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Testing Contextuality on Quantum Ensembles with One Clean Qubit The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Moussa, Osama et al. “Testing Contextuality on Quantum Ensembles with One Clean Qubit.” Physical Review Letters 104.16 (2010): 160501. © 2010 The American Physical Society. 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.Àk 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 corre- lations 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 Æ1, and 1 1 ensemble averages of correlations between measure- the ensemble average satisfies À 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. Æ1 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 Á Á 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 mea- surement 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 À1 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 À PÀ, where Pþ and PÀ are the projectors on the þ1 and À1 eigenspaces of S, the trans- ing outcomes Æ1 will be modified to pð1Þ¼ 1À þ trs½P , and the ensemble average to hX 1di¼ formation US is defined as US ¼ 12 Pþ þ Z PÀ. That 2 Æ is to say, if the system is in a À1 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 À Þ 2 þ jþihþj, provided the reduced dynamics on the probe qubit from preparation US ¼ 12 Pþ þ Z PÀ 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 ÀÞþ2ð 2 À Þð þ À ÀÞ 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 Ã.An-efficient measurement of X ¼ jþi  1 X 1 is in the þpffiffiffieigenstate of the Pauli operator, jþi ¼ hþj À jihj has two possible outcomes Æ with proba- ðj0iþj1iÞ= 2, the possible outcomes of Pauli X measure- bilities ment on the probe qubit is Æ1, with probabilities pð1Þ given by y pð1Þ¼traþs½USðjþihþj ÞUS ðjihj 1dÞ ¼ trs½hjð12 Pþ þ Z PÀÞðjþihþj Þ ð...Þyji ¼ trs½PÆ; FIG.

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