Abstract

Leggett-Garg inequalities are tests of macroscopic realism that can be violated by quantum mechanics. In this letter, we realise photonic Leggett-Garg tests on a three-level system and implement measurements that admit three distinct measurement outcomes, rather than the usual two. In this way we obtain violations of three- and four-time Leggett-Garg inequalities that are significantly in excess of those obtainable in standard Leggett-Garg tests. We also report violations the quantum-witness equality up to the maximum permitted for a three-outcome measurement. Our results highlight differences between spatial and temporal correlations in quantum mechanics.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2017 (4)

H. Katiyar, A. Brodutch, D. Lu, and R. Laflamme, “Experimental violation of the Leggett-Garg inequality in a 3-level system,” New J. Phys. 19, 023033 (2017).
[Crossref]

L. Xiao, X. Zhan, Z. H. Bian, K. K. Wang, X. Zhang, X. P. Wang, J. Li, K. Mochizuki, D. Kim, N. Kawakami, W. Yi, H. Obuse, B. C. Sanders, and P. Xue, “Observation of topological edge states in parity-time-symmetric quantum walks,” Nat. Phys. 13, 1117 (2017).
[Crossref]

X. Zhan, E. G. Cavalcanti, J. Li, Z. H. Bian, Y. S. Zhang, H. M. Wiseman, and P. Xue, “Experimental generalized contextuality with single-photon qubits,” Optica 4, 966 (2017).
[Crossref]

L. Xiao, K. K. Wang, X. Zhan, Z. H. Bian, J. Li, Y. S. Zhang, P. Xue, and A. K. Pati, “Experimental test of uncertainty relations for general unitary operators,” Opt. Exp. 25, 17904 (2017).
[Crossref]

2016 (8)

X. Zhan, X. Zhang, J. Li, Y. S. Zhang, B. C. Sanders, and P. Xue, “Realization of the contextuality-nonlocality tradeoff with a qubit-qutrit photon pair,” Phys. Rev. Lett. 116, 090401 (2016).
[Crossref] [PubMed]

K. Wang, X. Zhan, Z. Bian, J. Li, Y. Zhang, and P. Xue, “Experimental investigation of the stronger uncertainty relations for all incompatible observables,” Phys. Rev. A 93, 052108 (2016).
[Crossref]

J. J. Halliwell, “Leggett-Garg inequalities and no-signaling in time: A quasiprobability approach,” Phys. Rev. A 93, 022123 (2016).
[Crossref]

N. Lambert, K. Debnath, A. F. Kockum, G. C. Knee, W. J. Munro, and F. Nori, “Leggett-Garg inequality violations with a large ensemble of qubits,” Phys. Rev. A 94, 012105 (2016).
[Crossref]

L. Clemente and J. Kofler, “No fine theorem for macrorealism: Limitations of the Leggett-Garg inequality,” Phys. Rev. Lett. 116, 150401 (2016).
[Crossref] [PubMed]

G. C. Knee, K. Kakuyanagi, M.-C. Yeh, Y. Matsuzaki, H. Toida, H. Yamaguchi, S. Saito, A. J. Leggett, and W. J. Munro, “A strict experimental test of macroscopic realism in a superconducting flux qubit,” Nat. Commun. 7, 13253 (2016).
[Crossref] [PubMed]

J. A. Formaggio, D. I. Kaiser, M. M. Murskyj, and T. E. Weiss, “Violation of the Leggett-Garg inequality in neutrino oscillations,” Phys. Rev. Lett. 117, 050402 (2016).
[Crossref] [PubMed]

M. Navascués, S. Pironio, and A. Acín, “A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations,” New J. Phys. 10, 073013 (2016).
[Crossref]

2015 (10)

G. Schild and C. Emary, “Maximum violations of the quantum-witness equality,” Phys. Rev. A 92, 032101 (2015).
[Crossref]

P. Xue, R. Zhang, H. Qin, X. Zhan, Z. H. Bian, J. Li, and B. C. Sanders, “Experimental quantum-walk revival with a time-dependent coin,” Phys. Rev. Lett. 114, 140502 (2015).
[Crossref] [PubMed]

Z. H. Bian, J. Li, H. Qin, X. Zhan, R. Zhang, B. C. Sanders, and P. Xue, “Realization of single-qubit positive-operator-valued measurement via a one-dimensional photonic quantum walk,” Phys. Rev. Lett. 114, 203602 (2015).
[Crossref] [PubMed]

H. S. Poh, S. K. Joshi, A. Cerè, A. Cabello, and C. Kurtsiefer, “Approaching Tsirelson’s bound in a photon pair experiment,” Phys. Rev. Lett. 115, 180408 (2015).
[Crossref]

C. Robens, W. Alt, D. Meschede, C. Emary, and A. Alberti, “Ideal negative measurements in quantum walks disprove theories based on classical trajectories,” Phys. Rev. X 5, 011003 (2015).

