Testing Contextuality on Quantum Ensembles with One Clean Qubit

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As Published http://dx.doi.org/10.1103/PhysRevLett.104.160501

Publisher American Physical Society

Version Final published version

Citable link http://hdl.handle.net/1721.1/58715

Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. week ending PRL 104, 160501 (2010) PHYSICAL REVIEW LETTERS 23 APRIL 2010

Testing Contextuality on Quantum Ensembles with One Clean Qubit

Osama Moussa,1,* Colm A. Ryan,1 David G. Cory,2,3 and Raymond Laﬂamme1,3 1Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, N2L3G1, Canada 2Department of Nuclear Science and Engineering, MIT, Cambridge, Massachusetts 02139, USA 3Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2J2W9, Canada (Received 10 December 2009; published 23 April 2010) We present a protocol to evaluate the expectation value of the correlations of measurement outcomes for ensembles of quantum systems, and use it to experimentally demonstrate—under an assumption of fair sampling—the violation of an inequality that is satisﬁed by any noncontextual hidden-variables theory. The experiment is performed on an ensemble of molecular nuclear spins in the solid state, using established nuclear magnetic resonance techniques for quantum-information processing.

DOI: 10.1103/PhysRevLett.104.160501 PACS numbers: 03.65.Ud, 03.65.Ta, 76.60.k

Y Y The Bell-Kochen-Specker theorem [1–4] states that no S S ; fSkg ¼ ð kÞ¼ k (1) noncontextual hidden-variables (NCHV) theory can re- k k produce the predictions of quantum mechanics for correlations between measurement outcomes of some sets of irrespective of the product ordering. Repeating the prepa- observables. Any such set of observables constitutes a ration and measurement many times, and averaging over proof of the theorem. Recently, Cabello [5] and others the outcomes, one obtains an estimateQ of the ensemble S [6] used Bell-Kochen-Specker proofs to derive a set of average of the correlation h fSkgi ¼h k ð kÞi. inequalities that are satisfied by any NCHV theory but For the case where the coobservables fSkg are also are violated by quantum mechanics for any quantum state. dichotomic, with possible outcomes fðSkÞ¼1g, the These inequalities bound certain linear combinations of correlation (1) also takes on the possible values 1, and 1 1 ensemble averages of correlations between measure- the ensemble average satisfies h fSkgi þ . Note, ment outcomes of compatible observables, thus creating that in this case, these operators are Hermitian and unitary a separation between the predicted outcomes of quantum (also known as quantum Boolean functions). mechanics, and the bound that is satisfied by NCHV Consider any set of observables with possible outcomes theories. 1 arranged in a 3 3 table such that the observables in This provides an opportunity to test noncontextuality each column and each row are coobservable. It has been with finite-precision experiments—which has been the shown [5] that, for any NCHV theory, subject of contention for many years [7–9]—and without 4; the need for the creation of special quantum states [10–12]. ¼h r1 iþh r2 iþh r3 iþh c1 iþh c2 ih c3 i Already, two experiments, on a pair of trapped 40Caþ ions (2) [13], and with single photons [14], have demonstrated this where hr i is the ensemble average of the correlation state-independent conflict with noncontextuality. In this 1 Letter, we examine testing contextuality on quantum between outcomes of the observables listed in the first ensembles. row, and so forth. The above inequality is independent of This Letter is organized as follows. First, we sketch the the preparation of the ensemble, provided all terms are estimated for the same preparation. arguments leading to one of the inequalities derived in [5]. 1 Then we present an algorithm to estimate the expectation Now, consider a two-qubit system (e.g., 2 spin- 2 parti- value of the correlations of measurement outcomes for cles), and the set of observables listed in Table I. For any ensembles of quantum systems. And lastly, we report and NCHV theory, the inequality (2) holds for the correlations discuss the result of experimentally implementing the al- between measurement outcomes of the coobservables gorithm on a three-qubit ensemble of molecular nuclear listed in each row and column, where, e.g., h r1 i¼ Z1 1Z ZZ spins in the solid state. h fZ1;1Z;ZZgi¼h i, and so forth. Inequality.—For a quantum system prepared according On the other hand, according to quantumQ mechanics, the tr S to some state, , one can assign simultaneous outcomes ensemble average h fSkgi is given by ð k kÞ. Thus, for fðSkÞg of measurements of a set fSkg of coobservables a set of coobservables whose product is proportional to the (i.e., comeasureable; mutually compatible; commuting). In unit operator—as is the case for all rows and columns of this case, the correlation between the measurement out- Table I—the quantum mechanical prediction of the en- comes is given by semble average of the correlation is equal to the propor-

