Beyond the temporal Tsirelson bound: an experimental test of Leggett-Garg inequalities in a three-level system
Abstract
Macrorealism, as defined by Leggett and Garg, posits that a macroscopic system always exists in a well-defined state and that it can be measured without disturbing it. From these assumptions follow a set of inequalities, the Leggett-Garg inequalities, which hold under macrorealism but can be violated by quantum mechanics. The degree to which quantum systems can violate these inequalities is bounded and, in particular, if the measurements on the system are genuinely dichotomic, the bound for these temporal inequalities is the same as the Tsirelson bound of the corresponding spatial Bell inequality. In this paper 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 significantly in excess of the respective Tsirelson bounds. This underlines the difference between Bell and LG inequalities and hence spatial and temporal correlations in quantum mechanics. We also report violations of a second consequence of macrorealism, the quantum-witness equality, up to the maximum permitted for a three-outcome measurement.
In contrast to Bell inequalities which probe correlations between multiple spatially-separated systems Bell (1964); Giustina et al. (2015); *Shalm2015, the Leggett-Garg inequalities (LGIs) test the temporal correlations of a single system Leggett and Garg (1985); *LeggettGarg1987; Emary et al. (2015). The LGIs are based on two macrorealistic assumptions that intuitively hold in the world of our everyday experience: (i) macroscopic realism per se — that a system exists at all times in a macroscopically-distinct state; and (ii) non-invasive measurability — that it is possible to measure a system without disturbing it. Since both these assumptions fails under quantum mechanics, quantum systems can violate the LGIs. Hence the use of these inequalities as indicators of quantum coherence, in particular in macroscopic systems Leggett (2002).
The LGIs concern the correlation functions of dichotomic variable at measurement times . Typical three- and four-time LGIs can be written Emary et al. (2015)
(1) | |||
(2) |
as temporal analogues of the original Bell Bell (1964) and CHSH Clauser et al. (1969) inequalities. These inequalities, in particular the inequality and its close relatives, have been tested and violated in many experiments, with most studies having been performed on two-level quantum systems, e.g. Palacios-Laloy et al. (2010); Waldherr et al. (2011); Athalye et al. (2011); Xu et al. (2011); Goggin et al. (2011); Dressel et al. (2011); Knee et al. (2012); Groen et al. (2013); Katiyar et al. (2013); Asadian et al. (2014); Zhou et al. (2015); Knee et al. (2016); Formaggio et al. (2016). In such systems, the maximum quantum-mechanical value of the Leggett-Garg (LG) correlators are and in the three- and four-time case respectively. The derivation of these values is analogous Fritz (2010) to that of the Tsirelson bounds Cirel’son (1980); Poh et al. (2015); Navascués et al. (2016) of the corresponding inequalities for spatially separated observations and we will refer to these bounds as the temporal Tsirelson bounds (TTBs) of the LGIs ^{1}^{1}1In Ref. Budroni and Emary (2014), this bound was referred to as the Lüders bound.. It is known that bound the LGIs for quantum systems of arbitrary size provided that the measurements are genuinely dichotomic, i.e. can be modelled with exactly two projection operators Budroni et al. (2013). Recently, however, it was predicted that values of exceeding are possible for -level systems when the measurement apparatus provides more information than a single bit, and is thus modelled with orthogonal projectors Budroni and Emary (2014). In particular, for a three-level system with measurements decomposed as three projectors (each nevertheless associated with a value of either or ) it was predicted that the maximum value of the LG correlator under quantum mechanics is . A similar substitution of multi-outcome measurements into the Bell and CHSH inequalities leaves the (spatial) Tsirelson bounds unaltered Budroni and Emary (2014).
A small number of experiments have been performed on multi-level systems George et al. (2013); Robens et al. (2015); Katiyar et al. (2017), but no violations have yet been reported. In Ref. Robens et al. (2015) the decisive measurement at was only a two-outcome one. In Ref. George et al. (2013) three-outcome measurements were considered but the focus there was on “non-disturbing measurements”, rather than on maximising . Recently, a three-level NMR system was studied Katiyar et al. (2017) for which a theoretical maximum violation of was predicted. However, no evidence of violations greater than the TTB were found. Violations exceeding have recently been theoretically studied in multi-qubit systems Lambert et al. (2016). As far as we are aware, the CHSH-like LGI has not been tested experimentally.