M. Giustina, M. A. M. Versteegh, S. Wengerowsky, J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J.-Å. Larsson, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, J. Beyer, T. Gerrits, A. E. Lita, L. K. Shalm, S. W. Nam, T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger, “Significant-loophole-free test of Bell’s theorem with entangled photons,” Phys. Rev. Lett. 115, 250401 (2015).
[Crossref]

L. K. Shalm, E. Meyer-Scott, B. G. Christensen, P. Bierhorst, M. A. Wayne, M. J. Stevens, T. Gerrits, S. Glancy, D. R. Hamel, M. S. Allman, K. J. Coakley, S. D. Dyer, C. Hodge, A. E. Lita, V. B. Verma, C. Lambrocco, E. Tortorici, A. L. Migdall, Y. Zhang, D. R. Kumor, W. H. Farr, F. Marsili, M. D. Shaw, J. A. Stern, C. Abellán, W. Amaya, V. Pruneri, T. Jennewein, M. W. Mitchell, P. G. Kwiat, J. C. Bienfang, R. P. Mirin, E. Knill, and S. W. Nam, “Strong loophole-free test of local realism,” Phys. Rev. Lett. 115, 250402 (2015).
[Crossref]

C. Emary, N. Lambert, and F. Nori, “Leggett-Garg inequalities,” Rep. Prog. Phys. 77, 016001 (2015).
[Crossref]

Z.-Q. Zhou, S. F. Huelga, C.-F. Li, and G.-C. Guo, “Experimental detection of quantum coherent evolution through the violation of Leggett-Garg-type inequalities,” Phys. Rev. Lett. 115, 113002 (2015).
[Crossref] [PubMed]

X. Zhan, J. Li, H. Qin, Z. H. Bian, and P. Xue, “Linear optical demonstration of quantum speed-up with a single qudit,” Opt. Exp. 23, 18422 (2015).
[Crossref]

2014 (2)

A. Asadian, C. Brukner, and P. Rabl, “Probing macroscopic realism via Ramsey correlation measurements,” Phys. Rev. Lett. 112, 190402 (2014).
[Crossref] [PubMed]

C. Budroni and C. Emary, “Temporal quantum correlations and Leggett-Garg inequalities in multilevel systems,” Phys. Rev. Lett. 113, 050401 (2014).
[Crossref] [PubMed]

2013 (5)

C. Budroni, T. Moroder, M. Kleinmann, and O. Gühne, “Bounding temporal quantum correlations,” Phys. Rev. Lett. 111, 020403 (2013).
[Crossref] [PubMed]

R. E. George, L. M. Robledo, O. J. E. Maroney, M. S. Blok, H. Bernien, M. L. Markham, D. J. Twitchen, J. J. L. Morton, G. A. D. Briggs, and R. Hanson, “Opening up three quantum boxes causes classically undetectable wavefunction collapse,” Proc. Natl. Acad. Sci. 110, 3777 (2013).
[Crossref] [PubMed]

J. P. Groen, D. Ristè, L. Tornberg, J. Cramer, P. C. de Groot, T. Picot, G. Johansson, and L. DiCarlo, “Partial-measurement backaction and nonclassical weak values in a superconducting circuit,” Phys. Rev. Lett. 111, 090506 (2013).
[Crossref] [PubMed]

H. Katiyar, A. Shukla, K. R. K. Rao, and T. S. Mahesh, “Violation of entropic Leggett-Garg inequality in nuclear spins,” Phys. Rev. A 87, 052102 (2013).
[Crossref]

J. Kofler and Č. Brukner, “Condition for macroscopic realism beyond the Leggett-Garg inequalities,” Phys. Rev. A 87, 052115 (2013).
[Crossref]

2012 (6)

C. Emary, N. Lambert, and F. Nori, “Leggett-Garg inequality in electron interferometers,” Phys. Rev. B 86, 235447 (2012).
[Crossref]

M. Hendrych, R. Gallego, M. Mičuda, N. Brunner, A. Acín, and J. P. Torres, “Experimental estimation of the dimension of classical and quantum systems,” Nat. Phys. 8, 588 (2012).
[Crossref]

J. Ahrens, P. Badzia, A. Cabello, and M. Bourennane, “Experimental device-independent tests of classical and quantum dimensions,” Nat. Phys. 8, 592 (2012).
[Crossref]