0031-9007=10=104(16)=160501(4) 160501-1 Ó 2010 The American Physical Society week ending PRL 104, 160501 (2010) PHYSICAL REVIEW LETTERS 23 APRIL 2010

TABLE I. List of the two-qubit observables used to show that As shown in Fig. 1, one then applies the unitaries Sk in quantum mechanics violates inequality (2). This list has been succession to the system, controlled on the state of the used by Peres [15] and Mermin [16] as a Bell-Kochen-Specker probe qubit. Since, by definition, all Sk mutually commute, f1;X;Z;Yg proof for four-dimensional systems. are the single- then the order of their application has no bearing on the qubit Pauli operators, and, e.g., ZX :¼ Z X indicates a measurement of the Pauli Z operator on the first qubit and Pauli X measurement outcome. Repeating this procedure, and operator on the second. averaging the outcome of the measurement on the probe Q system, produces the correlation between this set of ob- c1 c2 c3 servables. Alternatively, one could prepare an ensemble of U r1 Z11ZZZþ1 systems according to , apply the transformations S in r2 1XX1 XX þ1 parallel to each member of the ensemble, and perform a r ZX XZ YY 1 bulk ensemble measurement to estimate h S i. This Q3 þ f kg þ1 þ1 1 alleviates the need for isolation of single quantum systems, and the repeated application of single shot, projective measurement. tionality constant, independent of the initial preparation of Since inequality (2) is valid for any preparation , then the system. Hence, the quantum mechanical prediction for one is free to choose to prepare the system according to the is 6, which violates inequality (2). maximally mixed state. In which case, only one qubit—the Algorithm.—To measure the correlation between a set of probe system—is not maximally mixed. This corresponds coobservables, consider introducing an ancillary (probe) to the model of computation known as deterministic quan- U tum computation with one clean qubit (DQC1) [18]. qubit, and applying a transformation Sk to the composite system for each observable Sk, in a manner reminiscent of Two models.—Suppose the measurement process on the coherent syndrome measurement in quantum error correc- probe qubit was -efficient, i.e., returning a faithful answer tion [17]. For an observable S with the spectral decom- fraction of the time, and otherwise a uniformly distrib- uted random outcome. The probabilities pð1Þ of obtain- position S ¼ Pþ P, where Pþ and P are the projectors on the þ1 and 1 eigenspaces of S, the trans- ing outcomes 1 will be modified to pð1Þ¼ 1 þ trs½P , and the ensemble average to hX 1di¼ formation US is defined as US ¼ 12 Pþ þ Z P. That 2 is to say, if the system is in a 1 eigenstate of S, apply a hSi. One can then estimate the expectation value hSi phase flip (Pauli Z operator) to the probe qubit, and if it is under an assumption of fair sampling and knowing the in a þ1 eigenstate, do nothing. This transformation can value of , which can be established from h1i. This also be expressed as a controlled operation dependent on model is equivalent to one where the probe system is 12 the state of the probe qubit, initially in the mixed state ð1 Þ 2 þ jþihþj, provided the reduced dynamics on the probe qubit from preparation US ¼ 12 Pþ þ Z P to measurement is represented by a unital map, i.e., a map 1 1 Z P P 1 1 Z P P ¼ 2ð 2 þ Þð þ þ Þþ2ð 2 Þð þ Þ that preserves the totally mixed state. To see this, suppose we prepare the probe qubit in some state a and then apply ¼j0ih0j1d þj1ih1jS; some transformation to the composite system, whose re- which is unitary for S unitary. If the system, denoted by s, duced dynamics on the probe qubit is described by a unital is initially prepared according to , and the probe qubit, a, linear map .An-efficient measurement of X ¼ jþi 1 X 1 is in the þpffiffiffieigenstate of the Pauli operator, jþi ¼ hþj jihj has two possible outcomes with proba- ðj0iþj1iÞ= 2, the possible outcomes of Pauli X measure- bilities ment on the probe qubit is 1, with probabilities pð1Þ given by y pð1Þ¼traþs½USðjþihþj ÞUS ðjihj 1dÞ

