitylocal oscillator [23]. This gives |h ˆ a o ˆ a o i| = η |h ˆ a ˆ a i| and |h ˆ a o ˆ a † o i| = η |h ˆ a ˆ a † i| . Therefore, η becomes anoverall factor in both sides of (8) and (10), which makesour scheme insensitive to detector efficiency. Multipartite systems —The inequalities (5) and (6) canbe further generalized to N -partite systems as (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)* k Y i =1 ˆ C i N Y j = k +1 ˆ C † j +(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ * N Y i =1 ˆ C † i ˆ C i + : 1st , (11) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)* N Y i =1 ˆ C i +(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ * k Y i =1 ˆ C † i ˆ C i + * N Y j = k +1 ˆ C † j ˆ C j + : 2nd , (12)where N -parties are divided into two groups of k - and N − k modes ( k = 1 , . . . , N − f = A + B Q ki =1 ˆ C i Q Nj = k +1 ˆ C ∗ j and ˆ f = A Q ki =1 ˆ C i + B Q Nj = k +1 ˆ C ∗ j , respectively, forthe condition h ˆ f † ˆ f i PT ≥
0. The N -mode GHZ statewith a mixture of vacuum, ρ = p S | GHZ ih GHZ | + (1 − p S ) | · · · ih · · · | , where | GHZ i = c |{ } k , { } N − k i + c |{ } k , { } N − k i is the superposition of k modes ( N − k modes) all occupying one (no) photon and vice versa, vi-olates the inequality (11). The multimode EPR state,produced by injecting single-mode squeezed states into aseries of beam splitters [24], violates the inequality (12).The violation can occur with any partition numbers( k = 1 , . . . , N − n -mode CV states for n ≥
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