In this paper, we report on LG experiments with single photons that implement a three-level quantum system measured with three orthogonal projectors. We investigate both and inequalities and our main result is the observation of maximum values of the LG correlators and , which clearly represent significant enhancements over the TTB. We also consider a quantum-witness Li et al. (2012) (or no-signalling-in-time Kofler and Brukner (2013)) test for our system, which is based on the same assumptions as the LGIs but is simpler and is in some ways preferable Clemente and Kofler (2016). In contrast to our results for the LGI, where the measured violations are still lower than the theoretical maximum, our maximum measured value for the quantum witness saturates the theoretical bound for a three-outcome test Schild and Emary (2015), and presents a significant enhancement over the hitherto-observed value for a two-outcome case Robens et al. (2015).
These values we obtain using ideal negative measurements (INMs) Leggett and Garg (1985). As in Refs. Knee et al. (2012); Robens et al. (2015); Katiyar et al. (2017), these allow us to acquire information about the system (here, the photon) without interacting with it directly, and thus take steps to address the “clumsiness loophole” Wilde and Mizel (2011). Our interferometeric set-up makes the designation of our measurements as INM extremely clear-cut.
Multiple-outcome LGI tests:- We consider a system in which we measure a variable with distinct outcomes. We connect with the standard LGI framework by introducing the mapping of a measurement of at times onto the value . The correlation functions are thus constructed , where is the joint probability to obtain and at times and . This way of constructing and its correlation functiosn clearly leaves the classical upper bounds in Eq. (1) and Eq. (2) unaffected. The maximum quantum-mechanical values for the and correlators are shown in Fig. 1(a) as a function of the number of levels of the quantum-mechanical system with number of outcomes . These values were obtained numerically as described in Ref. Budroni et al. (2013). For , violations significantly higher than the TTB are clearly possible. In the following, we will make use of a common experimental simplification of the LGIs Goggin et al. (2011); Robens et al. (2015); Lambert et al. (2016) and assume the coincidence of state preparation with measurement at , and simply define . This can reduce the maximum quantum violations, as shown in Fig. 1(a).
Experimental set-up:- We consider the smallest system that will admit a three-outcome measurement, a qutrit with states (), which we realise with single photons, Fig. 1(b). The basis states , , and are encoded respectively by the horizontal polarization of the heralded single photon in the upper mode, the horizontal polarization of the photon in the lower mode, and the vertical polarization of the photon in the lower mode. Heralded single photons are generated via a type-I spontaneous parametric down-conversion (SPDC). The polarization-degenerate photon pairsare produced in SPDC using a -barium-borate (BBO) nonlinear crystal pumped by a diode laser. With the detection of a trigger photon the signal photon is heralded for evolution and measurement Xue et al. (2015). The pump is filtered out with the help of an interference filter which restricts the photon bandwidth to nm.
Initial qutrit states are prepared by first passing the heralded single photons through a polarizing beam splitter (PBS), and a half-wave plate (HWP, H) before being split by a birefringent calcite beam displacer (BD) into two parallel spatial modes, upper and lower, with vertically-polarized photons directly transmitted through the BD in the lower mode, and with horizontal photons undergoing a mm lateral displacement into a upper mode. In the current set-up we set PBS and H to give vertically-polarized photons, which then remain in the lower mode through BD, thus initialising the qutrit in the state .
Time evolution from to time in our experiment is given by unitary operators (),
(3) |
in the basis (). The middle unitary between and is then set to . For the test the time evolution operator between and is taken the form shown in Eq. (3) and the evolution operator between and is set to . These are realised by a sequence of HWPs and BDs with and adjustable parameters Reck et al. (1994); Wang et al. (2016).
We identify our measurements as projections onto the three basis states with outcomes and thus the probability of obtaining outcome is the same as the probability of detecting the system in state . Projective measurement of the final photon state is performed with a PBS that maps the quitrit basis states of qutrit into three spatial modes and to accomplish the projective measurement. The photons are then detected by single-photon avalanche photodiodes (APDs), in coincidence with the trigger photons. The probability of the photons being measured in is obtained by normalizing photon counts in the certain spatial mode to total photon counts. The count rates are corrected for differences in detector efficiencies and losses before the detectors. We assume that the lost photons would have behaved the same as the registered ones (fair sampling). Experimentally this trigger-signal photon pair is registered by a coincidence count at APD with ns time window ^{2}^{2}2In the test, total coincidence counts were approximately over a collection time of s; in the test, they were approximately over s..