¼ trs½hjð12 Pþ þ Z PÞðjþihþj Þ ð...Þyji

¼ trs½P; FIG. 1. A quantum network to encode the correlation between and the ensemble average the outcomes of measurements fSkgk¼1...m on a d-dimensional X 1 p 1 p 1 tr P tr P system, in the phase of a probe qubit state. Repeating this h di¼þ ðþ Þ ð Þ¼ s½ þ s½ procedure for the same preparation and averaging the outcome ¼hSi: (3) of the measurement on the probe qubit gives the ensemble average hS1S2 Smi. Alternatively, for an ensemble of quan- Thus, to measure the ensemble average of the correlation tum systems initially prepared according to , on which opera- between a set of coobservables, one prepares a probe qubit tions are applied in parallel to the individual systems, an in the þ1 eigenstate of X, and the system according to . ensemble measurement readily produces hS1S2 Smi. 160501-2 week ending PRL 104, 160501 (2010) PHYSICAL REVIEW LETTERS 23 APRIL 2010

1 proton-decoupled 13C spectrum, following polarization pð1Þ¼ þ tr½jihjð Þ 2 a transfer from the abundant protons, for the particular ori- 1 entation of the crystal used in this experiment. A precise ¼ tr½jihj þ tr½jihjð Þ 2 a spectral fit gives the Hamiltonian parameters (listed in the inset table in Fig. 3), as well as the free-induction dephas- 1 ¼ tr½jihjð12Þ þ tr½jihjðaÞ ing times, T2 , for the various transitions; these average at 2 2ms. The dominant contribution [19]toT2 is Zeeman- 12 ¼ tr jihj ð1 Þ þ a ; shift dispersion, which is largely refocused by the control 2 pulses. Other contributions are from intermolecular 13 13 C- C dipolar coupling and, particularly for Cm, residual which are precisely the statistics one obtains in case the interaction with neighboring protons. The carbon control 1 12 probe qubit is initially in the state ð Þ 2 þ a, and the pulses are numerically optimized to implement the re- measurement process is faithful. quired unitary gates using the GRAPE [22] algorithm. Experiment.—We implement the algorithm described Each pulse is 1.5 ms long, and is designed [23] to have above to perform an experimental measurement of the an average Hilbert-Schmidt fidelity of 99.8% over appro- correlations as described in inequality (2) on an ensemble priate distributions of Zeeman-shift dispersion and control- of nuclear spins in the solid state using established nuclear fields inhomogeneity. magnetic resonance techniques for quantum information 1 The two spin- 2 nuclei C1 and C2, constituting the system processing [19,20]. Figure 2 shows the six experiments on which the measurements are performed, are initially required to estimate the six terms in (2). The pulse se- prepared according to the totally mixed state. Cm, repre- quence implementing the measurement of some observ- senting the probe qubit, is initially prepared according to able is the same whether it is being measured with the 12 ¼ð1 Þ 2 þ jþihþj. A spectrum is acquired for coobservables listed in its row or column. 12 12 this initial state, 2 2 , to serve as a reference. The experiments were performed in a static field of 7.1 T Then, the same initial preparation is repeated six more using a purpose-built probe. The sample is a macroscopic times, and the six experiments (shown in Fig. 2) represent- single crystal of malonic acid (C3H4O4), where a small ing the terms in of inequality (2) are performed, produc- fraction (3%) of the molecules are triply labeled with 13C ing six more spectra. These six spectra are then summed to form an ensemble of processor molecules. During com- with the appropriate signs (i.e., the spectrum from experi- putation, these processors are decoupled from the 100% ment c3 in Fig. 2 is subtracted) to produce what we denote abundant protons in the crystal by applying a decoupling pulse sequence [21] to the protons. Shown in Fig. 3 is a

C1 C2 H2

Hm1,2

data fit

Intensity [a.u.] C1 C2 Cm 10 5 0 −5 −10 Frequency [kHz]

FIG. 3 (color). Malonic acid (C3H4O4) molecule and Hamiltonian parameters (all values in kHz). Elements along the diagonal represent chemical shifts, !i, withP respect to the transmitter frequency (with the HamiltonianP i!iZi). Above the diagonal are dipolar coupling constants [ i

reference data 4

β 3 Intensity [a.u.]