A. Montina, “Dynamics of a qubit as a classical stochastic process with time-correlated noise: Minimal measurement invasiveness,” Phys. Rev. Lett. 108, 160501 (2012).
[Crossref] [PubMed]

G. C. Knee, S. Simmons, E. M. Gauger, J. J. L. Morton, H. Riemann, N. V. Abrosimov, P. Becker, H.-J. Pohl, K. M. Itoh, M. L. W. Thewalt, G. A. D. Briggs, and S. C. Benjamin, “Violation of a Leggett-Garg inequality with ideal non-invasive measurements,” Nat. Commun. 3, 606 (2012).
[Crossref] [PubMed]

C.-M. Li, N. Lambert, Y.-N. Chen, G.-Y. Chen, and F. Nori, “Witnessing quantum coherence: from solid-state to biological systems,” Sci. Rep. 2885 (2012).
[Crossref] [PubMed]

2011 (6)

M. M. Wilde and A. Mizel, “Addressing the clumsiness loophole in a Leggett-Garg test of macrorealism,” Found. Phys. 42, 256 (2011).
[Crossref]

G. Waldherr, P. Neumann, S. F. Huelga, F. Jelezko, and J. Wrachtrup, “Violation of a temporal Bell inequality for single spins in a diamond defect center,” Phys. Rev. Lett. 107, 090401 (2011).
[Crossref] [PubMed]

V. Athalye, S. S. Roy, and T. S. Mahesh, “Investigation of the Leggett-Garg inequality for precessing nuclear spins,” Phys. Rev. Lett. 107, 130402 (2011).
[Crossref] [PubMed]

J.-S. Xu, C.-F. Li, X.-B. Zou, and G.-C. Guo, “Experimental violation of the Leggett-Garg inequality under decoherence,” Sci. Rep. 1, 101 (2011).
[Crossref]

M. E. Goggin, M. P. Almeida, M. Barbieri, B. P. Lanyon, J. L. O’Brien, A. G. White, and G. J. Pryde, “Violation of the Leggett-Garg inequality with weak measurements of photons,” Proc. Natl. Acad. Sci. 108, 1256 (2011).
[Crossref] [PubMed]

J. Dressel, C. J. Broadbent, J. C. Howell, and A. N. Jordan, “Experimental violation of two-party Leggett-Garg inequalities with semiweak measurements,” Phys. Rev. Lett. 106, 040402 (2011).
[Crossref] [PubMed]

2010 (2)

A. Palacios-Laloy, F. Mallet, F. Nguyen, P. Bertet, D. Vion, D. Esteve, and A. N. Korotkov, “Experimental violation of a Bell’s inequality in time with weak measurement,” Nat. Phys. 6, 442 (2010).
[Crossref]

T. Fritz, “Quantum correlations in the temporal Clauser-Horne-Shimony-Holt (CHSH) scenario,” New J. Phys. 12, 083055 (2010).
[Crossref]

2006 (1)

A. Acín, N. Gisin, and B. Toner, “Grothendieck’s constant and local models for noisy entangled quantum states,” Phys. Rev. A 73, 062105 (2006).
[Crossref]

2002 (1)

A. J. Leggett, “Testing the limits of quantum mechanics: motivation, state of play, prospects,” J. Phys.: Condens. Matter 14, R415 (2002).

1994 (1)

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization and of any and discrete unitary and operator,” Phys. Rev. Lett. 73, 58 (1994).
[Crossref] [PubMed]

1987 (1)

A. J. Leggett and A. Garg, “Comment on “Realism and quantum flux tunneling”,” Phys. Rev. Lett. 59, 1621 (1987).
[Crossref]

1985 (1)

A. J. Leggett and A. Garg, “Quantum mechanics versus macroscopic realism: Is the flux there when nobody looks?” Phys. Rev. Lett. 54, 857 (1985).
[Crossref] [PubMed]

1980 (1)

B. Cirel’son, “Quantum generalizations of Bell’s inequality,” Lett. Math. Phys. 4, 93 (1980).
[Crossref]

1969 (1)

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880 (1969).
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[Crossref]

L. Xiao, K. K. Wang, X. Zhan, Z. H. Bian, J. Li, Y. S. Zhang, P. Xue, and A. K. Pati, “Experimental test of uncertainty relations for general unitary operators,” Opt. Exp. 25, 17904 (2017).
[Crossref]

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[Crossref]

K. Wang, X. Zhan, Z. Bian, J. Li, Y. Zhang, and P. Xue, “Experimental investigation of the stronger uncertainty relations for all incompatible observables,” Phys. Rev. A 93, 052108 (2016).
[Crossref]