INM of the qutrit state at earlier times is realised by placing blocking elements into the optical paths Emary et al. (2012). With, for example, the channels and blocked at , we obtain the probabilities without the measurement apparatus having interacting with the photon. In our experiment, this blocking is realized by a BD followed by an iris. The BD is used to map the basis states of qutrit to three spatial modes and the iris is used to block photons in two of the three spatial modes and let the photons in the rest one pass through. By changing the position of the iris, we can block any two of the channels and let the photons in the remaining one pass through for the next evolution. By using different sequences of blocking and unblocking as well as final detection at , all the necessary correlation functions can be constructed.
CHSH-type inequality:- We first consider with and and for . We choose the middle unitary according to , such that , where is a identity matrix and set and . In this case, the theoretical value of the LGI correlator as a function of and is for and which is the maximum value achievable under the condition that the -measurement coincides with initialisation. . This takes a maximum value of
Figure 2 shows our experimentally determined value of as a function of for several values of . Agreement with theory is close, and the maximum violations obtained are for and for . These values thus show clear experimental evidence of the super-Tsirelson-bound violations of the four-term CHSH-style LGI. Error bars indicate the statistical uncertainty, based on the assumption of Poissonian statistics. Compared with the theory, the measured value of is close to its theoretical prediction since the joint probability of is not affected by the imperfection of cascaded interferometers after the INM at . Furthermore, there is no cascaded interferometer in the measurement of and , such effects are reduced and the measured values of and are close to their theoretical predictions too. The main deviation from theory arises in the measurement of where, by construction, the final state should be the same as the initial state such that . However, due to imperfection in the cascaded interferometers in this setup, is smaller than and we obtain at .
Three term LGI and quantum witness:- The experimental geometry for the is essentially the same as above with one time-evolution step removed. In this case, and following Ref. Schild and Emary (2015), we take the second time evolution to be such that . This gives the correlation functions , , and
Whilst neither of the above LGI measurements saturate the maximum theoretical value of the LGI with shown in Fig. 1(a), we are able to saturate the bound for the quantum witness
(4) |
based on registering the outcome . Under macrorealism and non-invasive measurability, we have the equality . This witness can be constructed from the same probabilities as used in the test and these results are shown in Fig. 3(b). Theory Schild and Emary (2015) predicts a maximum value of this witness should occur for the parameters and and . At these points we observe the values and . This agrees well with the theoretical value of , which is the maximum possible value for a three-projector measurement (and thus, also for a three-level system).
We can relate this quantum witness directly to a LGI inequality if we choose measurement value assignments at (a blind measurement) Robens et al. (2015), and . In this case, the LGI of Eq. (1) reduces to
(5) |
Thus, the value by which the witness exceeds zero is the extent to which the corresponding LGI is violated. The maximum violation of this LGI with this measurement assignment is thus .
There is connection between enhanced violations of the LGI, quantum witness equality and dimension witnesses Budroni and Emary (2014); Schild and Emary (2015); Hendrych et al. (2012); Ahrens et al. (2012). However, from the view of experiment, the dimension of the system being measured is usually known before we can design an experimental setup. Therefore the dimensionless witness test is not considered here.
Discussion:- We have demonstrated experimental violations of LGIs in a three-level system and obtained values of the LGI correlators and greatly in excess of the TTBs, familiar from studies of two-level systems. We have also demonstrated a similar excess for the quantum-witness equality. These enhancements arise because the decisive - and -measurements here admit three distinct measurement outcomes, rather than the usual two. Under this measurement, the collapse of the wave function is greater than with two projectors and the resultant additional information gain enables the enhanced violation. In particular, in the case of the witness, the post-measurement state of the qutrit is the maximally mixed state . The corresponding violation is therefore up to the theoretical maximum for three-outcome measurements. These results provide an experimental demonstration of the difference between spatial and temporal correlations in quantum mechanics, since the (spatial) Tsirelson bound in the Bell and CHSH inequalities remains fixed at and , irrespective of the number of projectors.
In the future, it will be interesting to look at temporal analogues of different (spatial) Bell inequalities, in particular ones with multi-outcome measurements Acín et al. (2006); *Hirsch2016. We note that our measurements are within the standard quantum-mechanical framework and thus demonstrate that post-quantum effects are not needed to obtain enhanced LGI violations Dakić et al. (2013).
