Tests of Bell Inequality with Arbitrarily Low Photodetection Efficiency and Homodyne Measurements

Tests of Bell inequality with arbitrarily low photodetection efﬁciency and homodyne measurements

Mateus Araújo,1 Marco Túlio Quintino,1 Daniel Cavalcanti,2 Marcelo França Santos,1 Adán Cabello,3, 4 and Marcelo Terra Cunha5 1Departamento de Física, Universidade Federal de Minas Gerais, Caixa Postal 702, 30123-970 Belo Horizonte, MG, Brazil 2Centre for Quantum Technologies, National university of Singapore, 3 Science drive, Singapore 117543 3Departamento de Física Aplicada II, Universidad de Sevilla, E-41012 Sevilla, Spain 4Department of Physics, Stockholm University, S-10691 Stockholm, Sweden 5Departamento de Matemática, Universidade Federal de Minas Gerais, Caixa Postal 702, 30123-970 Belo Horizonte, MG, Brazil (Dated: November 5, 2018) We show that hybrid local measurements combining homodyne measurements and photodetection provide violations of a Bell inequality with arbitrarily low photodetection efficiency. This is shown in two different scenarios: when one part receives an atom entangled to the field mode to be measured by the other part and when both parts make similar photonic measurements. Our find- ings promote the hybrid measurement scenario as a candidate for loophole-free Bell tests beyond previous expectations.

PACS numbers: 03.65.Ud,03.67.Mn

I. INTRODUCTION with efficiency η0 = 1 and the other with efficiency η1, the threshold is η1 > 0 [17]. Albeit very alluring, this One of the greatest discoveries of modern science is last scenario lacks (until now) a physical system with that quantum mechanics predicts nonlocal correlations which to be implemented. [1]. Experiments have shown a good agreement with To overcome the detection loophole with photons, theory [2–8], but all of them have loopholes that prevent an alternative strategy is to use highly efficient homo- the definitive conclusion that nature is intrinsically non- dyne measurements (quadrature variables). However, local and allows for secure communication [9, 10] and to date, only small Bell violations were predicted using genuine randomness [11]. The “ultimate test of quan- feasible states and homodyne measurements [18, 19]. tum mechanics” [12], the loophole-free Bell experiment, Recently, Ji et al. [20] (see also [21]) introduced a promis- is still pending. ing idea: they proposed combining homodyne mea- Two requirements have never been simultaneously surements (quadrature variables) of high quantum ef- satisfied in a Bell test: spacelike separation between ficiency with photon number measurements. each observer’s measurement choice and the other ob- It has been shown that this new tool indeed allows for server’s measurement outcome [3, 5, 8], and overall de- Bell tests requiring lower threshold photodetection effi- tection efficiency η (defined as the ratio between de- ciencies in the photon-photon scenario: η > 0.71 with tected and emitted particles) beyond a threshold value feasible states [22] and even η > 0.29 for more general that rules out local models [6, 7]. Spacelike separation states [23]. The atom-photon case has also been recently is typically overcome by using photons as information discussed [24]. However, in practical terms, the advan- arXiv:1112.1719v3 [quant-ph] 5 Sep 2012 carriers. However, this makes it difficult to obtain η tage when comparing to previous proposals is, so far, higher than the threshold, as photodetection is usually too small to stimulate Bell experiments exploiting hy- inefficient. brid local measurements. For the Clauser-Horne-Shimony-Holt (CHSH) [13] In this paper, we show that the hybrid measurement inequality, three different thresholds are known, cor- scenario can be used to implement a variant of the responding to three different scenarios: (i) If we as- scheme proposed by Garbarino [17], and therefore al- sume that all measurements are made with efficiency lows a violation of the CHSH inequality even when the η, as is usual when, e.g., the qubits are the polarization photodetectors of Alice and Bob have an arbitrarily low of photons, this threshold is η > 2/3 [14]. (ii) If we efficiency. To achieve this result, we need to assume assume that Alice’s measurements are made with effi- ideal homodyne detection and use some states which ciency ηA = 1 and Bob’s with efficiency ηB, a condition are difficult to prepare (see, however, [25]). This find- that is close to what can be achieved in atom-photon ing promotes hybrid local measurements as a candidate experiments, the threshold is ηB > 1/2 [15, 16]. (iii) for photonic loophole-free Bell tests beyond previous If both Alice and Bob measure one of their observables expectations. 2