10 5 0 −5 −10 2 Frequency [kHz] 1

FIG. 4 (color). Summary of experimental results. Shown are 0 (in dashed blue) the spectrum produced by the initial preparation 0 0.2 0.4 0.6 0.8 1 1 1 t/T 2 2 2 procedure, 2 2 , establishing a reference for , and (in solid red) the sum of the six spectra corresponding to the six FIG. 5 (color online). Numerical simulation results of a simple terms in of inequality (2) with the appropriate signs, scaled by 1 model of decoherence (inset) showing the expected variation of for ease of comparison with the reference. 6 as a function of the ratio of the pulse length, t, to an effective dephasing time, T2. The dashed lines indicate bounds on the the final spectrum. Ideal closed-system quantum mechan- expected performance of the current experiment; for pulse length ics predicts the final spectrum to be identical to the refer- of 1.5 ms, and effective decoherence times of 2 ms (T2 ) and ence spectrum scaled by a factor of 6. Figure 4 shows the 30 ms ( intrinsic coherence times [19]), the value of is reference spectrum, and the final spectrum scaled by 1=6 expected to be 1.1 and 5.3, respectively. for comparison. Fitting the observable spectra, taking into consideration the effects of strong coupling, we estimate 5:2 0:1 the value of to be , in violation of inequality (2). This work benefited from discussions with J. Emerson, The uncertainty on is propagated from the goodness-of- A. Cabello, M. Laforest, and J. Baugh. This research was fit figure of merit ascribed to the spectral fitting process. supported by NSERC, CIFAR, Industry Canada, Ontario Decoherence, as it is wont to do, causes deviations from Ministry of Research and Innovation, and in part by the the idealized closed-system dynamics. To examine its ef- National Security Agency (NSA) under Army Research fect, we numerically simulate the dynamics of a simple Office (ARO) Contract No. W911NF-05-1-0469. model (shown in Fig. 5) in which each ideal transformation is followed by a symmetric error of a threefold tensor product of a single-qubit dephasing map, ð ÞP, given by y the operator sum representation ! ðÞ¼ AA, where *[email protected] 10 [1] E. P. Specker, Dialectical anthropology 14, 239 (1960). A0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi ; [2] J. S. Bell, Rev. Mod. Phys. 38, 447 (1966). 0 1 [3] S. Kochen and E. P. Specker, J. Math. Mech. 17, 59 (1967). 10 [4] N. D. Mermin, Rev. Mod. Phys. 65, 803 (1993). A1 ¼ pffiffiffiffi ; [5] A. Cabello, Phys. Rev. Lett. 101, 210401 (2008). 0 [6] P. Badzikag et al., Phys. Rev. Lett. 103, 050401 (2009). [7] D. A. Meyer, Phys. Rev. Lett. 83, 3751 (1999). and the parameter ¼ 1 expðt=T2Þ depends on the ratio of the pulse length, t, to an effective dephasing time, [8] A. Cabello, Phys. Rev. A 65, 052101 (2002). [9] N. D. Mermin, arXiv:quant-ph/9912081v1. T2. Using appropriate estimates [19] of this dephasing [10] C. Simon et al., Phys. Rev. Lett. 85, 1783 (2000). time, one is able to largely explain the deviation of the [11] Y.F. Huang et al., Phys. Rev. Lett. 90, 250401 (2003). experimental result from the prediction of quantum me- [12] H. Bartosik et al., Phys. Rev. Lett. 103, 040403 (2009). chanics in ideal conditions. [13] G. Kirchmair et al., Nature (London) 460, 494 (2009). Conclusion.—We have presented a protocol to directly [14] E. Amselem et al., Phys. Rev. Lett. 103, 160405 (2009). measure correlations between measurement outcomes, [15] A. Peres, Phys. Lett. A 151, 107 (1990). utilizing an ancillary (probe) two-dimensional system, [16] N. D. Mermin, Phys. Rev. Lett. 65, 3373 (1990). with the purpose of testing quantum contextuality. [17] D. P. DiVincenzo and P.W. Shor, Phys. Rev. Lett. 77, 3260 Conveniently, it can be used directly on ensembles of (1996). quantum systems, without the need for repeated projective [18] E. Knill and R. Laflamme, Phys. Rev. Lett. 81, 5672 measurement on single systems. Additionally, it can be (1998). [19] J. Baugh et al., Phys. Rev. A 73, 022305 (2006). straightforwardly extended to test similar inequalities on [20] C. A. Ryan, O. Moussa, J. Baugh, and R. Laflamme, Phys. higher-dimensional systems. Our experimental results Rev. Lett. 100, 140501 (2008). demonstrate—under the assumption of fair sampling— [21] B. M. Fung, A. K. Khitrin, and K. Ermolaev, J. Magn. that a three-qubit deterministic quantum computer with Reson. 142, 97 (2000). one clean qubit reveals correlations between measurement [22] N. Khaneja et al., J. Magn. Reson. 172, 296 (2005). outcomes that cannot be explained by any NCHV theory. [23] C. A. Ryan et al., Phys. Rev. A 78, 012328 (2008). 160501-4