Classical invasive measurements can give violations of the LGI, all the way up the algebraic bound Montina (2012). It is therefore important to ensure the non-invasivity of the measurements in any LGI test. Whilst no known scheme can completely rule out such invasivity ^{3}^{3}3Results such as Ref. Wilde and Mizel (2011) and Ref. Knee et al. (2016) reduce the “size” the clumsiness loophole, but do not close it altogether., we have used INMs here which rule out the direct influence of the measurements on the system itself. Nevertheless, our measurements of the quantum witness indicate that the correlations here are of the “signalling” type Kofler and Brukner (2013); Halliwell (2016), which then points to an interesting comparison between our work, where we have both signalling and , and Ref. George et al. (2013) where no-signalling was obeyed but the LGI violation was restricted to .
Acknowledgements.
We would like to thank Neill Lambert and George Knee for helpful discussions. We acknowledge support by NSFC (Nos. 11474049 and 11674056), NSFJS (No. BK20160024), the Open Fund from State Key Laboratory of Precision Spectroscopy of East China Normal University and the Scientific Research Foundation of the Graduate School of Southeast University.References
- Bell (1964) J. S. Bell, Physics 1, 195 (1964).
- Giustina et al. (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, Phys. Rev. Lett. 115, 250401 (2015).
- Shalm et al. (2015) 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, Phys. Rev. Lett. 115, 250402 (2015).
- Leggett and Garg (1985) A. J. Leggett and A. Garg, Phys. Rev. Lett. 54, 857 (1985).
- Leggett and Garg (1987) A. J. Leggett and A. Garg, Phys. Rev. Lett. 59, 1621 (1987).
- Emary et al. (2015) C. Emary, N. Lambert, and F. Nori, Rep. Prog. Phys. 77, 016001 (2015).
- Leggett (2002) A. J. Leggett, J. Phys.: Condens. Matter 14, R415 (2002).
- Clauser et al. (1969) J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys. Rev. Lett. 23, 880 (1969).
- Palacios-Laloy et al. (2010) A. Palacios-Laloy, F. Mallet, F. Nguyen, P. Bertet, D. Vion, D. Esteve, and A. N. Korotkov, Nat. Phys. 6, 442 (2010).
- Waldherr et al. (2011) G. Waldherr, P. Neumann, S. F. Huelga, F. Jelezko, and J. Wrachtrup, Phys. Rev. Lett. 107, 090401 (2011).
- Athalye et al. (2011) V. Athalye, S. S. Roy, and T. S. Mahesh, Phys. Rev. Lett. 107, 130402 (2011).
- Xu et al. (2011) J.-S. Xu, C.-F. Li, X.-B. Zou, and G.-C. Guo, Sci. Rep. 1, 101 (2011).
- Goggin et al. (2011) M. E. Goggin, M. P. Almeida, M. Barbieri, B. P. Lanyon, J. L. O’Brien, A. G. White, and G. J. Pryde, Proc. Natl. Acad. Sci. 108, 1256 (2011).
- Dressel et al. (2011) J. Dressel, C. J. Broadbent, J. C. Howell, and A. N. Jordan, Phys. Rev. Lett. 106, 040402 (2011).
- Knee et al. (2012) 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, Nat. Commun. 3, 606 (2012).
- Groen et al. (2013) J. P. Groen, D. Ristè, L. Tornberg, J. Cramer, P. C. de Groot, T. Picot, G. Johansson, and L. DiCarlo, Phys. Rev. Lett. 111, 090506 (2013).
- Katiyar et al. (2013) H. Katiyar, A. Shukla, K. R. K. Rao, and T. S. Mahesh, Phys. Rev. A 87, 052102 (2013).
- Asadian et al. (2014) A. Asadian, C. Brukner, and P. Rabl, Phys. Rev. Lett. 112, 190402 (2014).
- Zhou et al. (2015) Z.-Q. Zhou, S. F. Huelga, C.-F. Li, and G.-C. Guo, Phys. Rev. Lett. 115, 113002 (2015).
- Knee et al. (2016) G. C. Knee, K. Kakuyanagi, M.-C. Yeh, Y. Matsuzaki, H. Toida, H. Yamaguchi, S. Saito, A. J. Leggett, and W. J. Munro, Nat. Commun. 7, 13253 (2016).