We discuss two experimental setups. In the first sce- used for low intensity fields will be well represented, in nario, an entangled atom-light state is prepared. The the Fock basis, as electronic levels of the atom encode a qubit that can ∞ be very efficiently detected [26] and that remains with D := 0 0 ∑ n n .(3) Alice, while the light field is sent to Bob who chooses | ih | − n=1| ih | between highly efficient homodyning [27] or inefficient photodetection. Bob’s homodyne measurement is di- The quadrature measurement gives an infinite number of outcomes x R. So, to use the CHSH inequality we chotomized in order to fit the CHSH inequality, as shall ∈ be explained later. In this case, three out of the four pos- need to dichotomize them, i.e., to use a binning process. sible measurements are very efficient and we calculate The most general binning is that we output the value +1 if the X measurement returns x A+, where A+ is the threshold for Bob’s inefficient photodetection. In ∈ the second setup, both Alice and Bob use photonic sys- a subset of the real numbers. We output the value 1 if + − x A = R A . tems and perform homodyning (assumed to be nearly ∈ − \ perfect) or photodetection (efficiency threshold to be Introducing the projectors calculated). Z PQ := x x dx,(4) ± A± | ih | II. CHSH OBSERVABLES the dichotomized version of the X measurement can be written as Consider the quantum operator corresponding to the Q = P + P ,(5a) CHSH inequality Q − Q− := A (B + B ) + A (B B ),(1) and its matrix representation in the Fock basis will be B 0 ⊗ 0 1 1 ⊗ 0 − 1 = where A and B are observables with eigenvalues 1. m Q n m PQ+ n m PQ n i j ± h | | i Z − − When dealing with atomic systems we will consider = 2 ϕm(x)ϕn(x) dx δmn,(5b) that perfect Pauli measurements can be made. These A+ − measurements will be parametrized as where ϕn(x) = x n is the nth Hermite function. To find the adequateh | i two-qubit subspace, first note V(γ) := cos γ σz + sin γ σx.(2) that since Q2 = I, the subspace spanned by 0 , Q 0 {| i | i} The treatment of the photonic observables will be a is invariant under Q, and we can therefore write bit more involved, since they are the ones which are Q 0 = cos θ 0 + sin θ Ξ ,(6a) involved in the considerations of detection efficiency. | i | i | i More specifically, we want to use them to implement where the scheme from [17] that achieves arbitrarily low detection efficiency. However, this scheme is not appli- 2 ∞ Z Ξ := ϕ0(x)ϕn(x) dx n ,(6b) cable to the hybrid measurement scenario for two rea- ∑ + | i sin θ n=1 A | i sons: our quantum system – the occupation number of photons – is infinite-dimensional, while the scheme is and described for two qubits, and its definition of detection Z 2 efficiency – as the probability of detecting a particle – cos θ := 2 ϕ0(x) dx 1. (6c) + − is not applicable, since one of our measurements is pre- A cisely the presence or absence of a photon. A transla- Since both 0 and Ξ are eigenvectors of D, the restriction work is therefore required. tion of the| observablesi | i Q and D to this subspace still To begin, we shall use the results of [23], which has eigenvalues 1, and therefore it allows us to reach ± showed that it is possible to reach the Tsirelson bound the maximal violation 2√2. of 2√2 in the hybrid measurement scenario with states Also note that these restricted observables can be restricted to a two-qubit subspace. We shall see that written in the orthonormal basis 0 , Ξ in a simple precisely this same subspace allows the violation of way, namely {| i | i} the CHSH inequality with arbitrarily low detection efficiency. QR = cos θ σz + sin θ σx,(7a) Following [23], we define our photodetection and ho- DR =σz.(7b) modyning observables as follows: the output of our photodetection observable is defined as 1 in the case Now we shall consider the effects of the main error of a click and +1 when there is no click. Photodetectors− sources in this set up. First, let us suppose that the 3 photodetector has probability η of detecting a photon. with the expectation In this case, it is usual to model the imperfect photode- q 2 2 2 tection as an ideal detector preceded by an amplitude AΞ AΞ = 2H cos γ + 2 sin γ + (1 H) cos γ. h |BA| i − damping channel [28]. Thus we write (13) This is larger than 2 if η > 0 and γ 0, π , as can ∞ h i 2 k be seen from simple algebra. That is,∈ we have a viola- Dη = 0 0 + ∑ 2(1 η) 1 k k .(8) | ih | k=1 − − | ih | tion of a Bell inequality with arbitrarily low photodetec- Since we shall work with the restricted operators, we tion efficiency for any choice of noncommuting atomic observables. need to get the effects of η on DR. With the aid of the function To understand the effects of transmittance, we set η = 1 to find the critical value of t above which vio-