X. Zhan, X. Zhang, J. Li, Y. S. Zhang, B. C. Sanders, and P. Xue, “Realization of the contextuality-nonlocality tradeoff with a qubit-qutrit photon pair,” Phys. Rev. Lett. 116, 090401 (2016).
[Crossref] [PubMed]

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[Crossref] [PubMed]

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[Crossref] [PubMed]

X. Zhan, J. Li, H. Qin, Z. H. Bian, and P. Xue, “Linear optical demonstration of quantum speed-up with a single qudit,” Opt. Exp. 23, 18422 (2015).
[Crossref]

K. Wang, C. Emary, M. Xu, X. Zhan, Z. Bian, L. Xiao, and P. Xue, “Violations of a Leggett-Garg inequality without signalling for a photonic qutrit probed with ambiguous measurements,” arXiv:1711.05172 (2017).

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Z. H. Bian, J. Li, H. Qin, X. Zhan, R. Zhang, B. C. Sanders, and P. Xue, “Realization of single-qubit positive-operator-valued measurement via a one-dimensional photonic quantum walk,” Phys. Rev. Lett. 114, 203602 (2015).
[Crossref] [PubMed]

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K. Wang, X. Zhan, Z. Bian, J. Li, Y. Zhang, and P. Xue, “Experimental investigation of the stronger uncertainty relations for all incompatible observables,” Phys. Rev. A 93, 052108 (2016).
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[Crossref]

Zhang, Y. S.

L. Xiao, K. K. Wang, X. Zhan, Z. H. Bian, J. Li, Y. S. Zhang, P. Xue, and A. K. Pati, “Experimental test of uncertainty relations for general unitary operators,” Opt. Exp. 25, 17904 (2017).
[Crossref]

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Figures (3)

Fig. 1
Fig. 1 (a) Theoretical maximum values of K3 and K4 for M = N = 2 . . . 7. Results are shown both with and without restriction that the first measurement coincides with state preparation, Q(t1) = +1. Also indicated are the maximum experimental violations reported here. (b) Experimental set-up for the four-term CHSH-style LGI with the heralded single photons. The first PBS, HWP (H1) and BD1 are used to generate the initial qutrit state |C〉 at t1. Sets of HWPs and BDs are used to realize the evolution operators Ui. Projective measurement of the final photon state at t3 is realized via a PBS which maps the basis states of the qutrit into three spatial modes. Detecting heralded single photons means in practice registering coincidences between the trigger detector D0 and each of the detectors for measurement D1, D2, and D3. The ideal negative measurement at times t2 and t3 is realized by blocking channels, two at a time, such that detection at D1−3 implies that path taken was the non-blocked channel.
Fig. 2
Fig. 2 Experimentally-determined values of the four-time CHSH-style Leggett-Garg correlator K4 for our three-level, three-outcome set-up with time evolution described by parameters ϕ3 and θ3. Theoretical predictions are represented by different lines and the experimental results by symbols. Error bars indicate the statistical uncertainty based on the assumptions of Poissonian statistics. The maximum measured value of the Leggett-Garg correlator occurs at ϕ3 = π/4 (ϕ3 = 3π/4) and has value K4 = 2.96 ± 0.05 (2.95 ± 0.05), as compared with the theoretical maximum value of 3 (represented by the black solid line). This represents a significant enhancement over the temporal Tsirelson bound of 2 2 (represented by the orange dash-dot-dot line).
Fig. 3
Fig. 3 (a) As Fig. 2 but for the three-time Leggett-Garg correlator K3. The maximum measured value of the Leggett-Garg correlator occurs at {ϕ, θ} = {π/2, π/4} and has value K3 = 1.97 ± 0.06, as compared with the theoretical maximum value of 2. (b) The experimentally-determined values of the quantum witness W for our three-level, three-outcome setup. The maximum value of the quantum witness is W = 0.65 ± 0.02, which occurs for evolution parameters { ϕ , q } = { π / 4 , arccos 2 / 3 }. This saturates the theoretical maximum of W = 2/3 for a three-level system. The maximum value of the corresponding LGI, Eq. (5), is K3W = 1.65 ± 0.02. Both the maximum violation of K3 and K3W exceed the TTB K 3 TTB = 3 / 2.

Equations (5)

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3 K 3 1 ; K 3 C 21 + C 32 C 31
| K 4 | 2 ; K 4 C 21 + C 32 + C 34 C 41 ,
U i = ( cos θ i 0 sin θ i sin θ i sin ϕ i cos ϕ i cos θ i sin ϕ i sin θ i cos ϕ i sin ϕ i cos θ i cos ϕ i ) ,
W P 3 ( m 3 = C ) m 2 P 32 ( m 3 = C , n 2 ) ,
K 3 W = 1 + W 1 .

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