- Formaggio et al. (2016) J. A. Formaggio, D. I. Kaiser, M. M. Murskyj, and T. E. Weiss, Phys. Rev. Lett. 117, 050402 (2016).
- Fritz (2010) T. Fritz, New J. Phys. 12, 083055 (2010).
- Cirel’son (1980) B. Cirel’son, Lett. Math. Phys. 4, 93 (1980).
- Poh et al. (2015) H. S. Poh, S. K. Joshi, A. Cerè, A. Cabello, and C. Kurtsiefer, Phys. Rev. Lett. 115, 180408 (2015).
- Navascués et al. (2016) M. Navascués, S. Pironio, and A. Acín, New J. Phys. 10, 073013 (2016).
- (26) In Ref.\tmspace+.1667emBudroni and Emary (2014), this bound was referred to as the Lüders bound.
- Budroni et al. (2013) C. Budroni, T. Moroder, M. Kleinmann, and O. Gühne, Phys. Rev. Lett. 111, 020403 (2013).
- Budroni and Emary (2014) C. Budroni and C. Emary, Phys. Rev. Lett. 113, 050401 (2014).
- George et al. (2013) 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, Proc. Natl. Acad. Sci. 110, 3777 (2013).
- Robens et al. (2015) C. Robens, W. Alt, D. Meschede, C. Emary, and A. Alberti, Phys. Rev. X 5, 011003 (2015).
- Katiyar et al. (2017) H. Katiyar, A. Brodutch, D. Lu, and R. Laflamme, New J. Phys. 19, 023033 (2017).
- Lambert et al. (2016) N. Lambert, K. Debnath, A. F. Kockum, G. C. Knee, W. J. Munro, and F. Nori, Phys. Rev. A 94, 012105 (2016).
- Li et al. (2012) C.-M. Li, N. Lambert, Y.-N. Chen, G.-Y. Chen, and F. Nori, Sci. Rep. 2 (2012).
- Kofler and Brukner (2013) J. Kofler and Č. Brukner, Phys. Rev. A 87, 052115 (2013).
- Clemente and Kofler (2016) L. Clemente and J. Kofler, Phys. Rev. Lett. 116, 150401 (2016).
- Schild and Emary (2015) G. Schild and C. Emary, Phys. Rev. A 92, 032101 (2015).
- Wilde and Mizel (2011) M. M. Wilde and A. Mizel, Found. Phys. 42, 256 (2011).
- Xue et al. (2015) P. Xue, R. Zhang, H. Qin, X. Zhan, Z. H. Bian, J. Li, and B. C. Sanders, Phys. Rev. Lett. 114, 140502 (2015).
- Reck et al. (1994) M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, Phys. Rev. Lett. 73, 58 (1994).
- Wang et al. (2016) K. Wang, X. Zhan, Z. Bian, J. Li, Y. Zhang, and P. Xue, Phys. Rev. A 93, 052108 (2016).
- (41) In the test, total coincidence counts were approximately over a collection time of s; in the test, they were approximately over s.
- Emary et al. (2012) C. Emary, N. Lambert, and F. Nori, Phys. Rev. B 86, 235447 (2012).
- Hendrych et al. (2012) M. Hendrych, R. Gallego, M. M. čuda, N. Brunner, A. Acín, and J. P. Torres, Nat. Phys. 8, 588 (2012).
- Ahrens et al. (2012) J. Ahrens, P. Badzia, A. Cabello, and M. Bourennane, Nat. Phys. 8, 592 (2012).
- Acín et al. (2006) A. Acín, N. Gisin, and B. Toner, Phys. Rev. A 73, 062105 (2006).
- Hirsch et al. (2016) F. Hirsch, M. T. Quintino, T. Vértesi, M. Navascués, and N. Brunner, arXiv:1609.06114 (2016).
- Dakić et al. (2013) B. Dakić, T. Paterek, and Č. Brukner, arXiv:1308.2822 (2013).
- Montina (2012) A. Montina, Phys. Rev. Lett. 108, 160501 (2012).
- (49) Results such as Ref. Wilde and Mizel (2011) and Ref. Knee et al. (2016) reduce the “size” the clumsiness loophole, but do not close it altogether.
- Halliwell (2016) J. J. Halliwell, Phys. Rev. A 93, 022123 (2016).