H : [0, 1] [0, 1]; H(η) := (1 Ξ Dη Ξ )/2, (9a) lations can be found. We did a numerical search, again → − we conveniently write restricting ourselves to states in the subspace spanned by g , s 0 , Ξ . We found out that the criti- D = H(η)D + [1 H(η)]1.(9b) {| i | i} ⊗ {| i | i} Rη R − cal transmittance depends on the binning choice, and + = Another source of errors is the transmittance t 1 the best value we found was for the binning A 1 1 ≤ [ erf− 1/2, erf− 1/2], where erf is the error function. which affects both measurements: photodetection and − Then violations are obtained for t 0.55. Although the homodyning. It can also be modeled by an amplitude ≥ state A does serve as a proof of principle for the vio- damping channel. For the photodetection observable | Ξi the transmittance and efficiency effects can be com- lation of a Bell inequality with arbitrarily low efficiency, bined and we have it is not practical for a real experiment due to the un- physical nature of Ξ : for instance, the mean number D = H(ηt)D + [1 H(ηt)]1.(10) | i Rηt R − of photons Ξ N Ξ diverges. To avoid this problem, we could just truncateh | | i Ξ to any finite dimension, and ap- Unfortunately, the homodyning observable with trans- | i mittance t does not admit such a simple representation. proximate its properties arbitrarily well. However, this With this modeling of the detection efficiency, we truncated state would still be very hard to produce in a now have effective observables that are formally equal real experiment. It is physically sound to replace Ξ with a more fa- to two-dimensional observables, and therefore the anal- | i ysis done in [17] applies. miliar state. A good candidate is the so-called even cat state [29]: α + α III. ATOM-PHOTON SCENARIO cat := |qi |− i .(14) | i 2 α 2 √2 1 + e− | | In this scenario Alice measures atomic observables, The main reason behind this choice is that its fidelity described as Pauli matrices, and Bob measures pho- with Ξ can be made very high. Doing so, the criti- todetection and X quadrature. We set the effective cal efficiency| i will no longer be arbitrarily low, but will CHSH operator as still be very low. With this replacement, the state A | Ξi := V(γ) (D + Q ) + V( γ) (D Q ),(11) becomes BA ⊗ ηt t − ⊗ ηt − t where the observables were defined in sectionII. Acat = cos ν g 0 + sin ν s cat ,(15) | i | i | i As argued in the previous section we will restrict our where we shall optimize α and ν for each η and t, since attention to states belonging to the four-dimensional the functions defined in AΞ do not give the optimal subspace spanned by g , s 0 , Ξ , where g | i {| i | i} ⊗ {| i | i} | i result for this state. Figure 1 shows the boundary of the and s are two atomic levels. Also, we assume ideal region of parameters t and η for which a CHSH viola- transmittance| i (t = 1) and, for simplicity (since it + tion is observed for these states. The maximal CHSH is sufficient), only consider binnings A such that value for the state (15) is 2.60, reached when α 2.20i, R ( )2 = = ≈ A+ ϕ0 x dx 1/2, where QR σx. for η = 1 = t. In this case, the Bell operator AR is represented by a 4 4 matrix that can be easily diagonalizedB to find the × eigenstate of the maximal singular value IV. PHOTON-PHOTON SCENARIO

A ∝ | Ξi In this scenario both Alice and Bob measure photode- q (1 H) cos γ + sin2 γ + (1 H)2 cos2 γ g 0 tection and X quadrature. Now, the effective CHSH op- − − | i erator is + sin γ s Ξ ,(12) := D (D + Q ) + Q (D Q ),(16) | i BP ηt ⊗ ηt t t ⊗ ηt − t 4

1 1 A P | Ξi | Ξi A P 0.9 | cati | Hi 0.95 0.8 t t

0.7 0.9

0.6 0.85 0.5 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 η η

Figure 1. (Color online) Critical line on which = 2, con- Figure 2. (Color online) Critical line on which = 2, con- BA BP sidering the efficiency η of the photodetectorsh andi the trans- sidering the efficiency η of the photodetectorsh andi the transmittance t between the source and the photodetectors. For mittance t between the source and the photodetectors. A violations can be obtained above the critical efficiency | cati 0.066, for which α 2.29i, and above the critical transmittance ≈ 0.52, reached for α 2.87i. For AΞ the critical efficiency ≈ ≈ | i have also maximized PH P PH and found that the is zero, while the critical transmittance is 0.55. h |B | i ≈ binning choice that reaches the largest violation obeys cos θ = (√5 1)/2. 2 − with the observables defined in sectionII. Note that now the family of operators depends only on the binning choice, representedB by the parameter θ, in contrast V. CONCLUSIONS with the previous case where we could also optimize over the choice of the atomic observables, given by the γ parameter. We have discussed atom-photon and photon-photon As before, let us restrict ourselves to a four- Bell tests combining homodyne measurements and dimensional subspace, this time the one generated by photodetection. We have shown that, in both cases, a 2 Bell inequality can be violated even if the photodetec- 0 , Ξ ⊗ , and call PR the restriction of P to this {| i | i} B B tors used are arbitrarily inefficient, by assuming perfect subspace. By diagonalizing PR we can find a state B detection efficiency on the other observable. PΞ that has a violation for every η > 0. This state has a| complicatedi form and only provides a Bell violation The photon-photon scenario considered here mim- for t > 0.84. In Fig. 2, we present the region of param- ics, from the point of view of detection efficiencies, the eters for which a violation of the CHSH inequality can scheme studied in [17]. However, in our schemes the be found using this state. detection efficiencies play a different role in the Bell test. On the other hand, irrespective of optimizations, we While in [17], η is related to the probability of a third can take inspiration on Hardy’s paradox to find a sim- “no-click” outcome, in our scenarios it is related to an ple family of states that violates CHSH for all η > 0: attenuation process that cannot be directly noticed by the detectors. 1 h θ θ i Under the present results, we see that highly efficient PH := q cos ++ + sin ( + + + ) , | i 2 θ 2 | i 2 | −i |− i photodetectors are not required to achieve a loophole- 1 + sin 2 (17) free Bell test: indeed, the real problem is the trans- where are the eigenstates of Q . The expectation mission. Note that schemes to circumvent this prob- ± R value for| ti = 1 is given by lem have been recently proposed [30–33]. Also notice that under our scheme it is crucial that the propagat- 2 θ 4 θ 2 4 cos 2 sin 2 ing modes are very well matched to the modes to be PH P PH = 2 + H ,(18) h |B | i 1 + sin2 θ detected. Although we do not present here an exper- 2 imental recipe to produce the quantum states consid- and is easily seen to be larger than 2 for every η > 0 and ered (see [25]), we hope that the present fidings can nontrivial choice of θ. The critical transmittance of these guide future research towards feasible proposals within states is 0.92, and is reached in the limit θ 0. We the present Bell scenario. ≈ → 5

ACKNOWLEDGMENTS Research Foundation and the Ministry of Education of Singapore, the Projects No. FIS2008-05596 and No. FIS2011-29400 (Spain), and the Wenner-Gren Founda- This work was supported by the Brazilian agencies tion. D.C. is visiting UFMG under the program PVE- Fapemig, Capes, CNPq, and INCT-IQ, the National CAPES (Brazil).

[1] J. S. Bell, “On the Einstein-Poldolsky-Rosen paradox,” Experiment,” Phys. Rev. Lett. 98, 220402 (2007). Physics 1, 195–200 (1964). [16] N. Brunner, N. Gisin, V. Scarani, and C. Simon, [2] S. J. Freedman and J. F. Clauser, “Experimental Test of “Detection Loophole in Asymmetric Bell Experiments,” Local Hidden-Variable Theories,” Phys. Rev. Lett. 28, Phys. Rev. Lett. 98, 220403 (2007). 938–941 (1972). [17] G. Garbarino, “Minimum detection efficiencies for a [3] A. Aspect, J. Dalibard, and G. Roger, “Experimental Test loophole-free observable-asymmetric Bell-type test,” of Bell’s Inequalities Using Time-Varying Analyzers,” Phys. Rev. A 81, 032106 (2010). Phys. Rev. Lett. 49, 1804–1807 (1982). [18] R. García-Patrón, J. Fiurášek, N. J. Cerf, J. Wenger, [4] P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. R. Tualle-Brouri, and P. Grangier, “Proposal for a Sergienko, and Y. Shih, “New High-Intensity Source of Loophole-Free Bell Test Using Homodyne Detection,” Polarization-Entangled Photon Pairs,” Phys. Rev. Lett. 75, Phys. Rev. Lett. 93, 130409 (2004). 4337–4341 (1995). [19] H. Nha and H. J. Carmichael, “Proposed Test of [5] G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and Quantum Nonlocality for Continuous Variables,” Phys. A. Zeilinger, “Violation of Bell’s Inequality under Strict Rev. Lett. 93, 020401 (2004). Einstein Locality Conditions,” Phys. Rev. Lett. 81, [20] S.-W. Ji, J. Kim, H.-W. Lee, M. S. Zubairy, and H. Nha, 5039–5043 (1998). “Loophole-Free Bell Test for Continuous Variables via [6] M. A. Rowe, D. Kielpinski, V. Meyer, C. A. Sackett, Wave and Particle Correlations,” Phys. Rev. Lett. 105, W. M. Itano, C. Monroe, and D. J. Wineland, 170404 (2010). “Experimental violation of a Bell’s inequality with [21] D. Cavalcanti and V. Scarani, “Comment on efficient detection,” Nature 409, 791 (2001). “Loophole-Free Bell Test for Continuous Variables via [7] D. N. Matsukevich, P. Maunz, D. L. Moehring, Wave and Particle Correlations”,” Phys. Rev. Lett. 106, S. Olmschenk, and C. Monroe, “Bell Inequality Violation 208901 (2011). with Two Remote Atomic Qubits,” Phys. Rev. Lett. 100, [22] D. Cavalcanti, N. Brunner, P. Skrzypczyk, A. Salles, and 150404 (2008). V. Scarani, “Large violation of Bell inequalities using [8] T. Scheidl, R. Ursin, J. Kofler, S. Ramelow, X.-S. Ma, both particle and wave measurements,” Phys. Rev. A 84, T. Herbst, L. Ratschbacher, A. Fedrizzi, N. K. Langford, 022105 (2011). T. Jennewein, and A. Zeilinger, “Violation of local [23] M. T. Quintino, M. Araújo, D. Cavalcanti, M. França realism with freedom of choice,” PNAS 107, Santos, and M. Terra Cunha, “Maximal violations and 19708–19713 (2010). efficiency requirements for Bell tests with [9] J. Barrett, L. Hardy, and A. Kent, “No Signaling and photodetection and homodyne measurements,” J. Phys. Quantum Key Distribution,” Phys. Rev. Lett. 95, 010503 A: Math. Theor. 45, 215308 (2012). (2005). [24] N. Sangouard, J. D. Bancal, N. Gisin, W. Rosenfeld, [10] A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and P. Sekatski, M. Weber, and H. Weinfurter, “Large V. Scarani, “Device-Independent Security of Quantum violation of Bell inequalities using both particle and Cryptography against Collective Attacks,” Phys. Rev. wave measurements,” Phys. Rev. A 84, 052122 (2011). Lett. 98, 230501 (2007). [25] C. Teo, M. Araújo, M. T. Quintino, J. Mináˇr, [11] S. Pironio, A. Acín, S. Massar, A. Boyer de la Giroday, D. Cavalcanti, V. Scarani, M. Terra Cunha, and D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, M. França Santos, “Realistic loophole-free Bell test with L. Luo, T. A. Manning, and C. Monroe, “Random atom-photon entanglement,” arXiv:1206.0074 numbers certified by Bell’s theorem,” Nature 464, 1021 [quant-ph]. (2010). [26] F. Henkel, M. Krug, J. Hofmann, W. Rosenfeld, [12] Z. Merali, “Quantum Mechanics Braces for the Ultimate M. Weber, and H. Weinfurter, “Highly Efficient Test,” Science 331, 1380–1382 (2011). State-Selective Submicrosecond Photoionization [13] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Detection of Single Atoms,” Phys. Rev. Lett. 105, 253001 “Proposed Experiment to Test Local Hidden-Variable (2010). Theories,” Phys. Rev. Lett. 23, 880–884 (1969). [27] A. Ourjoumtsev, R. Tualle-Brouri, and P. Grangier, [14] J.-Å. Larsson and J. Semitecolos, “Strict “Quantum Homodyne Tomography of a Two-Photon detector-efficiency bounds for n-site Clauser-Horne Fock State,” Phys. Rev. Lett. 96, 213601 (2006). inequalities,” Phys. Rev. A 63, 022117 (2001). [28] L. Mandel and E. Wolf, Optical Coherence and Quantum [15] A. Cabello and J.-Å. Larsson, “Minimum Detection Optics. Cambridge University Press, 1995. Efficiency for a Loophole-Free Atom-Photon Bell [29] H. Jeong and T. Ralph, Quantum Information with 6

Continuous Variables of Atoms and Light, ch. Schrödinger for practical device-independent quantum key Cat States for Quantum Information Processing. distribution,” Phys. Rev. A 84, 010304 (2011). Imperial College Press, Sept., 2007. quant-ph/0509137. [32] J. B. Brask, N. Brunner, D. Cavalcanti, and A. Leverrier, [30] N. Gisin, S. Pironio, and N. Sangouard, “Proposal for “Bell tests for continuous-variable systems using hybrid Implementing Device-Independent Quantum Key measurements and heralded amplifiers,” Phys. Rev. A 85, Distribution Based on a Heralded Qubit Amplifier,” 042116 (2012). Phys. Rev. Lett. 105, 070501 (2010). [33] A. Cabello and F. Sciarrino, “Loophole-Free Bell Test [31] M. Curty and T. Moroder, “Heralded-qubit amplifiers Based on Local Precertification of Photon’s Presence,” Phys. Rev. X 2, 021010 (2012).